Lamb Dynamical Theory of Sound 1925

PU300KS ONADVANCED ..MATHEMATICS MathematicalAnalysis ofElectrical andOptical Wave-Motion, Harry Bateman $1.60 NumericalIntegration ofDifferential Equations,A.Bennett, W.Milne, HarryBateman $1.35 Almost Periodic Functions, A.Besicovitch $1.75 Non-Euclidean Geometry,R.Bonola $1.95 Introduction toBessel Functions, F.'Bowman $135 Theory ofGroups ofFinite Order, W.Burnside $2.45 Theory ofProbability, W.Burnside $1.00 AnElementary Treatise onFourier's Series andSpherical, Cylindrical, and Ellipsoidal Harmonics, W.E.Byerly $1.75 Contributions totheTheory ofTransfinite Numbers, G.Cantor$1.25 Introduction totheTheory ofGroups ofFinite Order, R.D.Carmichael $2.00 TheTheory ofNumbers andDiophantine Analysis, R.D.Carmichael $1.35 Introduction totheTheory ofFourier's Series andIntegrals, H.S.Carslaw $2.00 Statistics Manual, E.L.Crow, F.A.Davis., M.W.Maxfield $1.55 Introduction totheTheory ofNumbers, L.E.Dickson $1.65 Algebraic Theories, L.E.Dickson $1.50 LinearGroups,L.E.Dickson $1.95 Algebras andTheir Arithmetics, L.E. 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NEWYORK NEWYORK Published intheUnited Kingdom hyCon- stable andCompany Limited, 10Orange Street, London, W.C. 2 ThisnewDover edition, first published in 1960, isanunabridged andunaltered republi- cation ofthesecond edition published in1925. Itispublished byspecial arrangement with Edward Arnold Ltd. Manufactured intheUnited States ofAmerica Dover Publications, Inc. 180Varick Street NewYork 14,N.Y. ACOMPLETE surveyofthetheoryofsound would leadinto -OLmany fields, physical, physiological, psychological,aesthetic. Thepresenttreatise hasamore modest aim, inthat itis devotedmainlytothedynamical aspectofthesubject.Itis accordinglytoagreatextent mathematical, butIhave tried torestrictmyselftomethods andprocesseswhich shallheas simple anddirect asispossible, regard beinghadtothenature ofthequestionstreated. Ihopetherefore that thebookmay fairly bedescribed aselementary,andthat itmayserve asa steppingstone tothestudyofthewritingsofHelmholtz and LordEayleigh,towhich Iammyselfindebted foralmost all that Iknow ofthesubject. The limitation ofmethods hasinvolved some sacrifices. Varioustopicsofinterest have had tobeomitted, whilst others aretreatedonlyinoutline, butItrust thatenough remains toafford aconnected view ofthesubjectinatallevents itsmoreimportantbranches. Inthelatterpartofthebook anumber ofquestionsarisewhich itishardy possibletodeal withaccordingtothe stricter canons even ofmathematical physics. Some recourse tointuitional assumptionsisinevitable, and ifinorder tobringsuch questions within thescopeofthis treatise Ihave occasionallycarried this license alittle further than iscustomary,Iwouldpleadthat this isnotaltogethera defect, since attention isthereby concentrated onthose features which aremostimportantfrom thephysical pointofview. Althoughafewhistorical notes areinserted hereandthere, there isnoattemptatsystematiccitation ofauthorities. The reader whowishes tocarrythematter further willnaturally turn inthe first instance toLordRayleigh's treatise, where fullreferences, togetherwith valuable critical discussions, will befound. Imay perhapsbeallowed torefer also tothe article entitled"Schwingungenelastischer Systeme, insbeson- dere Akustik," inthefourth volume oftheEncyclopadie der mathematischenWissenschaften (Leipzig, 1906). iv PREFACE Ihaveregardedthedetailed descriptionofexperimental methods aslyingoutside myprovince.Itrust, however, that noonewillapproachthestudyofthesubjectashere treated without some first-handacquaintancewith theleading pheno- mena.Fortunately,agooddealcanbeaccomplishedinthis waywithvery simple andeasilyaccessibleappliances; and there is,moreover, nowant ofexcellentpracticalmanuals. Thebookhasbeen revised throughoutforthepresent edition, andanumber oferrors havebeen corrected. Ihave alsotaken advantageoftheopportunitytorectifysome omissions, andto notice some ofthemoreimportantrecentapplicationsofthe subject. H.L. CAMBRIDGE, May1925. CONTENTS INTRODUCTION ART. PAGE 1.Simple Vibrations andPureTones 1 2.Musical Notes , 3 3.Musical Intervals. Diatonic Scale..... 6 CHAPTER I THEORY OFVIBEATIONS 4.ThePendulum 8 5.Simple- Harmonic Motion 9 6.Further Examples......... 11 7.DynamicsofaSystem withOneDegree ofFreedom. Free Oscillations 12 8.Forced Oscillations ofaPendulum 16 9.Forced Oscillations inanySystem withOne Degree of Freedom. Selective Resonance 20 10.Superposition ofSimple Vibrations 22 11.Free Oscillations with Friction 24 12.General Dissipative System withOneDegree ofFreedom, Effect ofPeriodic Disturbing Forces.... 27 13. Effect ofDamping onResonance...... 32 14.Systems ofMultiple Freedom. Examples. TheDouble Pendulum , 34 15.General EquationsofaMultiple System.... 41 16.Free Periods ofaMultiple System. Stationary Properby. 44 17.Forced Oscillations ofaMultiple System. Principle of Reciprocity, 47 18.CompositionofSimple-HarmonicVibrations inDifferent Directions 48 19.Transition toContinuous Systems 52 20.OntheUse ofImaginary Quantities. 63 21. Historical Note ,, . , r 68 CHAPTER II STEINGS 22.EquationofMotion. Energy.-:-,. 59 23.Waves onanUnlimited Strings 61 24. Reflection. Periodic Motion ofaFinite String... 64 25.Normal Modes ofFiniteString. Harmonics... 68 CONTENTS CHAPTER III FOUBIEB'S THEOREM 32.TheSine-Series 89 33.TheCosine-Series 94 34.Complete Form ofFourier's Theorem. Discontinuities . 94 35.LawofConvergenceofCoefficients 96 36.Physical Approximation. Case ofPlucked String. . 98 37.ApplicationtoViolin String100 38. String Excited byImpact101 39.General TheoryofNormal Functions. Harmonic Analysis. 103- CHAPTER IV BAES 40.Elementary TheoryofElasticity. Strains.... 108 41. Stresses 110 42. Elastic Constants. Potential EnergyofDeformation .112 43.Longitudinal Vibrations ofBars 116 44.PlaneWaves inanElastic Medium 120 45.Flexural Vibrations ofaBar 122 46. Free-free Bar 126 47.Clamped-free Bar .129 48.SummaryofResults. Forced Vibrations.... 132 49.Applications 133 50. Effect ofPermanent Tension 134 51.Vibrations ofaRing. Flexural andExtensional Modes .135 CHAPTER V MEMBBANBS ANDPLATES 52.EquationofMotion ofaMembrane.Energy. .141 53.Square Membrane. Normal Modes 144 54. Circular Membrane. Normal Modes 14g 55.Uniform Flexure ofaPlate........ 152 56.Vibrations ofaPlate. General Eesults 154 57.Vibrations ofCurved Shells ._ IQQ. CHAPTER VI PLANE WAVES OFSOUND AKT. PAGE 58. Elasticity ofGases 160 59.Plane Waves. VelocityofSound 163 60.EnergyofSound- Waves 166 61. Reflection 171 62.Vibrations ofaColumn ofAir 173 63.Waves ofFinite Amplitude 177 64.Viscosity 186 65. Effect ofHeat Conduction 190 66.DampingofWaves inNarrow Tubes andCrevices . .193 CHAPTER VII GENERAL THEOEY OFSOUND WAVES 67. Definitions. Flux. Divergence 200 68.Equations ofMotion 203 69.Velocity-Potential 204 70.General EquationofSound Waves 207 71.Spherical Waves 208 72.Waves resulting from agivenInitial Disturbance . .215 73.Sources ofSound. Eeflection...... 217 74. Refraction duetoVariation ofTemperature. . . .219 75.Refraction byWind 222 75.Acoustic PropertiesofBuildings 225 15b.Doppler's Principle 226 CHAPTER VIII SIMPLE-HARMONIC WAVES. DIFFRACTION 76. Spherical Waves. Point-Sources ofSound.... 227 76a. Reflection ataPlane Surface 232 77.Vibrating Sphere234 78. Effect ofaLocal Periodic Force 239 79.Waves generated byVibratingSolid 240 80.Communication ofVibrations toaGas..... -241 81. ScatteringofSound Waves byanObstacle.... 244 82.Transmission ofSound byanAperture.... 248 83.Contrast between Diffraction Effects inSound mid Light. Influence ofWave-Length 252 vm CONTENTS AET. 84. 85. 86. 87.CHAPTER IX PIPES ANDEESONATOES Normal Modes ofRectangular andSpherical Vessels Vibrations inaCylindrical Vessel PreeVibrations ofaResonator. Dissipation Corrected Theory oftheOrgan Pipe. Resonator under Influence ofExternal Source. Reaction ontheSource , Mode ofAction ofan.Organ Pipe.Vibrations caused by Heat TheoryofReed-Pipes89. 90 90a.Multiple Resonance 906.TheHot-wire ResonatorPAGE 258 2(53 264 270 274 280 282 287 288 CHAPTEE X PHYSIOLOGICAL ACOUSTICS 91.AnalysisofSound Sensations. Musical Notes . . .289 92. Influence ofOvertones onQuality 291 93. Interference ofPure Tones. Influence ontheDefinition ofIntervals 292 94.Helmholtz Theory ofAudition 294 95.Combination-Tones 297 96. Influence ofCombination-Tones onMusical Intervals . .302 97. PerceptionofDirection ofSound 303 INDEX 305 THE DYNAMICAL THEORY OF SOUND THEDYNAMICAL THEORY OFSOUND INTEODUCTION 1.Simple Vibrations andPure Tones. Inanyordinary phenomenonofsound weareconcerned, firstwith thevibrating body, e.g.astringoratuningfork or acolumn ofair,inwhich thedisturbanceoriginates, secondly with thetransmission ofthevibrationsthroughthe aerial medium, next with thesensations which theimpactofthe waves onthedrum oftheearsomehow andindirectly produces, andfinally with theinterpretation which, guided mainly and perhaps altogether byexperience,weputuponthese sensations. Itisinsomethinglike thisnatural order that thesubject willbediscussed inthefollowing pages,butthelaterstages involving physiologicalandpsychological questionscanonlybe touchedupon very lightly. Asfewreaders arelikelytotakeupthisbook without someprevious knowledgeofthesubject wemay brieflyre- capitulateafewpointswhich willbemore orless familiar, with theview offixingthemeaningofsome technical terms which willbeofconstant occurrence. Manyofthematters here referred towillofcourse bedealt withmorefullylater. The frontier betweenphysicsandphysiologyisreached at thetympanic membrane, andfrom thephysical standpointitis tothevariations ofpressureintheexternalear-cavitythatwe must inthelast resort look, under normal(asdistinguished frompathological) conditions, forthecause ofwhatever sensations ofsound weexperience. These variations may conveniently beimaginedtobeexhibitedgraphically,liketheordinary variations ofbarometricpressure, byacurve inwhich the abscissaerepresenttimes andtheordinatcs deviations ofthe 2 DYNAMICAL THEORY OFSOUND pressureononesideorother ofthemean, theonlydifference beingthatthehorizontal andvertical scales arenowenormously magnified. Thevarietyofsuch curves isofcourse endless, and itis impossibletosupposethatadistinctprovisionismade inthe earfortherecognitionofeach, oreven ofeach ofthenumerous classes intowhich theymight conceivablybegrouped.Itis therefore necessarytoanalyse,asfaraspossible,both the vibration-forms and theresultingsensations into simpler elements which shall correspondeach toeach. Asregardsthe vibration-forms, there isonemode of resolution which atonce claims consideration ondynamical grounds. Thefundamental typeofvibration inMechanics is thatknown as"simple-harmonic,"which isrepresented graphic- allybyacurve ofsines(Fig. 3,p.10).This ismetwith in thependulum, andinallother cases ofafreely vibrating body ormechanical system having onlyonedegreeoffreedom. It canmoreover beshewn thatthemostcomplicatedoscillation of anysystemwhatever may,sofarasfriction canbeneglected,be regardedasmadeupofaseries ofvibrations ofthiskind, each ofwhich mightbeexcitedseparately bysuitableprecautions. Thereason forthepreeminent positionwhich thesimple- harmonictype occupiesinMechanics isthat itistheonlytype which retains itscharacterabsolutely unchanged whenever it istransmitted from onesystemtoanother. This willbeex- plainedmorefullyinthefollowing chapter. Theanalysisofsensations isamuch more delicate matter, and itwasagreat stepinAcoustics whenOhm* in1843 definitely propoundedthe doctrine that thesimplestand fundamentaltypeofsound-sensation isthatwhichcorresponds toasimple-harmonicvibration. Thisimpliesthat allother sound-sensations areinreality complex, being made upof elementary sensationscorrespondingtothevarioussimple- harmonic constituents intowhich thevibration-form canbe resolved. Thestatement issubjecttosomequalifications,in particularastothedegreeofindependenceofelementary *G.S.Ohm(17871854), professor ofphysicsatMunich 184954, known also astheauthor of''Ohm's Law"ofelectric conduction. sensationsverynear tooneanother inthescale, butthese need notdetain usatpresent.Itmayberegardedasinthemain fully established, chieflyinconsequenceofthelabours of Helmholtz*. Thesensation correspondingtoasimple-harmonic vibration iscalled "a"simpletone" ora"pure tone," ormerely a"tone." Thesound emitted byatuningfork fitted with asuitableresonator, orbyawidestopped organ pipe, givesthe bestapproachtoit. Since theform ofthevibration-curve isfixed, thedistinction between onesimpletoneandanother canonlybedue to difference offrequencyorofamplitude. The" frequency,"i.e. thenumber ofcompletevibrationspersecond, determines the " pitch," greater frequency correspondingtohigher pitch. The lower andupperlimits offrequencyfortones audible tothe human earareputatabout 24and24,000 respectively;the range employedinmusic ismuch narrower, andextendsonly fromabout 40to4000. Asbetween tones ofthesamepitch, theamplitude,orrather itssquare, determines therate of supplyofenergytotheearand sotherelative"intensity," but itwillbeunderstood that itisphysicalrather than subjective intensitythat ishere involved. Between tones of differentpitch onlyavague comparisonofloudness ispossible, andthismayhave little relation tothesupplyofenergy. Near thelimits ofaudibilitythesensation maybefeeble, eventhough theenergy-supplyberelativelyconsiderable. 2.Musical Notes. From thechaos ofmorecomplexsounds there stands outa special class, viz.thatofmusical "notes." The characteristic of such sounds isthat thesensation issmooth, continuous, and capable (atleast inimagination)ofindefiniteprolongation withoutperceptible change. Thenature ofthecorresponding vibrations iswell ascertained. Ifweinvestigate anycontrivance *Hermann Helmholtz(1821 94), successively professor ofphysiology (Konigsberg 1849), anatomy (Bonn 1855), physiology (Heidelberg 1858) and physics (Berlin 1871). Reference willoften bemade tohisclassical work :Die Lehre vondenTonempfindungenalsphysiologische Grundlagejilr dieTkeorie der Musik, Brunswick, 1862. AnEnglish translation from thethird edition(1870) waspublished byA.J.Ellisunder thetitleSensationsofTone, London, 1875. bywhich anote ofgoodmusicalqualityisactually produced, wefindthatthevibration canberesolved intoaseries ofsimple- harmoniccomponents whosefrequenciesstand tooneanother inacertainspecial relation, viz.theyareproportionaltothe numbers 1,2,3,....Individual members oftheseriesmaybe absent, andthere ispracticallyalimit ontheascending side, butnoother ratios areadmissible. Itisevident from the above relation that theresultant vibration-form isnecessarily periodicincharacter, recurring exactlyatintervalsequaltothe periodinwhich the firstmember oftheseriesgoesthroughits phases.Itmust beremembered, however, thattheearhasno knowledgeoftheperiodiccharacter assuch,and itmust notbe supposedthatevery periodicvibration willnecessarily producea sensation which ismusicallytolerable. Thesuperpositionof simple-harmonicvibrations toproduce periodicvibration-forms isillustrated bysome ofthediagrams givenbelow in ChapterIII. Onemusical notemaydiffer from another inrespectof pitch, quality, and loudriess. Thepitchisusuallyestimated asthat ofthefirstsimple-harmonicvibration inthe series, viz. that oflowestfrequency,but iftheamplitudeofthis first componentberelatively small, andespeciallyifitfallnear the lower limit oftheaudible scale, theestimatedpitchmaybe that ofthesecondcomponent. By" quality" ismeant thatunmistakable character which distinguishesanoteononeinstrument from thenote ofthe samepitchasgiven byanother. Everymusical instrument hasasarule itsownspecific quality *,which isseldomlikelyto beconfused with that ofanother.Everyone recognizesfor instance thedifference incharacter between thesound ofa flute, aviolin, atrumpet, andthehuman voice, respectively. Itisobvious that difference ofquality,sofarasitisnotdueto adventitious circumstancesf,canonlybeascribed todifference ofvibration-form, andsotodifferences intherelativeamplitudes andphasesofthesimple-harmonicconstituents.Accordingto *French timbre;German Klangfarbc. fSuch asthemanner inwhich thesound setsinandceases;this isdifferent forinstance intlieviolin andthepiauo. sounds ofthesamequality andabout thesamepitch. Itfollows from thepreceding that, sofarasOhm's law is valid, thesensation ofamusical notemustbecomplex,andmade upofthesimpler sensations, ortones, whichcorrespondtothe varioussimple-harmonic elements inthevibration-form. This doctrine has tocontend withstrongand tosome extent instinctiveprepossessionstothecontrary, andsomepreliminary trainingisusually necessarybefore itisacceptedasafactof personal experience. Weshall return tothisquestion later; at present wemerelyrecord thatthatelement inthesensation which correspondstothegravest simple-harmonicconstituent iscalled the"fundamental tone," andthattheothers aretermed its"overtones"or"harmonics." 3.Musical Intervals. Diatonic Scale. There arecertainspecial relations, familiar totrained ears, inwhich twonotes ortwosimpletonesmaystand toone another. These arethevarious consonant andother "intervals." Physically theyaremarked bythepropertythatthefrequencies correspondingtotherespective pitchesare inadefinite numerical ratio,which canbeexpressed bymeans oftwosmall integers. Thenames ofthemore importantconsonant intervals, with therespective ratios, areasfollows : Unison 1 :1 Octavo 1 :2 Fifth 2 :3 Fourth 3 :4 MajorThird 4 :5 Minor Sixth 5 :8 Minor Third 5 :6 MajorSixth 3 :5. The earhasofcourso noappreciationofthenumerical relations themselves; buteach interval ismore orlesssharply "defined," inthesense thataslight mistimingofeither note is atonce detected bythe beats, andconsequentsensation of roughness,which arcproduced.Theexplanationofthese latter peculiaritiesmust bedeferred furthepresent. 6 DYNAMICAL THEOEY OFSOUND Thenamesgiventothevarious intervals areinasense accidental, andrefer totherelativepositionsofthenotes on theordinary"diatonic scale." This isbased onthe"major chord," which isacombination ofthree notes formingaMajor andaMinor Third;i.e.theirfrequenciesareas4 :5 :6.Ifwe start from,anyarbitrary note,which wewill callC,askeynote, thetwonotes which lieaFifth above andbelow itarecalled the"dominant"(G)andthe"subdominant" (F,) respectively. Ifweform themajor chord fromCwegetthenotesE=|C, andG=|G.Againifweformthemajorchord fromGweget thenotesB=fG=J0,andd=fG=fC.The latter falls outside theoctavebeginningwithC;thecorrespondingnote within theoctave isD=fC.Lastly, formingthemajorchord from F,wegetA/=fF=fxfC=fC,theoctave ofwhich is A= -JC,andCitself.Wethusobtain thescale ofseven notes whosefrequenciesareproportionaltothenumbers heregiven: CDEFGAB 195435151S ? IF ? 3 -F 24 27 30 32 36 40 45 This iscontinued upwardsanddownwards inoctaves; thesame letters arerepeatedasthenames ofthenotes, butthevarious octaves maybedistinguished bydifference oftype,andby accents orsuffixes. Theprecise pitchofthekey-noteissofar arbitrary;itdetermines, and isdeterminedby,that ofany other note inthe scale. Amongmusicians thestandard has varied indifferentplacesandatdifferent times, thegeneral tendency beinginthedirection ofarise.Germanphysical writers, including Helmholtz, have followed astandard which assignstoacertainAafrequencyof440 *.Onthis basiswe have thefollowing frequenciesforacertainrangeofthe scale : *Thismakes c'=264. Physical instrument makers nowoften take c'=256, which isconvenient onaccount ofitscontinueddivisibility by2. c' d' e'f g'a'V G"d" e"f"g"a" 2G4 297330 352 396440 495 528594660704792880990 Underneath theordinarymusical symbols wehaveplaced theconvenient literal notation employed byGerman writers. Thismaybecontinued upwards bymeans ofadditional accents (G'", clv ,...),anddownwardsbysuffixes(C,, C,,, ...). Ifintheconstruction ofthescalewehadused, instead of themajor,theminor chord, which consists ofaMinor anda MajorThird inascending order, thefrequencies beingas 10 :12 :15,weshould haverequiredthree notes notincluded intheabove scheme. Andif,startingfromanynotealready obtained (otherthan C)asanew key-note,weproceedto construct amajororaminor scale, further additional notes are required.Inthecaseoftheviolin, orofthehuman voice, orof some other wind-instruments which allow ofcontinuous varia- tionofpitch,thispresentsnodifficulty. But ininstruments likethepianoororganthemultiplicationoffixed notes beyond amoderate limit isimpracticable.Itisfound, however, that byaslight tamperingwith thecorrect numerical relations the requirementsofmostkeyscanbofairlywellmetbyasystem oftwelve notes ineach octave, which areknown as CC#DDtfEFnGGttAAftB. Thisprocessofadjustment,orcompromise,iscalled"tempera- ment"; ontheusualsystemof"equal" temperamentthe intervals between thesuccessive notes aremadeequal,the octave being accordinglydivided intotwelvestepsforeach of which thevibration-ratio is2".Thus theratio ofGtoCis made tobe2"=T4983 instead ofT5. CHAPTER I THEORY OFVIBRATIONS 4.ThePendulum. Avibrating body,such asastringorabaroraplate, cannotgiverise toasoundexceptinsofarasitactsonthe surrounding medium, which inturn exerts acertain reaction onthebody. Thereaction ishowever inmanycases soslight that itseffectsonlybecome sensible after alargenumber of oscillations. Hence, tosimplify matters, webegin byignoring it,andinvestigatethenature ofthevibrations ofamechanical system considered ascompletelyisolated. Thetheoryofvibrationsbegins, historically andnaturally, with thependulum. With thissimple apparatus weareable toillustrate, inallessentials, many important principlesofacoustics, themere differ- ences ofscale asregards amplitudeandperiod, enormous asthey are,being unimportant from the dynamical pointofview. AparticleofmassM,suspendedfrom afixed point byalight stringoflength I,issupposed tomake small oscillations, inaverticalplane, about itspositionofequilibrium.Iftheinclina- tion ofthestringtothevertical never exceeds afewdegrees,the verticaldisplacementofthe particle may (toafirstapproximation) beneg- lected, andthetension (P)ofthestringmaybe equatedtothegravity Myoftheparticle.Since thehorizontal displacement (*)isaffectedonlybythehorizontalcomponent ofthetension, wehaveFig. 1. 7(1) Ifweput n2=g/l, (2) thisbecomes-T7+w2a?=0; (3) andthesolution is x=Acosnt+Bsinnt, (4) where theconstants A,Bmayhave anyvalues. That this formulareallysatisfies(3)isverified atoncebydifferentiation; andsince itcontains twoarbitraryconstants A,B,weareable toadaptittoanyprescribedinitial conditions ofdisplacement andvelocity. Thus if,when t=0,wearetohave <c=xQ) dx/dt=u0)wefind n cc=#cosnt+--sinnt (5)n' Itisofcoursenecessary,intheapplicationtothependulum, thattheinitial conditions should besuch asareconsistent with theassumed"smallness"oftheoscillations. Thus in(5)we mustsupposethattheratios x/landu/nlareboth small. In virtue of(2)thelatter ratio isequalto^(ufjgl),sothatUQ must besmallcomparedwith thevelocity"dueto"halfthe lengthofthependulum. 5.Simple-Harmonic Motion. Ifin4(4)weput (1) asisalways possible byasuitable choice ofaande,weget x=acos(nt+e) (2) Theparticular typeofvibration represented bythisformula isoffundamental importance. Itiscalled a"simple-harmonic," or(sometimes)a" simple" vibration. Itscharacter isbest exhibited ifweimaginea geometrical pointQtodescribe acircle of1 radius awith the constantangular velocityn. Theorthogonal projection Pof Qonafixed diameter AOA' willmoveexactly accordingtoFig.2. 10 DYNAMICAL THBOEY OPSOUND theformula(2),provideditbestarted attheproperinstant. Theanglent+e(AOQ)iscalled the"phase"; and the elements a,earecalled the"amplitude"andthe"initial phase," respectively. The interval%Tr/nbetween two suc- cessive transitsthroughtheorigininthesame direction is called the" period." Inacoustics, where wehave todealwith very rapid vibrations, itisusual tospecify,instead ofthe period,itsreciprocalthe" frequency" (IV),i.e.thenumber of completevibrationspersecond;thus JVn/27r*. Inthecase ofthependulum,where n=*J(g/l),theperiod is27r^(l/g). Asinthecase ofallother dynamical systems which weshallhave occasion, toconsider, this isindependent oftheamplitudesolongasthelatter issmall*[. ThevelocityofPinanypositionis dsc dt=nasin(nt+e)=n .PQ, .(3) asappearsalsobyresolvingthevelocity (no)ofQparallel toOA. Inallcases ofrectilinear motion ofapointthemethod of graphical representation bymeans ofacurve constructed with *Thewant ofaseparate name fortheangular velocity nintheauxiliary circle issometimes felt. Inthetheory ofthetides theterm"speed" was introduced byLord Kelvin. Asanalternative term inacoustics theword "rapidity" mayperhaps besuggested. tThis observation wasmade byGalileo in1583, thependulum being a lamp which hangsinthecathedral ofPisa. thetime tasabscissa andthedisplacement xasordinate is ofgreatvalue. This iscalled the"curve ofpositions,"orthe "space-timecurve." Inexperimentalacoustics numerous mechanical andopticaldevices havebeen contrived bymeans ofwhich such curves canbeobtained. Inthepresentcase ofasimple-harmonic vibration, theformula(2)shews thatthe curve inquestionisthewell-known"curve ofsines." 6.Further Examples. Thegoverningfeature inthetheoryofthependulumis that theforceactingontheparticleisalways towards the positionofequilibriumand(toasufficientapproximation) proportionaltothedisplacementtherefrom. All cases of thiskind arecovered bythedifferential equation andtheoscillation istherefore ofthetype (2)of5,with nz=KjM. Themotion isthereforesimple-harmonic, with thefrequencyK determinedsolelybythenature ofthesystem, andindependent oftheamplitude. The structure ofthisformula should be noticed, onaccount ofitswideanalogies. Thefrequency varies asthesquareroot oftheratio oftwoquantities,one ofwhich (-/)measures theelasticity,orthedegreeofstability, ofthesystem,whilst theother isacoefficient ofinertia. Consider, forexample,thevertical oscillations ofa^ massMhangingfrom afixedsupport byahelical spring.Inconformity with Hooke's lawofelasticity, weassume that the force exerted bythespringis equaltotheincrease oflength multiplied byacertain constant K,which maybecalled the"stiffness" of theparticular spring.Inthepositionofequilibrium thetension ofthespring exactlybalances thegravity Mg;and ifMbedisplaced downwardsthrougha space x,anadditional forceKxtowards thisposition iscalled intoplay,sothat theequationofmotion isof 12 DYNAMICAL THEOEY OFSOUND thetype (I). The inertia ofthespringitself ishere neglected*. Again, suppose wehave amassMattached toawirewhich istightlystretched between twofixedpoints with aten- sionP.Weneglect gravity andtheinertia ofthewirelg>' itself; andwefurther assume thelateral displacement (x)to besosmall thatthechangeintension isanegligiblefraction ofP.Ifa,bdenote thedistances oftheattachedparticle from thetwo ends,wehave j|/* P*_p* ..................(3)dt2a b^ which isofthesame form as4(3),with n?=P(a+b)/Mab. Thefrequencyistherefore ab This case isofinterest because acoustical frequenciescan easilyberealized. Thus ifthetension be10kilogrammes, andamass of5grammesbeattached atthemiddle, the wirebeing50cm.long,wefindN=63. 7.Dynamics ofaSystem withOneDegree ofFreedom. Free Oscillations. Theaboveexamplesare allconcerned with therectilinear motion ofaparticle,butexactlythesametypeofvibration ismetwith ineverycase ofadynamical systemofonedegree offreedomoscillating freely, throughasmallrange,about aconfigurationofstableequilibrium. Asystemissaid tohave"onedegreeoffreedom" when thevariousconfigurationswhich itcanassume can allbe specified byassigningthepropervalues toasinglevariable element or"coordinate." Thus, thepositionofacylinder (ofanyform ofsection) rollingonahorizontalplaneisdefined bytheangle throughwhich ithasturned fromsome standard position. Asystemoftwoparticles attached atdift'ercntpoints ofastringwhose endsA,Barefixed hasonedegreeoffreedom *Acorrection onthisaccount isinvestigated in 7. ifitberestricted todisplacementsinthevertical plane through A,B,fortheconfiguration maybespecified bytheinclination ofanyoneofthestringstothehorizontal. Again,thecon- figurationofasteam-engine andofthewhole train ofmachinery which itactuates isdefined by theangularcoordinate ofthe flywheel. Thevarietyofsuch systemsisendless, but ifwe exclude frictional orother dis- sipativeforces thewhole motion ofthesystem when startedg anyhow and left toitself is governed bytheequationofenergy. And inthe case of small oscillations about stableequilibrium,the differential equationofmotion, asweshall see,reduces alwaystothe type6(1). Wedenote byqthevariable coordinate whichspecifies theconfiguration. Asinthecase ofFig. 6,thismaybe chosen invariousways,buttheparticularchoice made is immaterial. From the definition ofthesystemitisplain that eachparticleisrestricted toacertainpath.Ifin consequenceofaninfinitesimal variationBqofthecoordinate aparticleindescribes anelement 8sofitspath, wehave Ss=a.8q,where aisacoefficient which isingeneraldifferent fordifferentparticles, and alsodependsontheparticular configuration qfrom which thevariation ismade. Hence, dividing bythetime-element Bt,thevelocityofthisparticle isv=adq/dt,orinthefluxional notation*, v=aq. Hence thetotal kineticenergy, usually denoted byT,is T=$Z(mtf)=$atft (1) where a=S(ma2 ), (2) thesummation 2embracingalltheparticlesofthesystem. The coefficient aisingeneralafunction ofq;itmaybe called the"coefficient ofinertia"fortheparticular configura- tionq.Forexample,inthecase oftherolling cylinderreferred *Theuseofdots todenote differentiations with respecttotwasrevived by Lagrange intheMecanique Analytique (1788), andagain inlater times by Thomson and Tait. "Wewrite qfordqjdt andqfor 14 DYNAMICAL THEOEY OPSOUND toabove, itisthe(usually variable) moment ofinertia about the line ofcontact with the horizontalplane, provided q denote theangularcoordinate. Thepotential energyofthesystem,since itdepends onthe configuration,willbeafunction ofqonly.Ifwedenote it by F",theconservation ofenergy gives |ag2+F= const., (3) providedthesystembefreefrom extraneous forces. The value oftheconstant isofcourse determined bythe initial circumstances. Ifwedifferentiate (3)with respecttot,the resulting equationisdivisible by q,andweobtain ..1da .dV ,.//f,a2+23/+^= ' W which mayberegardedastheequationoffreemotion ofthe system,with theunknown reactions between itspartselim- inated. Intheapplicationtosmall oscillations itgreatly simplifies. Inorder that theremaybeequilibriumtheequation (4) mustbesatisfied byq=const. ThisrequiresthatdVfdq= ; i.e.anequilibrium configurationischaracterised bythe fact that thepotential energyis"stationary"invalue forsmall deviations from it.Byaddingorsubtractingaconstant, we canchoose qsoastovanish intheequilibrium configuration which isunder consideration, whence, expandinginpowersof thesmallquantity g,wehave F=const.+%cq-+ ..., (5) thefirstpowerofqbeingabsent onaccount ofthestationary property.Theconstant oispositiveiftheequilibriumcon- figurationbestable, andVaccordinglythen aminimum*. Ib maybecalled the"coefficient ofstability." Ifwesubstitute from(5)in(4),andomit terms ofthe second order inq,q,weobtain aq+cq=0, (6) where amaynowbesupposedtobeconstant, and tohave the value correspondingtotheequilibrium configuration. *Intheopposite casethesolution of(6)below would involve realexponen- tials instead ofcircularfunctions, indicating instability. Since (6)isofthesametypeas6(1),with n*=c/a, ...........................(7) thevariation ofqissimple-harmonic, Sivy- 2=Gcos(nt +e),.....................(8) withthefrequency Moreover, since thedisplacementofany particleofthe system alongitspath, from itsequilibrium position,ispro- portionaltoq(being equaltoaqintheabovenotation), wesee thateach particlewillexecute asimple-harmonicvibration of theabove frequency, andthatthedifferentparticleswillkeep stepwith oneanother, passing throughtheirmeanpositions simultaneously. Theamplitudesoftherespective particlesare moreover infixed ratios tooneanother, theabsoluteamplitude, andthe phase, beingalonearbitrary,i.e.dependentonthe particularinitial conditions. The kinetic andpotential energiesarerespectively 21=W=Ka2sin2 (nt+e), ,(} thesumbeing ............... (11) invirtue of(7). Since themean values ofsin2 (??i+e)and cos2(nt+e)areobviously equal,and therefore each= |-,the energyisontheaveragehalf kinetic and halfpotential. Theapplicationofthetheorytoparticularcasesrequires onlythecalculation ofthecoefficients aand c,thelatterbeing (inmechanicalproblems) usuallythemore troublesome. In thecase ofabodyattached toavertical wire,andmaking torsional oscillations about theaxisofthewire,aisthemoment ofinertia about this axis,and cisthemodulus oftorsion, i.e.cqisthetorsionalcouple when thebodyisturnedthrough anangle q. Againinthecase ofamasssuspended byacoiledspring (Fig. 4),ifweassume that theverticaldisplacementofany pointofthespringisproportionaltoitsdepthzbelow the 16 DYNAMICAL THEORY OFSOUND pointofsuspensionintheunstrained state, thekineticenergy isgiven by { J ...... ...............(12) ifpbethelinedensity,Ztheunstretchedlengbh,andqthe displacementoftheweight. The inertia ofthespringcan therefore beallowed forbyimaginingthesuspendedmass tobe increased byone-third that ofthespring. 8.Forced Oscillations ofaPendulum. The vibrations sofarconsidered are" free," i.e.thesystem issupposed subjecttonoforcesexceptthose incidental toits constitution and itsrelation totheenvironment. Wehave now toexamine theeffect ofdisturbing forces, andinparticular that ofaforce which, isasimple-harmonicfunction ofthetime. This kind ofcase arises when onevibrating bodyactson another under such conditions that thereaction onthe first bodymaybeneglected. Fordefiniteness wetake thecase ofamass movable ina straight line,thesubsequent generalization (9)beingavery simplematter. Theequation (1)of6isnowreplaced by (1) thelasttermrepresentingthedisturbing force,whoseamplitudeFtandfrequency p/%7r,areregardedasgiven*.Ifwewrite na ,F/M=f,..................(2) wehave -+n-x=fco&pt...................(3) Thecompletesolution ofthisequationis fx=Acosnt+Bsinnt+^-cospt, ...... (4)n*p* asiseasilyverified bydifferentiation. The firstpartofthis,with itsarbitraryconstants A,B, representsafreevibration ofthecharacterexplainedin5, *Theslightly more general casewhere theforce isrepresented byFcos(pt-|-a) canbeallowed forbychanging theorigin fromwhich tisreckoned. with thefrequency n/27r propertothesystem. Onthis is superposeda"forced vibration" represented bythe lastterm. This isofsimple-harmonic type,with thefrequency p/Z-rrofthe disturbing force, andthephaseisthesame asthatoftheforce, ortheopposite, accordingasp$n,i.e.accordingastheimposed frequencyislessorgreater thanthenaturalfrequency. Theabovetheoryiseasilyillustrated bymeans ofthe pendulum.Iftheupperendofthestring,instead ofbeing fixed, ismade toexecute ahorizontal motion inwhich the displacementattime tis(Fig. 7),theequationofmotion(1) of4isreplaced by rPrMa ill-775= ai2 .(5) or.(0) This isthesame asiftheupper endwere fixed, andthebob were subjecttoahorizontal forcewhose accelerative effect is -n?g. Ifasaparticularcasewetake =a.cospt, (7) wegettheform(3),with/=??,2a.TheannexedFig.8repre- sents theforced oscillation inthetwocases ofp<nandp>n, respectively. Thependulum oscillates asifCwere afixed 18 DYNAMIC Ala TJtUfiUJtlX UF point,thedistance GPbeing equaltothelengthofthesimple pendulumwhose freeperiodisequaltotheimposed period Sir/jp. ThisexampleisduetoYoung*, whoappliedittoillustrate thedynamical theoryofthetides, where thesamequestionof phasearises. Itappearsfrom thistheorythatthetides inan imagined equatorialbeltofocean, ofabreadth notexceeding afewdegreesoflatitude, and ofanydepth comparablewith theactualdepthofthe sea,would be"inverted," i.e.there would belowwater beneath themoon, andhighwater in longitudes90E.andW.from it,thereason beingthat the periodofthedisturbingforce(viz.12lunar hours)islessthan thecorrespondingfreeperiod,sothat there isoppositionof phase. Thearbitraryconstants inthecompletesolution (4)are determined bythe initial conditions. Suppose,forexample, thatthebodystarts from restinthezeropositionattheinstant t=0.Wefind ffc=-^ (cosntcosfit). (8) P-n** asmaybeimmediatelyverified. When theimposed frequency p/%7risnearly equaltothe naturalperiod,the lastterm in(4)becomes very large,and it maybethattheassumptionastothesmallness of a;onwhich theequation (1)isusuallybased(asinthecaseofthependulum) istherebyviolated. The resultexpressed by(4)isthennotto beacceptedwithout reserve, butwehave atallevents anindica- tionofthereason whyanamplitudeofabnormal amount ensues whenever there isapproximate agreementbetween thefreeand theforcedperiod. Inthecase(pn)ofexact coincidence-between thetwo periods,thesolution(4)becomesaltogether unmeaning,butan intelligibleresultmaybeobtained ifweexamineanyparticular *DrThomas Young (1773 1829), famous forhisresearches onlight, and other branches ofphysics. Theelementary theory offreeandforced oscilla- tionswasgiven byhim inanarticle on"ATheory oftheTides, including the consideration ofResistance," Nicholson's Journal, 1813; Miscellaneous Works, London, 1855, vol. n.,p.262. 111WJUUU U11B 111 XllUO,JL.U. caseof(8),theformula maybewritten fsini(). /n\...... (9) andaspapproaches equalitywithnthistends tothelimiting form (10) Thismaybedescribed (roughly)asasimplevibration whoseamplitudeincreases proportionallyto t.Forareason justindicated this isonlyvalid asarepresentationoftheearlier stagesofthemotion. The case ofadisturbingforce ofmoregeneralcharacter maybebrieflynoticed. The differentialequationisthen of theform Themethod ofsolution, byvariation ofparameters,or otherwise, isexplainedinbooks ondifferentialequations. The result, whichmay easily beverified, is a;=-sinntf If(t)f If(t]si cosntdt cosnt If(t]sinntdt.(1.2) Itisunnecessarytoaddexplicitlyterms ofthetype Acosnt+Bsinnt,whichexpressthe free vibrations, since those arealready presentinvirtue ofthearbitrary constants impliedintheindefiniteintegrals. Iftheforcef(t)isonlysensible foracertain finiterangeof t,and iftheparticlebeoriginallyatrest inthepositionof equilibrium,wemaywriteIf* If*.x-sinntl f(t)cosntdt cosntl f(t)smntdt, (13)n J oo n J ao since thismakes &=(), dx/dt=Qfor t= oo .The vibra- tionwhich remains after theforce hasceased tobesensible is accordingly x=Acosnt+Bsinnt,...............(14) where ir i/"" f(t)s'mntdt, B=~f(t)coantdt. (15) otbJ_.QQ Forexample,let thisrepresentsaforcewhich issensible foragreaterorless interval onboth sides oftheinstant t=0,accordingtothe value ofT,theintegralamount orimpulse being /A*.By makingTsufficientlysmallwecanapproximateascloselyas wepleasetothecaseofaninstantaneous impulse.Since coantdt TT"~ U wehave x=-- sinnt................... (18)n Theexponentialfactor shews the effect ofspreadingout theimpulse.This effect isgreater,thegreaterthefrequency ofthenatural vibration. 9.Forced Oscillations inanySystem withOneDegree ofFreedom. Selective Resonance. Thegeneralizationofthese results offers nodifficulty. When givenextraneous forces actonasystemwith onedegreeof freedom, whose coordinate isq,theworkwhichtheyperformin aninfinitelysmallchangeofconfiguration, being proportionalto 8q,maybedenoted byQ$q. Thequantity Qiscalled the "force"actingonthesystem,"referred tothecoordinate q" Forinstance, ifqbetheangularcoordinate ofabodywhich can rotate about afixed axis,Qisthemoment oftheextraneous forces about this axis. Itfollows thatinanyactual motion ofthesystem therate atwhich extraneous forces aredoingAvork isQq.Theequation ofenergy nowtakes theform whence, insertingthevalue ofTfrom 7(1),wehave *Thegraph ofthisfunction isgiven, foranother purpose, inFig. 14,p.33. tTheformer oftheseintegralsisevaluated inmost books ontheIntegral Calculus. second oraer asoeiore. nence, suusuiouuiuguuevaiue 01 Y from 7(5),wefind aq+cq=Q......................... (3) "WhenQisofsimple-harmonic type, varying (say)ascospt, theforced oscillation isgiven by which isofcourse merelyageneralizedform ofthelastterm in 8(4). Twospecialcasesmaybenoticed. When pisvery small, (4)reduces toq=Qfc.Thismaybedescribed asthe" equili- brium" value* ofthedisplacement,viz. itisthe statical displacement which would bemaintained byaconstant force equaltotheinstantaneous value ofQ.Inother words, itis thedisplacement which would beproducedifthesystem were devoid ofinertia(ci=0).Denotingthisequilibrium valueby q,wemaywrite(4)intheform q=ST^p]n*'.....................(5) where, asin7,ndenotes thespeedofafreevibration. When, ontheother hand,pisverygreat compared withn, (4)reduces to q=-Q/p*a,.....................(G) approximately. This isalmost thesame asifthesystem wore devoid ofpotential energy,theinertia alonehaving anysensible influence. When twoormoredisturbingforces ofsimple-harmonic typeactonasystem, theforced vibrations duetothemmaybe superposed bymere addition. Thus adisturbingforce Q=/cos(pj 4-O+/ acoa(p+O+......(7) willproduce theforced oscillation fCOS *Thename istaken from thetheory ofthetides, whore theequilibrium tide-height isdefined asthatwhich would bemaintained bythedifiturbiugforces ifthese were toremaitipermanently attheir instautaneous values. 22 DYNAMICAL THEOEY OFSOUND Itwillbeobserved that theamplitudesofthevarious terms arenotproportionaltothose ofthecorresponding terms inthe value ofQ,owingtothedifference inthedenominators. This isanillustration ofaremark made in 1that the simple-harmonic typeistheonlyonewhich isunaltered in character when itistransmitted, thecharacter ofthecomposite vibrationrepresented by(8)beingdifferent from that ofthe generatingforce. Inparticularit'oneoftheimposed speeds pi}p2,...benearlycoincident with thenaturalspeed n,the corresponding element inthe forced vibration may greatly predominateover the rest. This isthetheoryofselective "resonance," sofarasitispossibletodevelopitwithout reference todissipativeforces. 10.Superposition ofSimple Vibrations. Thesuperpositionofsimple-harmonicmotions inthesame straightlinehasmany important applications.For instance, theheightofthetideatanystation isthealgebraicsum ofa number ofsimple-harmoniccom- ponents,themost considerable (atmany stations) beingthose whoseperiodsarehalfalunar andhalfasolarday,respectively. Thecompositionoftwo simplevibrations maybeillus- trated bythegeometrical method ofFig.2.Iftwo points Ql,Q2describe concentric circles with theangularvelo- cities TCI,n2 ,theirprojections onafixed diameter willexecutesimple-harmonicvibrations oftheforms<18' xl=&!cos(?2ji+6j),#2=a2cos(nzt+e2),......(1) where alsazaretheradii ofthetwo circles, and 6j,e2arethe initial inclinations oftheradii OQi,OQ Ztotheaxisofx.The result ofthesuperpositionis x=#1+aia,........................(2) and itappearsthatthevalue ofasistheprojectionofOR,the diagonaloftheparallelogramdetermined byOQ 1;OQ Z. IfWj=n,thetwocomponentvibrations havethesameperiod, theangle QiOQ 2isconstant, andtheresultant vibration is simple-harmonicofthesameperiod. Bufcifnl}w2areunequal,theangle QiOQ. 2willvarybetween and180,andORwill oscillate between thevalues a^az. InLord Kelvin's "tidal clock," the"hands" 0Q1}OQZrevolve inhalfalunar andhalfasolarday, respectively, andthesides Q-tR,Q2Roftheparallelogramarcformed ofrodsjointedto these andtooneanother. TheprojectionofRthen indicates thetide-heightduetothesuperpositionofthelunarand solar semidiurnal tides. Iftheperiods STT/W!, 27r/n2arevery nearly thoughnot exactly equal,theangle QfiQ^ willvaryverylittle inthecourse ofasinglerevolution ofOQ lorOQ Z,andtheresultant vibration maybedescribed, ingeneral terms, asasimplevibration whose amplitudefluctuates between thelimits!+a2.Theperiod ofafluctuation istheinterval inwhich onearmOQ^gainsfour right angles ontheother, or27r/(ttjn2).Inverting, wesee thatthefrequencyofthefluctuations isthedifference ofthe frequenciesofthetwoconstituent vibrations. Wehave here thereason forthealternation of"spring" and"neap" tides, accordingasthephasesofthelunar and solar semidiurnal tidesagreeorareopposed.Inacoustics wehave theimportant phenomenonof"beats"between twotones ofslightlydifferent pitch.The contrast between themaximum andminimum amplitudesisofcoursegreatest when theamplitudesa1;aaof Fig. 10. theprimaryvibrations areequal. Wethenhave x=acos(n,i+e^)+a2cos(nj+ea) =2acos{(w,-)$+&(<!-<?2)}cos {(n,+7?2)t+$(e,+e8)}.(3) Thismaybedescribed, inthesamegeneral manner asbefore, asasimplevibration whoseperiodis2?r/^(^+n2},andwhose amplitudeoscillates between thelimits and 2a,inthetime 77./1. (??1n2).This isillustratedgraphically,withxasordinate and tasabscissa, inFig. 10,forthecaseof%:?i2=41 :39. 11.Tree Oscillations with Friction. Theconceptionofadynamical systemasperfectlyisolated andfreefromdissipative forces, which wasadapted provisionally in410,isofcourse anideal one.Inpracticetheenergyof free vibrations isgraduallyusedup,orrather converted into other forms, althoughinmost cases ofacoustical interest the processisacomparativelyslow one, inthesense that the fraction oftheenergywhich isdissipatedinthecourse ofa single periodisveryminute. Torepresenttheeffects ofdissipation, whether thisbedue tocauses internal tothesystem,ortothecommunication of energytoasurrounding medium, weintroduce forces ofresist- ancewhich areproportionaltovelocity. The forces inquestion arebyhypothesisfunctions ofthevelocity*', andwhen the motion issmall, the firstpower onlyneed beregarded. Theequationoffreemotion ofaparticleabout apositionof equilibriumthusbecomes M _ifr 7? C\\ Mdt*~KXHdb*..................(' whereRisthecoefficient ofresistance. Ifwewrite k, ..................(2) The solution ofthisequation maybemade todependon that of4(3)bythefollowingartificef.Weput (4) *Weshall seeatalater stage (Chap. VIII)thattheresistance ofamedium may introduce additional forces depending ontheacceleration. These have theeffect ofaslight apparent increase ofinertia, andcontribute nothing to thedissipation. Itisunnecessary totakeexplicit account ofthem atpresent. fAnother method ofsolution isgiven in20. andobtain, onsubstitution, (5) Wehavenowthree cases todistinguish.Ifthefriction be relatively small, morepreciselyifk<2>itwemayput ri*=nz-%kn -, .....................(6) andthesolution of(3)is yAcosn't+jBsin n't, ............... (7) whence~-"*x=e(Acosn't+Bsmn'tf) (8) Changingthearbitrary constants, andputting ........................... (9) wehave so=ae~l'T cos(n't+e)................(10) Thismaybedescribed asamodified simple-harmonicvibration inwhich theamplitude (ae~^T )sinksasymptoticallytoast increases. Thetime Tinwhich theamplitudeisdiminished inthe ratioIjeiscalled the"modulus ofdecay." The relation between xand tisexhibitedgraphicallyinFig. 11, where thedotted linesrepresent portionsoftheexponential curvesac=ae~t'r .Forthesake ofclearness therapidity ofdecayisheretaken tobemuchgreaterthan itwould bein anyordinaryacousticalexample. 2C DYNAMICAL THEORY OFSOUND Wehave seen thatatruesimple-harmonic beregardedastheorthogonal projectionofuniJ acircle.Ananalogous representationofthemo< isobtained ifwereplacethe circle byanequ described with constantangular velocityn'abou thedirection inwhich theradius vector rde formula(10)isinfactequivalenttoc=rcos6, j Eliminatingtwehave =a.e where$(n'r)~l ,a.=ae^ .This isthepolare spiralinquestion.Thecurve inFig.12corres withFig.11. Inmost acousticalapplicationsthefraction 7c averysmallquantity. Inthis case, the dif- ference between nand n'isasmallquantity ofthesecond order, andmayusuallybeig- nored;inother words, theeffect offriction on theperiod,isinsensible. Itmaybenoted that thequantity I/nr,whose squareisneglected,is theratio oftheperiod ^. ,_ Fig.12. 27T/ntothetime 2?rT inwhich theamplitudeisdiminished intherati If 7cbegreaterthan 2?itheform ofthe sol altered, viz.wehave Theparticle comes asymptoticallytorestbutdoesnotosc infactwemay easilyseethat itpassesonce atmost th itszeroposition. Thistypeofmotion isrealized inthe c apendulum swinginginaveryviscousliquid,andin"dead galvanometers andother electrical instruments, but it little interest inacoustics. Ifk=2n,exactly,thesolution of(3)isoftheform -nt < astowhich similar remarks maybemade. 12.General Dissipative System withOneDegi Freedom. Effect ofPeriodic Disturbing Forces. The effect ofdissipationonthefreemotion ofanys havingonedegreeoffreedom isallowed forbytheassur thatthere isalossofmechanical energyataratepropoi tothesquareofthegeneralized velocity,sothab inthenc of7 d whence aq+bq-\-cq=.................. This isofcourse thesame asifwehadintroduced afric forceQ=-bqin9(3). Theequation (1)hasthesame form as11(3),ai results willcorrespondifweput When thedissipationissmall, therate ofdecay< amplitudecanbeestimated byanindependent method,i Stokes*, which weshall often find useful. Theperiod practicallyunaffected'by vicosity,aconsiderable nuin oscillations canbefairly represented by qGcos(nt+e),..................., provided Cand ebegradually changedsoastolitthea circumstances. Theaverage energyoversuchanintcrv be^n2a(72 ,approximately, by7(li) ;andtherate ofd tion willbe bq*=%n*bCn- {1-cos2(nt4-e)}, *SirGeorge Gabriel Stokes (18191903), Lucasian Professor ofMatl atCambridge (18191903). Wehave seen that atruesimple-harmonicvibration may beregardedastheorthogonal projectionofuniform motion in acircle. An.analogous representationofthemodified type (10) isobtained ifwereplacethe circle byanequiangular spiral described with constantangular velocityn'about thepole 0,in thedirection inwhich theradius vector rdecreases*. The formula (10)isinfactequivalenttoSBrcos6,provided Eliminatingtwehave r=e-^ (12) where$=(wV)"1 ,a=ae^e .This isthepolar equationofthe spiralinquestion.Thecurve inFig.12correspondsinscale withFig.11. Inmost acousticalapplicationsthefractionk/2n,orIJnr,is averysmallquantity. Inthis case, the dif- ference between nand n'isasmallquantity ofthesecond order, andmayusually beig- nored;inother words, theeffect offriction on theperiodisinsensible. Itmaybenoted that thequantity 1/nr,whose squareisneglected,is theratio oftheperiod j,.12 2-Tr/wtothetime ZTTT inwhich theamplitudeisdiminished intheratio e*"orjfa. IfIcbegreaterthan 2ntheform ofthesolution of(3)is altered, viz.wehave whence ifas= . .(14) .(15) *Thistheorem wasgivenin1867byP.G.Tait(18311901).Professor of Natural PhilosophyatEdinburgh (18GO 1901). Theparticle comes asymptoticallytorestbutdoesnotoscillate; infactwemay easilyseethat itpasses once atmostthrough itszeroposition. Thistypeofmotion isrealized inthecase of apendulum swinginginaveryviscousliquid,andin"dead-beat" galvanometers andother electrical instruments, bub itisof little interest inacoustics. Ifk=2n,exactly,thesolution of(3)isoftheform astowhich similar remarks maybemade. 12.General Dissipative System withOneDegree of Freedom. Effect ofPeriodic Disturbing Forces. The effect ofdissipationonthefreemotion ofanysystem havingonedegreeoffreedom isallowed forbytheassumption thatthere isalossofmechanicalenergyatarateproportional tothesquareofthegeneralized velocity,sothat inthenotation of7 ft bq2 , (1) whence aq+bq+cq=(2) This isofcourse thesame asifwehadintroduced africtional forceQ=bqin9(3). Theequation (1)hasthesame form as11(3),andthe results willcorrespondifwepub ?ia=c/a,r=2a/b (3) When thedissipationissmall, therate ofdecayofthe amplitudecanbeestimated byanindependent method, due to Stokes*, which weshall often find useful. Theperiod being practicallyunaffected'by vicosity, aconsiderable number of oscillations canbefairly represented by q=Qcos(nt+e), (4) provided Gand ebegradually changedsoastolitthealtering circumstances. Theaverage energyoversuchaninterval will be^n*aCz ,approximately, by7(11); andtherate ofdissipa- tion willbe bq*=%n*bC2 {1-cos2(nt+e)}, *SirGeorge Gabriel Stokes(1819 1903), Lucasian Professor ofMathematics atCambridge (18191903). 28 DYNAMICAL THEORY OFSOUND themean value ofwhich is^??26(72 .Equatingthemean rate of decayoftheenergytothemeandissipation, weget -%tfbC\..................(5) Cit whence-Jr+l~=Q>.....................(6>dt 2a or C=Ctie-tlT )........................(7) ifr=2a/6,asin(3). When there aregivenextraneous forces inaddition tothe dissipative influences, theequationofenergytakes theform ^aqz+W)=-W+Q<i>............... (8) whencenq+bq+cqQ......................(9) Thisequation givesatoncetheforcenecessarytomaintain aprescribed simple-harmonic vibration, say q=Acospt......................(10) Thus Q=A{(c-p*a)cospt pbsinpt]......... -(H) The firstpartofthisexpressionhasthesame form asin 9, anddepends onlyontherelation between theinertia andthe elasticityofthesystem. Thesecondpartisrequiredtocompen- satethedissipation.Ifweput cp-a=jRcosa, pl=Rsina............(12) (11)becomes Q=ARcos(pt +a)................... (13) Thesolution forthecase ofaprescribed force Q**Ccoapt........................(14) follows bywriting C/RforAand ta.jpfor t.The forced vibration dueto(14)isaccordingly* G q=jicos(pt-a)..................(15) Thevalues ofRandaaredetermined by JRa=(c-_pa a)s+p263 ,tan=p ,....... (16) ^* /r>cp-a^ ' Ristobetakenpositively,andamaybeassumed tolie between and IT. *Another wayofobtainingthissolution isexplainedin 20. Theequation (9)isstill satisfied ifweadd to(15)terms representingafree oscillation; and these added terms are necessaryinorder toconstitute acompletesolutioncapableof adjustmenttoarbitraryinitial conditions. The freevibration dies out,however, asymptotically,sothat after thelapseofa sufficient timetheforced vibration (15)isalone -sensible. Thecircumstances which affect theamplitude andphaseof thisforced vibrationrequirecareful attention. Theamplitude isamaximum whenR2isleast, i.e.when andthemaximum amplitudeisaccordingly n r7,2\"46f- b^ .(18)nb\ 4iaoJ Inmost cases ofinterest 62 /aft *sasmallquantityofthe second order; themaximum isthenCfnb, andoccurs when p=n,veryapproximately. Again,itappearsfrom(15)and(16)thatthephaseofqlags behind that ofthedisturbingforcebyanangle a,which lies between and^TT,orbetween ^TTandTT,accordingaspzisless orgreaterthanc/a,i.e.accordingastheimposed frequency islessorgreaterthan thenaturalfrequency. If,keeping p constant, wediminish thedissipation-coefficient 6,atends to thelimit orTT,respectively,inaccordance with 8,where we found exact agreementoroppositionof-phasein.theabsence ofresistance. Butif,keepingbconstant, wemakepapproach thevalue n(=\/(c/a))which determines thefrequencyinthe absence ofdissipation,atends tothelimit^TT,andthephases ofqandQdifferbyanamountcorrespondingtoaquarter-period. Thismeans thatthemaxima ofthedisturbingforce arenow synchronouswith themaxima ofthevelocity </. Somelightisthrown onthese relations ifweexamine thecase ofapendulum whose bob receivesequal positive andnegativeinstantaneous impulses alternatelyatregular intervals. Itisseen atonce fromFig.13thatanimpulsein thedirection ofmotion accelerates orretards thephaseofan otherwise free vibration, accordingas itprecedesorfollows velocity. Thus ifwhen theparticleisatP,onitswayto0, thevelocitybeincreased intheratio ofPQtoPQ 1;thephase isaccelerated bytheangleQOQ 1}whilst asimilar impulseatP' would retard thephase bytheangleQOQ\. Inorder thatnoeffectmaybeproducedonthephaseit isnecessarythat theimpulsebedelivered attheinstant of passing through0.Ifweimaginethat asmallassisting impulseisgivenateverysuchpassage,asinthecase ofthe ordinaryclockescapement, wehave anillustration ofthe circumstances ofmaxi- mum resonance. The periodofthedisturbing force isexactly equalto thenaturalperiod,and the force synchronizes withthevelocity. The amplitudeisdeter- mined bytheconsidera- tionthattheworkdone bytheimpulsesmust balance that lostby friction. The result is notessentiallydifferent iftheimpulsebedif-Fig.13. fused symmetricallyabout 0,asinthe case ofasimple- harmonic force, since theacceleration ofphase ononeside of iscancelled bytheretardation ontheother. Next supposethat theassisting impulsesaregiven eachtime thebobpassesthesymmetricallysituatedpoints P,P'inwards. There isanacceleration ofphaseateach impulse,andtheperiodisshortened. This illustrates thecase ofadisturbingforce whoseperiodislessthan thenatural period,andwhose maxima andminimaprecede themaxima and minima ofthevelocity.Ifontheother hand theimpulsesare givenasthebobpassesthepointsPandP'outwards, there is arepeatedretardation ofphase,andtheperiodislengthened. Thiscorrespondstothecase ofadisturbingforcewhosoperiod THEORY OPVIBRATIONS 81 isgreater than thenaturalperiod;themaxima andminima oftheforcenowfollowthose ofthevelocity. The reader is recommended tofollow outindetail theargumenthere sketched, andtoexamine theeffect ofsubstitutingacontinuoussimple- harmonic force fortheseries ofdisconnectedimpulses. An explanation mayalsobefound, onthesameprinciples,ofthe factthatasmall frictional force varyingasthevelocityhasno sensible effect onthefreeperiod. Wereturn totheanalyticaldiscussion. Adifference of phase between theforce andthedisplacementisessential in order thatthedisturbingforcemaysupply energytocompensate thatwhich iscontinually beinglostbydissipation. When, as in 9,there iscomplete agreement (oropposition)ofphase betweenqand Q,theforce is,inastronomicalphrase,"in quadrature"withthevelocity q,that is,thephasesdifferby\tr, andthetotalworkdone inacomplete periodiszero. Under thepresentcircumstances thedisturbingforce isatanyinstant doing work attherate Qq=1-Q-sin(pt a.)cospt ==^2 {sina_sin(2^-a)}, (19) themean value ofwhich is Thesame expressionisofcourse obtained asthemean value ofbq*,since theenergy supplied bythedisturbingforcemust exactly compensate,ontheaverage,thatwhich iscontinually beinglostbydissipation,themean energystored inthesystem beingconstant. Itfollows from (16)and(20)thatthedissipationisgreatest when a=ITT,orp=n,i.e.when theimposed frequencycoincides with that ofthefreevibration intheabsence ofresistance. Themaximum value is%C2 Jb,being greater,ofcourse, the smaller thevalue ofb. Theabnormalamplitude anddissipation which ensue whenever theimposed periodisequal,ornearly equal,tothe naturalperiodconstitute thephenomenonof"resonance," alreadyreferred toin8,ofwhich weshall havemany acousticalexamplesinthesequel.Itmaybeillustrated mechanically bygivingaslightto-and-fro motion ofsuitable periodtothepointofsuspensionofasimple pendulum,or betterbymeans ofadouble pendulum (14),i.e.anarrange- ment inwhich twoweightsareattached atdifferentpointsto astring hanging verticallyfrom afixedpoint.Iftheupper- weight (M)beconsiderable, whilst thelower one(m)isrelatively small,Mwillswing almost exactlylikethebobofasimple pendulum,the reaction ofmbeing slight. Under these conditions themotion ofmispracticallythat ofapendulum whosepointofsuspensionhasanimposed simple-harmonic vibration(8),and ifthelengthofthelowerportionofthe stringbeproperly adjusted, aviolent motion ofmmayensue. Oneveryimportant pointremains tobementioned. Asthe interval p/nbetween theforced andthenaturalfrequencies divergesfromunity (oneitherside), thedissipationfalls off from, itsmaximum themorerapidly,thesmaller thevalue of the frictional coefficient 6.Inother words, thegreaterthe intensityoftheresonance inthecase ofexact coincidence of frequencies,thenarrower therangeoverwhich itisapproxi- mately equaltothemaximum. Forexample,atuning fork,even whenmounted ona"resonance box," requires veryperfect tuning inorder that itmaybeexcitedperceptibly bythevibrations of another forkintheneighbourhood, whereas thecolumn ofair inanearlyclosed vessel(e.g.abottle oranorgan pipe)will respond vigorouslytoamuch wider rangeoffrequencies. To elucidate thepoint, wenotice thattheexpression (20)of12 forthedissipation maybewritten where ft=%nblc= LJnr,,(2) THEORY OFVIBBATIONS 33 inthenotation of12(3).Thesecond factor hasitsmaximum value 1/pwhenp=n,andevidentlydiminishes morerapidly, aspfndeviates fromunity, thesmaller thevalue of/3.The question maybeconvenientlyillustratedgraphically bycon- structingacurve which shallshew thedissipation corresponding todifferentfrequencies.Asregardstheabscissa, itwould in strictness bemost propertotake, nottheratiop/n,but its logarithm,sinceequalintervals (inthemusicalsense) then correspondtoequal lengthsofthe axis of as.Wemight therefore write .(3) butwhen, asusually happens,thesensible resonance isconfined toasmallrangeofp/n,wemayusethesimpler formulae /v\ .-..-f__ _..._ w(*) Thecurverepresented bythelatterequationissymmetrical about theaxisofy,andapproachestheaxisofxasymptotically as ?increases. Itisevident that if@boincreased inany -pop Fig. 14. ratio, thenewcurve isobtained byincreasingalltheabscissae in that ratio,anddiminishingtheordinates intheinverse ratio, thearea (TT)included between thecurve andtheaxisofxbeing 34 DYNAMICAL THEORY OFSOUND unaltered. Theintensitysinks toone-half itsmaximum when a?=ft*,or -1+-!(5)n~~m Thus ifthedampingbesuch that afreevibration would have itsamplitude diminished intheratio1/ein10,100,1000 periods*, respectively,thecorrespondingvalues oftheinterval pjnatwhich thedissipationwould bereduced toone-half the maximum would be1'016, 1'0016, 1'00016. Thecurve in(4)isshewn inFig.14. Theaboveargumentdeals withthedissipation,which isthe mostimportantfeature. Theconsideration ofthesquareofthe amplitude,oroftheenergystored inthesystem,leads tovery similar results, especially when thedampingisslight. 14.Systems ofMultiple Freedom. Examples. The Double Pendulum. Weapproachtheconsideration ofsystems having anyfinite number ofdegreesoffreedom. Asystemissaid tohavem suchdegrees whenmindependent variables, or"coordinates," arerequired andaresufficient tospecifythevariousconfigura- tions which itcanassume. The notion, firstbroughtinto formalprominence byLord Kelvinf,hasawideapplication inmechanism and intheoretical mechanics. Inthecase of thetelescopeofanaltazimuth instrument orofanequatorial wehavem=2;inthegyroscope,or(more generally)inany case ofarigid bodyfree toturnabout afixedpoint,m=3; forarigid structure orframe movable intwodimensions m=3;forarigid structurefreely movable inspacem=G. The choice ofthecoordinates inanyparticularcasecanbe made inanendlessvarietyofways, butthenumber isalways determinate. Thus intechnical mechanics wehave thepro- positionthatarigidframe movable inoneplanecanbefixedby *Inanexperiment byLordRayleigh, thenumber ofperiods foraparticular tuning fork of256v.s.wasabout 5900.When aresonator wasusedthenumber fellto3300.Theory ofSound, vol.n.,p.436. tWilliam Thomson, afterwards Lord Kelvin (18241907), Professor of NaturalPhilosophyatG-lasgow 184699. Thematter isexplained inThomson andTail's NaturalPhilosophy, 2nd ed.,195201(1879). IJIOClllO UlU111CC 1J.JU1YO (JUJ.1.U three fixedpointsintheplane*. Similarly anyrigidthree- dimensional structure canbeanchored firmly bysixlinks connectingsixpointsofitwith sixpointsfixedrelatively totheearth. Proceedingtothevibrations ofamultiple systemabout aconfigurationofequilibrium,webeginasbefore with the examination ofafewparticularcases. Take firsttheoscillations ofaparticleinasmooth bowl of anycontinuousshape. Bymeans ofsuitable constraints, the particle mayberestricted tooscillate inanygivenvertical plane throughthelowestpoint 0,e.g.byconfiningitbetween two frictionlessguides infinitelyclose tooneanother. In generalthere willbealateralpressure ononeorother ofthese guides,which willhowever vanish iftheplaneinquestion passes througheither oftheprincipaldirections ofcurvature at0.Hence twomodes offreesimple-harmonic vibration, in perpendicular directions, arepossible,withspeeds whereHl}_R2,aretheradii ofcurvature oftheprincipalsections at0.Onaccount oftheassumed smallness ofthemotion, these vibrations maybesuperposed. The result is,ifas,ybe horizontalrectangularcoordinatesthrough 0, XA!cosn-f,+A2sinn^t,} y=B}cosn%t4-B.2sinn Since thiscontains fourarbitrary constants, wecanadjust thesolution togiveninitial values ofx,y,x,y. This case isvery neatlyillustratedbyBlackburn'spen- dulum f(Fig. 15).Aweight hangs byastringCPfromapoint CofastringAGBwhose endsA,Barefixed. Thestrings being supposeddestitute ofinertia, thepointPwillalwaysbeinthe sameplane withA,B,C.Under thiscondition thelocus of Pisthering-shapedsurfacegenerated byrevolvingacircle *Provided thedirections ofthethree links benotconcurrent (orparallel). There isaproviso ofamorecomplex character inthecasewhich follows;but such details need notoccupy ushere. fH.Blackburn, Proi'essor ofMathematics atGlasgow 1849 79. 3t> DYNAMICAL THEORY OFSOUND with centreGandradius CP,intheplaneAGE, aboutABas anaxis;andtheprincipalradii ofcurvature atthelowestpoint arejSj.=CO,JRZ=EO,where .Z?isthepointofABvertically above 0.Thecorrespondingdirections ofvibration arere- spectivelyinandperpendiculartotheplaneABO, 21, Fig. 16. PO Fig. 15. Fig. 17. Anotherverysimplecase isthat oftwoequal particlesM attachedsymmetricallyatdistances afrom theends ofatense string, whose totallength is,say,2(a+6),sothat 26denotes thelengthofthe centralportion.One obvious mode of simple-harmonic vibration isthatinwhich thedeflections ofthe twoparticles arealways equal andofthesamesign (Fig. 16). IfPbethetension ofthestring,theequationofmotion of eitherparticleisthen .(3) andthespeedistherefore (4) Inanother mode thetwo deflections areequalinmagnitude andoppositeinsign,sothatthemiddlepointofthestringis stationary (Fig. 17).Thecircumstances arethereforeexactly thesame asin 6,andthespeedis ' greater,asweshouldexpect, than n:.Ifwedenote the VLbJBATIONS 37 . .deflections ofthetwoparticles byx,y,thesuperpositionof thetwomodesgives x=Acos(nj+O+Bcos(nzt+e2),| y=Acos(n^t+e^)Bcos(n$t4-e2)J where thefourconstants A,B,e15e2arearbitrary. Inthecase ofthree attachedparticlesthenature ofthe various modes isnotsoimmediately obvious, even inthecase ofsymmetry. We willsupposethat themasses areequal, andthattheydivide the line into fourequal segmentsa. Denotingthedeflections byx,y,z,wehave J.i"""""" J~~~-*"""~"-*" * a -P--Paa dt2a a' , Ifweput,forshortness, /t=P/Ma,thesemaybewritten .(8) Toascertain theexistence ofmodes ofvibration inwhich themotion ofeachparticleissimple-harmonic,with thesame periodandphase, weassume, tentatively, x=Acos(tit+ ),y=Bcos(nt+e),z=Ccos(nt-fe).(9) Itappears,onaubstibution in(8),that theequationswillbe satisfiedprovided (n*-2yu)A+p,B=0; p.A+(n2-2/A)B+fj.C=0,-............(10) These three equationsdetermine thetwo ratiosA :7? :C andthevalue ofw2 .Eliminatingtheformer ratios wehave .............(11) 38 DYNAMICAL THEORY OFSOUND This isacubicinn2 .One root is^=2^,andwefindon reference to(10)that thismakes^-0,A^-C,,andthere- fore #=Acos(77^+0,2/=0,*=- 4:cos(*!+ ,)...-(12) Thismode mighthave been foreseen, and itsfrequency determined atonce, asinthepreceding example.The remainingroots of(11)are and itappearsfrom (10)thatthesemake A2=Z,Bz=-*/2Az,andA3=Cs,BS respectively.Thecorrespondingmodes aretherefore as=Aycos(njt+e.,)>y~-^Azcos(n2t+ea), z=A2cos(?izt+e2),...(13) and a:A3cos(nAt4-e3),y=V2-43cos(nst+e3), ^=^48cos(w3i4-e3)....(14) These areshewn, alongwith theformer mode, inFig.18. Thecompletesolution oftheequationsisobtained bysuper- positionof(12), (13)and(14),andcontains thesixarbitrary constants A-L,Az,A3,e1}eaes- Fig.18. Weconclude these illustrations withthecaseofthedouble pendulum,where weareentirely dependentongeneral method. AmassMhangsfrom afixedpoint byastringoflength, andasecond massmhangsfromMbyastringoflengthb. Forsimplicity wesupposethemotion confined toonevcrticul THEOKY OFVIBRATIONS 39 plane. The horizontal excursionsto,yofM,mrespectively being supposed small, thetensions oftheupper andlower stringswillbe(M4-m)gandmg,approximately. Theequa- tions ofmotion aretherefore Tofindthepossible modes ofsimple-harmonicvibration we assume x=Acos(nt4e),y=Bcos(nt+ e) (1C) Theequationsaresatisfiedprovided (i- ...(17) wherefjt=m/(M+m) (18) Eliminating theratioA :B,wefind 2_l_#!=0, (19)ab which isaquadraticinn\Thecondition forreal roots, viz. .(20) isobviously alwaysfulfilled. Itisfurther easilyseen thatboth roots arepositive,so thatnalso isreal. Theproblem includes anumber ofinter- esting special cases, butwewillonlynotice one ortwo. Iftheratio//,,=m/(M+m), besmall, thetworoots of(19)aren^=ff/a, v.?=g/b, approximately.Intheformer caseMoscillates likethebob ofasimple pendulumoflength a,whilstmexecutes whatmayberegardedasaforced oscillation 40 DYNAMICAL THEOBYOFSOUND ofthecorresponding frequency;this case hasalready been referred toin 13.Inthesecond mode theratioA :Bissmall, asappearsfrom thesecond ofequations (17);Misthennearly atrest, whilstmoscillates likethebob ofapendulumof length6. Since theexpressionontheleft-hand side of(20)cannot vanish, thetwofrequenciescanneverexactly coincide, butthey becomeapproximately equalifa=b,nearly,andpissmall. Acurious phenomenon maythenpresentitself. Themotion ofeachmass, beingmadeupoftwosuperposed simple-harmonic vibrations ofnearly equal period, mayfluctuategreatlyin extent, and iftheamplitudesofthetwovibrations areequal wehaveperiodsofapproximate rest, asexplainedin10.The motion thenappearstobetransferredalternatelyfrommtoM, andfromMtom,atregularintervals*. If,ontheother hand,Missmallcompared withm,&isnearly equaltounity, andthetworoots of(19)arenz=g/(a+b)and w2=mffjM.(a+b)lab, approximately. Theformer rootmakes B/A=(a+fy/a, nearly,sothat thetwomasses arealways nearlyinalinewith thepointofsuspension, mnowoscillating likethebob ofapendulumoflength a+b.Inthesecond mode theratioB/Aissmall, sothatmisapproximatelyat rest;themotion ofMisthen likethat ofaparticleattached toastring which isstretched between fixedpointswith a tension mg (cf. 6). Another case ofinterest isobtained ifwemakeainfinite. Onerootof(19)then vanishes, andtheother is - ff-~ which makesAjB~m/M.This indicates that ifthesupport ofasimple pendulum yield horizontally, butwithoutelasticity, thefrequencyisincreased inacertain ratio which isofcourse *Theinfluence ofdissipationisofcourse here neglected. Ifmbesubject toafrictional resistance, andespeciallyifthemodulus ofdecaybelessthan theperiodofthefluctuation given bytheabove theory, thephenomena are modified, andtheillustration ofthetheoryofresonance(12)isimproved. There isnowacontinual, though possiblyaalow, drain ontheoriginal energy oi'JI. smaller thegreatertheinertia ofthesupport.This ishowever moreeasilyseendirectly. 15.General Equations ofaMultiple System. Thegeneral theoryofthesmall oscillations ofamultiple system canonlybegivenhe.re inoutline. Inthecaseofonedegree offreedom(7)itwaspossibletobasethetheoryontheequation ofenergy alone, butwhenwehavemore than onedependent variable this isnolonger sufficient, andsome furtherappeal must bemade toDynamics.Forbrevityofstatement wewill suppose that there aretwodegreesoffreedom, butthere is nothingintheargumentwhich cannot atoncebeextended to thegeneralcase. Weimagine, then, asystemsuch thatevery configuration which weneed consider canbespecified bymeans oftwo independent geometricvariables or"coordinates"qltq^Ifin anyconfiguration (q1}qa)thecoordinateql(alone)receive an infinitesimal variationqj}any particlemofthesystemwill undergoadisplacement8sL=o^S^inacertain direction. Similarlyifqzalone bevariedmwillbedisplaced througha spaceSs2=a2Sg2inacertain direction, different ingeneralfrom theformer. TheresultantdisplacementSswhen both variations aremade isgiven by s2=Ssj2+2SsaSs2cos6+Ss22 =tfSqf+2a1a2cos6SqlSq2+of%2 , (1) where 6denotes theangle between thedirections ofBslt8s.,. Ifwedivide by Stf,weobtain thesquareofthevelocityv oftheparticle m,inanymotion ofthesystem throughthecon- figuration (q},q2}}interms ofthegeneralized "componentsof velocity" q1}qz>thus v*=afq?+21a2cosOqfo+ff88 g3" (2) The total kinetic energyofthesystemisthereforegiven by 2T=*(my2 )=an^2+2aIBg,&4-a^2 , (:J) where On=S(iny-f), a12=S(ma^cos9\a22=2(ma./},...(4) thesummation 2extendingover alltheparticles mofthe 42 DYNAMICAL THEORY OFSOUND system.The coefficients au,a12 ,a22areingeneralfunctions of qltq2;theyarecalled the"coefficients ofinertia" forthepar- ticular configurationconsidered. Next, letFldenote thetotal force actingonm,resolved inthedirection ofSsj,andletFthave thecorresponding meaning forthedirection ofSs2.Theworkdoneonthesysteminany infinitesimal displacementwilltherefore be 2(F.Bs,+F2SsJ=2(F&) Bq,+2<T22)Bq2.......(5) Ifthere arenoextraneous forces, thiswork isaccounted for byadiminution inthepotential energyVofthesystem. When extraneous forces actwehave inaddition theworkduetothese, which wemaysuppose expressedintheform The coefficients Qj,Q2arecalled, byanobviousanalogy,the generalized "componentsof(extraneous)force." Hence whence Intheapplicationtosmall oscillations weassume thatqltq* aresmallquantities vanishingintheconfigurationofequi- librium, and forconsistency wemust alsosupposethat the disturbingforces Q1}Qzaresmall. Thequantities al5 2and therefore alsoan,a12,a22maynowbetreated asconstants. Thevelocityoftheparticlemismadeupofcomponents a^, ctzqzinthedirections Ssiand^2,respectively; and ifweneglect thesquaresofsmallquantitiesitsacceleration ismadeupin likemanner ofcomponentsaag1;a2g2*.Henceresolvinginthe direction of8sLtheforcesactingoninwehave m(;$!+ 2^2cos6)=Fi,} andsimilarly m(a^cos 6+a^)=Fz.)............ *Theformer ofthese twoquantitiesis(tothefirstorder) theacceleration calculated onthesupposition thatq^alone varies, andthelatter istheaccelera- tionwhenq-2alone varies. Itisonlyonthehypothesis ofinfinitely small motions thattheresultant acceleration isobtained bysuperpositionofthese. THEOEY OPVIBEATIONS 43 Ifwemultiplytheformer oftheseequations by or,andthe secondby 2,andsum for alltheparticles ofthesystem, we find,with thenotation of(4), 3F ,,..,,andsimilarly a21+ where a21isofcourse identical witha,,. When there arenoextraneous forces theseequations areby hypothesis satisfiedby&= <),&=0.Theconfigurationof equilibrium istherefore characterizedbythepropertythat inotherwords, thepotential energyisstationaryforallinfini- tesimaldisplacements therefrom. Hence ifVbeexpanded in powers ofq1}qz,theterms ofthe firstorder willbeabsent, andwemaywrite with sufficientapproximation 2V=c11q1t4-2^^+c.,^2 ,............(11) aconstant termbeing omitted. Thequantitiescn,c12,c^are called the"coefficients ofstability." Hence(9)maybewritten i-fa,2q,+cnqi+c12q,=Q, ,-v ^' where c21=c12. Ifwelookback toanyofthespecial problems of14we shallrecognize thattheequations ofmotion areinfeetofthis type. Forexample,inthecase ofthedouble-pendulum we have Theformulae thereforecorrespondifweput ^=x, q,=y,. an=M, a,a=0,a22=m,(...(14) c,,=(M-I-m)ff/a+mgjb,cja=-mg/b,cS2 44 DYNAMICAL THEORY OFSOUND Thegeneralcase ofmdegreesoffreedom hardlydiffers exceptinthelengthoftheformulae. Wehavethenmequations ofthetype a?i+a,^[ a+...+asmqm+c&l^+cs2g2+...+c8Ulqm=Q*,(15) where sisanyoneoftheintegers 1,2,3,...m. 16.Free Periods ofaMultiple System. Stationary Property. Inthecase offreevibrations wehaveQl0.Q2=0,and thesolution of15(12)then followsexactlythesame course asintheparticular examples already given. Weassume ql=A-icos(nt+e), q^=A&cos(nt-\- e), (1) andobtain (cnn2an)AI4-(c12n2aia)A2=0,] .(2) (c22w2a22)A2O.j EliminatingtheratioA^:A,,,weobtain =0, (3) where(itistobenoticed) thedeterminant isofthe"sym- metrical"type.Thisequation givesthetwoadmissible values ofnz .Adoptingeither oftheseweobtain asolution inwhich theratio ofA^toA2isdetermined byeither oftheequations (2).Themode ofvibration thusascertained involves therefore twoarbitrary constants, viz.theabsolute value of(say)Alsand the initial phasee.Thesecond root of(3)leads toanother solution oflikecharacter. Theextension ofthemethod tothegeneralcase isobvious, but itmaybewell tostate theresultsformally.Inany conservativesystemofmdegreesoffreedom there arein generalmdistinct "normal modes" offree vibration about aconfigurationofstableequilibrium,thefrequenciesofwhich aregiven byasymmetrical determinantalequationofthemth order in /i.2 ,analogousto(3),andsodepend solelyonthecon- stitution ofthesystem.Ineach ofthese modes thesystem oscillatesexactlyasifithadonlyonedegreeoffreedom, the coordinates q1}q,...qmbeinginconstant ratios tooneanother, andthedescriptionof 7thereforeapplies. The directions of motion ofthevariousparticlesandtherelativeamplitudesare inanyonemode determinate, though usuallydifferent for different modes, theonly arbitrary elementsbeingtheabsolute amplitude andthephase-constant. Theequationsofmotionbeing necessarily linear, since products andsquaresofthecoordinates andtheir differential coefficients withrespecttothetime areexpressly excluded, it follows that the different solutions maybesuperposed by addition ofthecorresponding expressions. This hasbeen sufficiently illustrated inthepreceding examples. Bysuper- posinginthiswaythemnormal modes, each wifch itsarbitrary amplitude andphase, weobtain asolutioninvolving 2m arbitrary constants, which isexactly theright number to enable ustorepresentthe effect ofarbitraryinitial values of thecoordinatesql}qz,...qmand velocitiesqltq2,...qm.In other words, themostgeneralfreemotion ofthesystem about aconfigurationofstableequilibrium mayberegardedasmade upoftheTOnormal modes with suitableamplitudes and initial phases.Thisprincipledates fromD.Bernoulli* (1741). Inparticularcases itmayhappenthattwo(ormore)ofthe naturalperiodsofthesystemcoincide. There isthen acorre- sponding degreeofindeterminateness inthecharacter ofthe normal modes. Thesimplest exampleisfurnished bythe spherical pendulum,orbyaparticle oscillatinginasmooth sphericalbowl. Thenormal modes maythen betaken to correspondtoanytwohorizontal directionsthroughtheposition ofequilibrium. From thetheoreticalstandpointsuch coinci- dences mayberegardedasaccidental, since theyaredestroyed bytheslightestalteration intheconstitution ofthesystem (e.g.ifthebowl intheabove illustration beintheslightest degree ellipsoidal),butinpractice theyoften lead tointeresting results. Of.53below. Animportantcharacteristic ofthenormal modes, first pointedoutbyLordKayleighin1883, has stilltobereferred *Daniel Bernoulli (17001782), one oftheyounger members ofthe distinguished family ofSwiss mathematicians. Professor ofmathematics at StPetersburg (172533),andofphysicsatBale(175082). Hischiefwork wasonhydrodynamics, onthetheoryofvibrating strings, andontheflexure ofelastic beams. 46 DYNAMICAL THEOBY OFSOUND to. If,bytheintroduction offrictionless constraints which do nowork, thesystem berestricted tovibrate inamode only slightlydifferent from oneofthese, theperiodwillbealtered onlybyasmallquantityofthesecond order. Inother words theperiodsoftheseveral normal modes are" stationary"when comparedwith those ofslightlydifferent constrained modes. Suppose,forinstance, thatthenormal mode inquestionissuch that initthecoordinateq1alone varies. Wehave, then, in(2), aI2=0,c,2=0,and thenaturalfrequencyisdetermined by n8=cn/an.Iftheconstraint beexpressed byq2=\q1}the condition that theconstrainingforces shall donowork, viz, Qi&+&2 0,orQ1+XQ2=0,leads to (an+X2a22)q,+(GU.+Vfcja) qi=0,............(4) andthespeed (p)isaccordingly givenby When TV,issmall, this differs from n2byasmallquantityof thesecond order. Theproof, althoughlimited totwodegrees, iseasily generalized. Owingtoourlibertyofchoice ofthe coordinates, wecanalways arrangethatqlshall betheonly coordinate which varies inthemode inquestion, andthat theconstraint shallbeexpressed byasystemofrelations of thetypeq^\q,, qs-^qi, q*=vqi ..... Foranobvious illustration wemay have recourseagain totheparticleonasmooth surface. Iftheconstrainedpath beavertical sectionthroughthelowestpoint,theperiodis 27iV(-R/Sr),whereRistheradius ofcurvature ofthesection, and itisknown thatRisamaximum orminimum fortheprincipal sections. Theequation (5)shews further thattheconstrainedperiodis (asintheparticular case) intermediate between thetwonatural periods;thispropertycanalsobegeneralized. Itfollows thatevenwhen itisnoteasytoascertain the precisecharacter ofaparticularnormal mode, acloseapproxi- mation tothefrequencycanoftenbeobtained ontheassumption ofanassumedtypewhichwecanjudgeonindependent grounds tobeafairly good representation,ofthetrue one.And inthe M Jtr JL o (14),andconsider anassumedtypeofsymmetricalvibration inwhichxz-=\y. Thekineticenergyisthengiven by 2T=N(d?+if+-z2 )=M(l+2\2 )f, (6) sothattheinertia-coefficient isM(1+2X2 ).Forthepotential energy wehave T1 T1 2 ,a)2V=- {tf2+(y-xf+(z-y)2+z*\=-(4A2-4^ C& 6t asisfoundbycalculation oftheworkrequiredtostretch the string (asin22) ;orotherwise. The coefficient ofstabilityis thereforeP/a.(4X.24\+2).Forthespeed (p)wethenhave (8) This isstationaryforX=^2,andthecorresponding speeds areasin 14.Inthiscase itwasevident beforehand that the assumedtypewould include thetrue natural modes ofsym- metrical character. Itisunnecessaryforthepurposesofthisbook todiscuss in detail thetheoryofdissipationinamultiple system. The generaleffect isthesame asin 12;the free vibrations graduallydieout,but ifthedissipativeforces berelatively small theperiodsarenotsensiblyaffected. 17.Forced Oscillations ofaMultiple System. Prin- ciple ofReciprocity. Thetheoryofforced oscillations issufficientlyillustrated if in15(12)weassume that Q^varies ascospt,whilst()2=0. Theequationswillbesatisfied ifweassume thatqlandqaboth varyascospt, provided (ou-p*a,n)qi+(CM-p"a lz}q2=Qi, ^ These determine the(constant)ratios ofq^andg2toQl;thus AOr)!.Q,, (2) 48 DYNAMICAL THEORY OFSOUND whereA(p*)isthedeterminant ontheleft-hand side of16 (3),withp*written forn?.Thegeneralconclusion isthatwhen aperiodicforce ofsimple-harmonic typeactsonanypartof thesystem, every partwillexecute asimple-harmonicvibration ofthesameperiod,with synchronismofphase,but the amplitudewillofcourse bedifferent indifferentparts. When theperiodoftheforced vibration nearlycoincides with that ofoneofthefreemodes, anabnormal amplitudeofforced vibration will ingeneral result, owingtothesmallness ofthe denominator intheformulae (2). Foracompleteaccount of thismatter weshould have totakedissipativeforces into consideration, asin 12. Aremarkable theorem ofreciprocity,firstprovedbyHelmholtz foraerial vibrations, andafterwardsgreatlyextended byLord Bayleigh,follows from(2).Ifweimagineasecond case of forced vibration(distinguished byaccents)inwhich Qi= whilst Q2'varies ascospt,weshallhave Comparing with(2),weseethat fcrQi-fc'^' (4) Theinterpretationismosteasily expressed when the"forces" QiandQ2'areofthesame character, e.g.bothordinarystatical forces, orbothcouples,inwhich casewemayputQ:=Q/,and obtainqz=ql'.Inwords: The vibration oftype2duetoa given periodicforce oftype1agreesinamplitudeandphase withthevibration oftype1duetoanequalforce oftype2. Anexamplefrom thetheoryofstringswillbefound in 28. Theaboveproofiseasily extended tothegeneralcase ofmdegreesoffreedom. 18.Composition ofSimple-Harmonic Vibrations in Different Directions. Werecur tothesubjectofcompositionofsimple-harmonic vibrations which, thoughnotsoimportantasinOptics, claims a little further attention. Ifinafreely vibrating system wefix ourattention onaparticular particle, thedirections inwhich it oscillates intheseveral normal modes willingeneral bedifferent. Thesuperposition then takesplaceofcourseaccordingtothe lawofgeometrical orvector addition. Itwill suffice toconsider thecaseoftwodegreesoffreedom, where wehaveindependent simple-harmonicvibrations inthe directionscorrespondingtotheSs1}8s2of 15.Theresult isa plane orbit, usuallyofacomplicatedcharacter. Forinstance, inthecaseofBlackburn'spendulum (14),wehave xAcos(n^t+ej),y=Bcos(n.2t+e>), ......(1) wherex,yarerectangularcoordinates. The orbit ishere contained within therectanglebounded bythelines cc=A, y=B.Ifn-i,nzarecommensurable, thevalues ofas,yand x,ywillrecur after thelapseofanintervalequaltothe least commonmultipleofthetwoperiods,andthepathwillbe re-entrant. Theresulting figures,obtained inthisandinother ways, areassociated with thename ofLissajous*, whohashad many followers inaregionwhich isveryattractive from the experimental pointofview. Thesimplestcase isthat ofn^=nz.Ifweeliminate tin (1)wethenobtain Thisrepresentsanellipse which, iftheinitialphasesel3e2coincide, ordiffer by TT,degeneratesintoastraightline(Fig. 20). The simplestmechanical illustration isfurnished bythespherical pendulum. When therelation isthatoftheoctave (^=2n2) wehave acurve withtwoloops, whichmaydegenerateintoone orother oftwoparabolicarcs(Fig. 21).Thecurves inthese and inother cases ofcommensurabilityareeasilytraced from, the formulae(1)with thehelpoftables.Asimple geometrical construction isindicated inFig.22,where thecircumferences of theauxiliarycircles aredivided intosegments correspondingto equalintervals oftime inthetwosimple-harmonicmotions which aretobecompounded.Ifwestart atacorner ofany *J.A.Lissajous (182280).Professor ofphysicsattheLyce'e StLouis 1850 74;rector oftheAcademyofChambe'ry 18745,and ofBesancon 1875 9.His chief memoir, Etude optiqua desnwitvements vibratoires, was publishedin1873. 50 DYNAMICAL THEOEY OFSOUND oneoftherectanglesinthefigure,andproceed diagonally,we passthroughasuccession ofpoints, equidistantintime, ona curve ofthesystem. Fig. 22. Another conceptionofthesefigures,alsodue toLissajous, maybementioned. Ifwewrite 6fornj,andadjusttheorigin oft,theformulae (1)areequivalent,onthehypothesisof commensurability,to x=acos0,73 2;=Z>cosM-a), where pfqisafraction initslowest terms. Theseequations, when combined with .................... .....(4) representacurve ofsines traced onthesurface ofthecircular cylinder andgoing throughitsperiod ptimes inqsuccessive circuits ofthecylinder.The Lissajouscurve (3)istheorthogonal projectionofthiscurve onaplane (^=0)throughtheaxis of 52 DYNAMICAL THEORY OFSOUND thecylinder.This isillustrated byFig. 22,where thedotted branch mayberegardedastheprojectionofthatpartofthe sine-curve which liesontherearhalfofthecurved surface. A changeintherelativephasein(1)isequivalenttoachangein theangle a,andmayberepresented byarotation ofthecylinder about itsaxis, ofcorrespondingamount. This, again,maybe illustrated fromFig.22bystartingthecurve onestepfurther to therightorleft.When theratio oftheperiodsisnearly, but notexactly,that oftwointegers,theorbitgradually passes throughthevariousphasesofthecommensurable case, ina recurring cycle*. Thus inthecase ofapproximate unison, or approximate octave, thecycleincludes thephasesshewn in Fig.21or22,followed bythesame inreverse order. Thesame result isobtained byacontinuous rotation ofLissajous' cylinder. 19. Transition, toContinuous Systems. Thespacewhichwehavedevoted tothestudyofdynamical systemsoffinite freedom isjustified bytheconsideration that weheremeet withprinciples,intheirprimitive andmosteasily apprehended forms, which runthroughthewhole oftheoretical acoustics. Inthesubsequent chapters weshall beconcerned withsystems such asstrings, bars,membranes, columns ofair, where thenumber ofdegreesoffreedom isinfinite. Mathematic- ally,itissometimespossibletopassfromoneofthese classes to theotherbyasortoflimiting process,aswhen D.Bernoulli(1732) discussed thevibrations ofahangingchain asalimitingform oftheproblem where alargenumber ofequal andequidistant particlesareattached toatensestring whose ownmass is neglected. Inany case, there canbenoquestionthat the general principlesreferred toretain theirvalidity. Themain qualificationtobenoticed isthat thenormal modes arenow infinite innumber. Itisusual toconsider them asarranged *InLissajous' method thevibrations which areoptically compounded are those oftwotuning forks. Thefigures obtained when thetones sounded bythe forksformanyoneofthesimpler musical intervals giveabeautiful verification ofthenumerical relations referred toin 3.Inthecase ofunison, when the tuningisnotquite exact, thecycle ofchanges synchronises with thebeats which areheard;see 10. maystillbecalled the"fundamental," and isgenerallythe mostimportant. Beforeleavingthegeneral theoryitmaybedesirable to emphasize oncemore theimportanceofthesimple-harmonic typeofvibration fromthedynamical pointofview.Wehave seenthat itisthecharacteristic typeforafrictionlesssystemofone degreeoffreedom, or(more generally)forasystem oscillating asifitpossessed onlyonedegree,asinthecase ofthenormal modes. Itisalsotheonlytypeofimposedvibration which is accurately reproduced,onalargerorsmaller scale, inevery partofthesystem.Ifaforce ofperfectly arbitrary typeactat anypoint,thevibrations producedinotherpartsofthesystem have asarulenospecialresemblance tothisortooneanother; itisonlyinthecase ofaperiodicforcefollowingthesimple- harmonic lawofvariation with thetime thattheinduced vibrations areexactly similar, andkeep stepwith the force. Moreover itisonlyinsofarasthedisturbingforce issimple- harmonic, orcontainssimple-harmonic constituents, that itis capableofgeneratingaforced vibration ofabnormalamplitude when acriticalfrequencyisapproached.Itisinthese circum- stances thatHelmholtz found theclue tohistheoryofaudition, towhich weshallhave torefer atalaterstage. 20.OntheUseofImaginary Quantities. Thetreatment ofdynamical equations canoftenbegreatly simplified bytheuseofso-called"imaginaries." Asweshall occasionallyhaverecourse tothisprocedure,itmaybeconvenient toexplain brieflytheprinciplesonwhich itrests. Thereader willbefamiliar withthegeometrical representa- tion ofa"complex" quantitya4-ib,wherea,barerealand i stands forV(~1)byavector drawn from theorigintothe pointwhose rectangularcoordinates are(a,b) }andwith the factthat addition ofimaginaries correspondstogeometrical addition (orcomposition)oftherespectivevectors. The symbola+ibwhenappliedasamultiplying operatortoany vector denotes thesameprocess bywhich thevector a+ibmay besupposedtohave been derived from thevector1,viz. it 6*DYNAMICAL THBOBY OFSOUND t/ Hersthelengthinacertainratio r,and turns it througha certain anglea.These quantitiesaredenned by or>>(2) thequadrantinwhich liesbeingdeterminedhythesign attributedtocosa orsinaby(1).Wehave then ' -'"- ")(3) Henceasymboloftheform cos+1sinadenotes theopera- tionofturningavector throughanangleawithout alterationoflength;inpar- ticularthesymbolidenotes tbeoperationofturning througharight angleinthe positive (counter-clockwise) direction. Thesymbol w=cos+isin6(4) mayberepresented byaunit vectorOPdrawn fromi thedirection 0.Ifweregard thisasafunction of0,and if +8*berepresented byO/*,&&P()I* f11*^ to80Thevector PP'which reiirwwntHawwill therefore have alength 80,andmncoitinUml througharight angle relativelytoOP,itoayiubulwillb,iM.w. Hence Itiseasily hewn that,U nlyw.lutiun fthisequatioi, whichfulfils thenucttwuryr.onHtion that IL-1lor00,u i/;w............................ (0) where e*IBtoltak.n an.IHiiuMl byth..nlinary exponential series. Thus..,rt. e:-.('OH0+ #...................(') Wemayaddthat th"additiut^th-utvin"ofihoexponential function cannow l,d.'rivd imuu-diaK-lyfruin thgeoraetal representation. THEOBY OFVIBEATIONS 55 Ithasbeenthought worth while torecapitulatethese ele- mentarymatters becausetheyhaveinterestingillustrations in thepresent subject. Thus ifas,yberectangular coordinates, andwewrite *=4-iy, ...........................(8) theequation z=Ceint ,...........................(9) whereGmayofcourse becomplex, expressesthatthevector C isturned inatime tthrough ananglentinthepositivedirec- tion. Itthereforerepresents uniform motion inacircle, with angular velocity n,inthepositivedirection. Theradius ofthis circle isgiven bythe"absolute value" ofG,which isoften denoted by|G|;thus ifG=A+iB,whereAandBarereal,we have |G \=*J(A2+.B2 ).Inthesamewaytheequation z=C'e-i1*........................(10) representsuniform motion inacircle, with angular velocity n, inthenegative (orclockwise) direction. Wecome now totheapplicationtolinear differential equationswith constant coefficients. From ourpointofview thesimplestcase istheequation of 4.Inorder thatevery stepoftheworkmayadmit of interpretation,weassociate with thistheindependent equation fl+'ty-o,.....................(12> asinthetheoryofthespherical pendulum.Thetwomaybe combined intheoneequation whichmayindeed beregardedasrepresenting directly,without theintermediaryof(11)and(12),thelawofacceleration inthe spherical pendulumand similarproblems.Tosolve (13)we assume z=CeKt ,andwefind that theequationissatisfied providedX2 -I-n-0,orX=+in.Since different solutions can beadded, weobtain theform z=Ceint+G'e~in\ certainanglea.rnesequantitiesareaennea ay rcosa=a,rsina=6, (1) or r=VO2+62 ),a=tan~1 (&/a), (2) thequadrantinwhich aliesbeingdetermined bythesign attributed tocosaorsinaby(1),Wehavethen a+ib=r(cosa+isina) (3) Hence asymboloftheform cosa+isinadenotes theopera- tion ofturningavector throughanangleawithout alteration oflength;inpar- ticular thesymbolidenotes theoperation ofturning througharight angleinthe positive (counter-clockwise) direction. Thesymbol w=cos+isin(4) mayberepresented byaunit vectorOPdrawn from in thedirection 6.Ifweregard p)g.23. thisasafunction of0,and if w+Bwberepresented byOP',theanglePOP' willbeequal toS6.Thevector PP'whichrepresentsSwwilltherefore have alength 80,and since itisturnedthrougharight angle relativelytoOP, itssymbolwillbeiSO.w. Hence dw . de=(5> Itiseasily shewn that theonlysolution ofthisequation which fulfils thenecessarycondition thatw=~L for6=0,is w=eie , (6) where eioistobetaken asdefined bytheordinary exponential series. Thus e*9=cos-Msin(7) Wemayaddthat the"addition-theorem" oftheexponential function cannowbederivedimmediatelyfrom thegeometrical representation. Ithasbeen thoughtworth while torecapitulatethese ele- mentarymatters because theyhaveinterestingillustrations in thepresent subject. Thus ifx,yberectangular coordinates, andwewrite st=*as+iy,........................... (8) theequation z-Geini ,........................... (9) whereGmayofcourse becomplex, expressesthatthevectorC isturned inatime tthroughananglentinthepositivedirec- tion. Itthereforerepresentsuniform motion inacircle, with angular velocity n,inthepositivedirection. Theradius ofthis circle isgiven, bythe"absolute value" ofG,which isoften denoted by |G\;thus ifGA4-iB,whereAandBarereal,we have |C \=*J(A2+B2 ).Inthesamewaytheequation z~C'e-int........................ (10) representsuniform motion inacircle, with angular velocity n, inthenegative (orclockwise) direction. Wecomenow totheapplicationtolinear differential equations with constant coefficients. From ourpointofview thesimplestcase istheequation of 4.Inorder that every stepoftheworkmayadmit of interpretation, weassociate with thistheindependent equation (12) asinthetheoryofthespherical pendulum. Thetwomaybe combined intheoneequation whichmayindeed beregardedasrepresenting directly, without theintermediaryof(11)and(12),thelawofacceleration inthe spherical pendulum and similarproblems. Tosolve (13)we assume z=GeM ,andwefind that theequationissatisfied providedA,2+n*=0,orX=+in.Since different solutions can beadded, weobtain theform *=Ceint+G'e~int , 56 DYNAMICAL THEOEY OFSOUND with twocomplex arbitraryconstants G,G'.These canbe determined soastoidentifyaandz,attheinstant t=0,with thevectors whichrepresenttheinitialpositionandvelocityof thepoint (x,y).Itappearsfrom (14)that themostgeneral motion ofapoint subjectto(13)maybeobtained bythe superpositionoftwouniform circular motions inoppositedirec- tions. Thesame problem (virtually)hasbeen treated in 18, where thepathwasfound tobeanellipse.This resolution ofan"ellipticharmonic" vibration intotwocircular vibrations inoppositedirections hasimportant applicationsinOptics. Thesolution oftheequation (11)maybederived from(14) bytakingthe"real" partofboth sides,i.e.byprojectingthe motion ontotheaxisofx.Since G,G'areoftheforms C=A+iB, G'=A'+iB', (15) itmight appearatfirst that theresult would involve four arbitraryconstants. These occur, however, insuch away thattheyarereally equivalent onlytotwo. Thuswefind x=(A+A')cosnt-(B-B')smnt (16) The kinematical reason forthis isthat, asregardstheir projectionsonastraight line, right-handedandleft-handed circular motions areindistinguishable. Animportant practical corollaryfollows. Weshould have obtainedequal generality, sofarasthesolution of(11)isconcerned, ifwehadcontented ourselves with either solution of(13),forexample z=Ceint , (17) andtaken therealpart x=Acosnt Bsinnt(18) This conclusion isobviouslynotrestricted totheparticular differentialequation (11)withwhich westarted. Theuseof anadjunct equationsuch as(12)hasonlybeen resorted to inorder toremove thesuspicionofanythingthatcantruly becalled"imaginary"inthework. Such assistance can always beinvokedmentally, but itisasunnecessaryasit would betediousalways formallytointroduce it.Ifinany case ofalinear differentialequation between soandt,with constant real coefficients, weseek forasolution ofthetype a**Co1 ,theimaginaryvalues(ifany)of A,willoccur in VltfKATIONS 57 conjugate pairs oftheformmin,andwemayassert that thepartofthesolutioncorrespondingtothispairofroots willbegiven with sufficientgeneralityifwemake use ofone onlyofthese, writing, forinstance, andtakingtherealpart. Wemayapply theseconsiderations, forexample,tothe equation CZj,& _~,g_. at Cut ofresisted motion about anequilibrium position (11).Ifwe putx=GQl ,wehave X2+k\+fji=(21) Hence \=-$kinf , (22) where nf=V(/"--i^2 ), (23) provided W<4^.Ontheaboveprincipleasufficient solution is or,inrealform, #=e~Va(Acosn't-Bsinn'4 (24) which isequivalentto11(8). The Hatno method canbefollowed with regardtothe equationofforcedoscillations, say dx,. /0,v (2o> In.stoad ofthinwotake theequation d*z vdz .(26) thoimplied adjunct equation beingofthetype (25)with fmptin-stoad of/cos ptontherighthand.Aparticular solution ia fJPt /-O^X*=t>e, (tf) provided(p. p*+ikp)G=f. (28) A*?4 "^p'TiTp<29> 58 DYNAMICAL THEOEY OFSOUND Ifweput fjt,-p*=Rcosa}kp=Rsinct, (30) thisbecomes *=W> (31) therealpartofwhich is *=|cos(p-)(32) Thismaybecompared,forbrevity,with theprocessof12. 21. Historical note. Thetheoryofvibrations hasalongandrather intricate history,inwhich Pure Mathematics andMechanics have reacted ononeanother withgreat advantagetotheprogress ofboth sciences. Variousspecial problemsofgreatinterest hadbeen solved bytheBernoullis, Euler*, andother mathe- maticians, but itistoLagrangefbhatweowethegeneral theoryofthesmall oscillations ofasystemoffinite freedom treated bymeans ofgeneralizedcoordinates. Thework of Lagrangewaspurposelysomewhat abstract informj;the fulldynamical interpretationwasreserved forThomson and Taifc (Natural Philosophy; 1867),towhom wealsoowethe now current terminologyofthesubject. Thetheoryhas been very greatlyextendedbyLordRayleigh,andsystematic- allyappliedtoacoustics aswell asother branches ofphysics, invariouswritings,most ofwhich (downtotheyear 1896) areincorporatedinhisTheory ofSound. *Leonhard Euler, born atBale 1707, died atStPetersburg 1788.Hewrote extensively onmost branches ofmathematics andmechanics, and fixed to agreat extent thenotations now inuse. tJoseph Louis Lagrange, born atTurin 173C, died atParis 1813,"tho greatest mathematician since thetime ofNewton." J"Onnetrouvera point deFigures dans cetOuvrage. Lesme'tliodes qua j'yexpose nedemandent niconstructions, niraisonuemensg^omdtriqnes ou me'chamques, mais seulemeut desoperations alge"briques, aesujeties auue marche reguliereefcuniforme."(PrefacetotheMScanique Analytiquc, 1788.) 1sted.London 1877, 2nd ed.London 189i 6.Seealso hisScientific Papers, Cambridge 18991922. CHAPTER II STRINGS 22.Equation ofMotion. Energy. Weproceedtothemore orless detailedstudyofthe vibrations ofvarioustypesofcontinuoussystems. Amongsb these the firstplace must formany reasons beassignedto thetransverse vibrations ofauniform tensestring. Historically, thiswasthefirstproblemofthekind tobetreatedtheoretically. Themathematicalanalysisissimple, andvariouspointsofthe general theory sketched inthepreceding chapterreceive interesting illustrations, which aremoreovereasilyverified experimentally. Again,thesequenceofthenaturalperiods offreevibration hasthespecial"harmonic"relation which haslongbeenrecognizedasinsomewayessential togood musicalquality, althoughthetrue reason, which isultimatelya matter ofphysiology,hasonlyinrecent times beeninvestigated. Themathematicaltheoryhasfurthersuggested someremarkable theorems, astotheresolution ofavibration ofarbitrary type intosimple-harmonic constituents, which arcoffar-reaching significance. Finallyitistobenoted that inthepropagation ofadisturbancealongauniformstring wehave the firstand simplest typeofwave-motion. Thestringissupposedtobeofuniformline-density p, and tobestretched with atension P.The axis ofxistaken alongtheequilibrium position, andwedenote byythetrans- verse deflection atthepoint x,attime t.Itisassumed that thegradient dy/dxofthecurve formed bythestringatany instant issosmall that thechangeoftension may be 60 DYNAMICAL THEOBY OFSOUND neglected.Under these conditions theequationofmotion ofanelement Bxis n, .................. (1) where^denotes theinclination ofthetangentline tothe axis ofx.Theright-handside is,infact, thedifference of thetensions onthetwoends oftheelement, when resolved in thedirection ofy.Invirtue oftheassumption justmadewe maywrite sinA/T=tan-v/r=dy/dx,sothat (1)becomes wherec*=P/p............................ (3) Itiseasilyseen that theconstant chasthedimensions of avelocity. Thekinetic energyofanyportionofthestringisgiven by T=lpjpda!..................... (4) taken between theproperlimits. Thepotential energy may becalculated intwoways.Inthe firstplacewemayimagine thestringtobebroughtfrom restinitsequilibrium position torest inanyassignedformbymeans oflateralpressures appliedtoit.Forsimplicity supposethat atanystageofthe processtheordinates allbear thesame ratio(&)totheir final valuesy,sothat thesuccessive forms assumed bythestring differonlyinamplitude. The forcewhich must beappliedto anelement 8$tobalance thetensions onitsends is wheresin-^isnow tobeequatedtokdy/das; andthedisplace- mentwhen kincreasesbyBkisy8k.The totalworkdoneon thiselement istherefore -PyifSx.fkdk=- where theaccents indicate differentiations withrespect to as. Thepotential energyisaccordingly (5) STRINGS 61 Inthealternative method wecalculate thework done in stretching thestring againstthetension P.The increase in lengthofanelement Sxis V(1+y'"}Sx-Sx =%y'n~Bx, approximately,sothat F=iPjy2^...................... (6) Theformulae(5),(6)lead toidentical results whenappliedto thewhole disturbed extent ofthestring. Forbyapartial integration wehave where the firstterm refers tothelimits. Itvanishes atthe extremities ofthedisturbedportion,since yisthere=0. 23.Waves onanUnlimited String. Thesolution of22(2)is y=f(ct-x) +F(ct+x\............... (1) where thefunctions/,Farearbitrary.Itiseasilyverified bydifferentiation that thisformula does infactsatisfythe differentialequation, andweshall seepresentlythatbymeans ofthetwoarbitraryfunctions which itcontains weareable to representtheeffect ofanygiveninitial distribution ofdisplace- ment(y)andvelocity (i/).Itwaspublished byd'Alemberb* in1747. Thetwoterms in(1)admit ofsimple interpretations. Takingthe firstterm alone, weseethat sofarasthis is concerned thevalue ofyisunaltered when asand ctare increased byequalamounts;thedisplacementtherefore which exists attheinstant tatthepointxisfound atalater instant t+rinthepositionas+CT.Hence theequation y=f(ct-x}.....................(2) representsawave-formtravelling unchanged with thevelocity cinthedirection of^-positive. Theequation y=F(ct+a)..................... (3) representsinlikemanner awavetravelling with thesame velocityinthedirection of^-negative. And itappearsthat *J.leRond d'Alembert (1717 83),encyclopaedist andmathematician;lie made important contributions todynamics andhydrodynamics. triemostgeneraltreemotion 01uue madeupoftwosuchwave-systems superposed. Theformoftheexpression V(P/p)forthewave-velocityis tobenoticed. Asinallanalogouscases thewave-velocity appearsasthesquareroot oftheratio oftwo quantities,one ofwhichrepresents (inageneral sense) the elasticity,andthe other theinertia, ofthemedium concerned. Asimple proofoftheformula forthewave-velocityhas beengiven byProf. Tait*. Imagineastringtobedrawn with constantvelocityvthroughasmooth curved tube, theportions outside thetubebeing straightand inthesame line. Since there isnotangentialacceleration thetensionPisuniform. Also theresultant ofthetensions ontheends ofanelement 8s,atanypointofthetube, willbeaforcePSs/Rinthe direction ofthenormal, whereRistheradius ofcurvature. This willbalance the"centrifugalforce" p8s.v2/Rifv-=P/p. Under thiscondition thetubemaybeabolished, since itexerts nopressure,andwehave astanding wave onamoving string. Ifwenowimpressoneverythingavelocityvintheopposite direction totheformer, wehave awaveprogressingwithout changeofform, onastringwhich isotherwise atrest,with the velocity */(P/p).Itwillbenoticed that thisinvestigationdoes notrequirethedisplacementstobesmall. Themotion ofanunlimitedstring consequentonarbitrary initial conditions y=*(*),y^(), P-o], (4) maybededuced from(1),but itwillbesufficient towritedown theresult, viz. irx+ci ;It<X>^- (5)AbJ %-ct Thismaybeimmediatelyverified. Ifthe initial disturbance berestricted toafinite extent ofthestring,themotionfinallyresolves itself intotwo distinct wavestravellingwithoutchangeinopposite directions. Intheseseparatewaves wehave $=+cy', (G) *Encyc.Brit. 9thed.Art."Mechanics." STRINGS G3 asisseen atoncebyconsidering twoconsecutivepositionsofthe wave-form. Thus ifinFig.24thecurves A,Brepresentthe positionsattheinstants t,t+St,wehavePQ=cSt,RP=y$t, MP/PQ=y',whence theformer oftherelations(6).Thesame thingfollows ofcourse from differentiation of(2). Con- versely,itiseasilyseen from (5),orotherwise, that ifthe initial conditions beadjustedsothat either oftherelations(6) iseverywhere satisfied, asingle progressive wave will result. When thestringisstarted with initialdisplacement, but noinitialvelocity,theformula(5)reduces to y-ifo(*-c0+0(<e +oO}............. CO Thetwocomponentwave-forms resemble theinitialprofile,but areofhalftheheightatcorresponding points.Itiseasilyseen withoutanalysisthat thishypothesissatisfies thecondition of zero initialvelocity. Itappearsfrom(6)that inanycaseofasingle progressive wave theexpressions (4)and(6)of22forthekinetic and potential energiesareequal.LordRayleighhaspointedout that thisvery generalcharacteristic ofwave motion maybe inferred otherwise asfollows. Imaginethewave asresulting from aninitial condition inwhich thestringwasatrest,and theenergyEtherefore allpotential,inthemannerjust explained. Thetwoderived waves have halftheamplitude (at corresponding points)oftheoriginal form, andthepotential energyofeach istherefore\E.Since thetotalenergyofeach wave must \)Q^E,itfollows that thekineticenergyofeach must be\E. Inmathematicalinvestigationsitisnotunusual tofindthe effect ofdissipation represented bythehypothesisthat each element ofthestringisresisted byaforceproportionaltoits velocity,sothatthedifferentialequationtakes theform Asregardsthetheoryofstringed instruments thisparticular 64 DYNAMICAL THEORY OPSOUND correction hasnoimportance,thedirect influenceofthe ail- being quite insignificant;butthesolution of(8)when kis small isofsome interest from thestandpointofwave-theory, andmaytherefore findaplacehere. Ifthesquareof&bo neglected,theequation maybewritten This isofthesame form as22(2),andtherefore y=e~^t f(ct-x} +e-^tF(ct^x)....... (10) Thisrepresentstwo wave-systems travellinginopposite directions withvelocity c;butthere isnowagradualdiminu- tionofamplitudeineach case astimegoes on,asisindicated bytheexponentialfactor.Again,since thefunctions are arbitrary,wemay replace f(ct-x)andF^ct+as) by e*k(t-*le>f(ct-x)and e*k(t+*F(ct +a>), respectively,sothatthesolution mayalsobewritten y=rifa/Y(c*-*) +eWc^(c*+a)....... (11) Thisform isappropriatewhen aprescribedmotion ismaintained atagiven pointofthestring.Thus iftheimposedcondition bethaty=<f>(t}for cc0,thewavespropagatedtotheright oftheoriginaregivenby y<-***/'**-- ................ (12) Theexponentialshews thedecrease ofamplitudeasthewaves reachportionsofthestringfurther andfurther awayfrom tho origin. 24. Reflection. Periodic Motion ofaFinite String. Ifapointofthestring, saytheorigin 0,befixed,wenmnfc havey=atthispointforallvalues oft.Hence, in23(1.), = ,or =-*. Thesolution therefore takes theform y=f(ct-a;)-f(ct +x)................(1) Asappliedforexampletotheportionofthestring which liestothe leftof0,thisindicates thesuperpositionofadirect or"incident waverepresented bythe firstterm,anda"re- flected" waverepresented bythesecond. Theamplitudeof thereflected wave isequal,atcorresponding points,tothat oftheincident wave, sothat there isnoalteration inthe energy, butthesignofyisreversed. Itisotherwise obvious that ifonanunlimitedstring westart twowaves which are antisymmetrical withrespectto0,inopposite directions, the Fig.25. pointofthestring which isat willremain atrest,even if itbefree. Hence bythecrossingofthewaves thecircum- stances ofreflection atafixedpointareexactly represented. Itwillbenoticed thatalateral force isexerted onthefixed point duringtheprocessofreflection. Inthecase ofafinitestring whose ends are(say)atthe pointsx=0,oc=I,wehave thefurther condition that /(c*-J) -/(<*+=(2) forallvalues oft.Ifwewrite zforctI,thisbecomes /(*)=/<* +20, (3) sothatf(z)isaperiodic function, itsvaluesrecurring when- ever zincreases by2^.Itfollows thatthemotion ofthestring isperiodicwithrespecttot,theperiod 2l/cbeingthetime which awave would take totravel twice thelength.Itis otherwise evident that adisturbancestartingfromanypoint Pofthestring,ineither direction, will aftertwosuccessive reflections attheendspassPagain,inthesame direction as atfirst,with itsoriginal amplitudeandsign. 66 DYNAMICAL TtUfiOUY UJb'SOUND When the initial data areofdisplacement only,i.e.with zero initialvelocity,thesuccessive forms assumed bythestring inthecourse ofaperiodcanbeobtainedbyagraphicalcon- struction. Wesupposetheinitial formy=<(#),where <j>('#)is originally definedonlyforvalues ofxrangingfrom toI,tobe continuedindefinitely bothways, subjecttotheconditions Ifweimagine curves ofthetypethus obtained totravel bothwayswithvelocity c,and ifwetake ateach instant the arithmetic mean oftheordinates, inaccordance with 23(7),it isevident thatthevaryingform thusobtained willrepresent Fig. 26. apossible motion onanunlimitedstring,inwhich thepoints as=0,as= I,x=21,...remain atrest.Theportion between #=and<e=l will thereforesatisfyalltheconditions ofthe question. Theprocessisillustrated intheannexedFig.26; theinitial formhere consists oftwostraight pieces meetingat anangle, andtheresult afteranintervalZ/8cisascertained. Inthiswaywemighttrace(after Young) thesuccessive forms assumedbyastringexcitedby"plucking,"onepointof thestring being pressed aside outofitsequilibrium position,andthen released fromrest,buttheactual construction canin such acasebegreatly simplified. Itiseasily seen that the form ofthestringatanyinstant consists ingeneralolthree portions; theouterportions have thesamegradients asthe twopieces intowhich thestring wasinitially divided, whilst thegradientofthemiddleportionisthearithmetic mean of Fig. 27. IntheannexedFig. 27,whichcorrespondswithFig. 26,the pluckingissupposedtotakeplaceatadistance ofone-fourth thelength from oneend,andthephases shewn follow one another atintervals ofone-sixteenth ofacomplete period,the successive formsbeingAPB,AQ^B, AQJ5,AQ SR3B,AQJRJ3, andsoon. Itisevident oninspectionofthefigurethatany pointofaplucked string moves backwards andforwards with constantvelocity between twoextremepositions,inwhich it restsalternately during (ingeneral) unequalintervals. The space-time diagramsofthemiddlepoint, and ofthepoint plucked, under theconditions ofFig. 27,aregiveninFig.28. Fig. 28. 68 DYNAMICAL THEORY OFSOUND Inthelatter caseoneoftheintervals ofrestvanishes*. Ibisofcourse with thevibrations ofafinite stringthat wearechieflyconcerned inacoustics. Thestringisusually stretched withconsiderable tension between thetwopoints which limit thevibrating portion.Atoneatleast ofthese points the string passesoverabridge restingonasounding-board,whose function itistocommunicate thevibrations tothesurrounding air.The direct action ofthestringingeneratingair-waves isquite insignificant,butbythealternating pressureonthe bridgethewhole area ofthesounding-boardissetinto forced vibration. Thisimpliesofcourse acertain reaction onthe string itself, which ishowever, inthe firstapproximation, usually negligible,forthereason givenin 4. Forexperimental purposesanarrangementcalled a"mono- chord" isused. Thesounding-boardhereforms theupperface ofarectangular"resonance chamber." Thedistance between thebridgescanbevaried andmeasured, andthetension, being produced byaweightattached tooneendofthewire,which passesover asmoothpulley,canberegarded,atallevents approximately,asknown. Forpurposesofcomparisononeor more additional wiresmaybestretchedalongside theformer, their tensionbeing adjusted,asinthepianoforte, bymeans of pegsattheextremities. 25.Normal Modes ofFinite String. Harmonics. Thepreceding investigationshavebeengiven onaccount of their historicalimportance, andforthesakeoftheanalogies with othertypesofwave-motion which weshallmeetwith later. From thepurelyacousticalpointofviewtheyarehowever of secondaryinterest. The earknowsnothingoftheparticular geometrical forms assumedbythestring, and isconcerned solelywith thefrequencies and intensities ofthesimple- harmonic constituents intowhich thevibration canberesolved. *The theoretical vibration- forms have been verifiedexperimentally by Krigav-Menzel andRaps, Wied. Ann., vol. L.,1893, sofarastheinitialstage's ofthemotion areconcerned. After afewvibrations theform isseen tobe undergoing agradual change. This isattributed toaslight yielding ofthe supports ofthestring,inconsequence ofwhich thenormal frequencies arenot exactly commensurable, and theresulting motion therefore notaccurately periodic. Theconstruction inFig.27isalsodue tothese writers. Toascertain thenormal modes ofvibration ofafinitestring wemayhave recourse tothegeneral procedure explainedin ChapterI.Inanysuchmode ywillvaryasasimple-harmonic function ofthetime, saycos(nt+e).Thismakes y=n?y, andtheequation (2)of22therefore assumes theform dx2 (1) The solution ofthis,exhibitingthetime-factor, is / ,nxn.nx\ /. N ,~^ y=(Acos-- 1-Bsm )cos(nt+e)..........(2)\ c c/ The fixed ends ofthestring beingatas=0,as=I,wemust haveA 0,sin(nl/c) 0,whence nl[7rc=l,2,3, ......................(3) Thisgivestheadmissible values ofn.Inanyonenormal mode wehave, therefore, STTX cosfSTTCt ( where sisaninteger, andtheamplitude Csand initialphase egarearbitrary. Thegravest,orfundamental mode, which determines thepitchofnote sounded, correspondstos=1. Fig. 29. Thestringthen oscillates intheform ofthecurve ofsines between thetwoextremepositions shewn intheupper partof Fig.29.Thefrequencyis 70 DYNAMICAL THEORY OFSOUND and sovariesinverselyasthelengthandasthesquarerootof theline-density, anddirectlyasthesquareroot ofthetension. These statements, which were formulated asthe result of experiment longbefore themathematical theory hadbeen developed,areknown asMersenne's laws*. Thedetermination ofabsolutepitchbytheformula(5)does notadmit ofvery great accuracy owingtothedifficultyinmeasuringthetension, which isapt(owingtofriction)tobeslightlydifferent onthe twosides ofabridge. Theprinciplesthatthefrequencydiminishes with increase oflengthandwith increase ofline-density,have afamiliar illustration inthepianoforte,where longerandintrinsically heavierstringsareused forthegravernotes. Iftherelation ofpitchwereadjusted bylengthalone thestrings corresponding tothelower notes would have tobeatleast100times aslong asthosebelongingtothehighest.Inorder tosecure asuffi- cientlylowpitchwithinpracticallimits oflength,andwith asufficientdegreeoftension, thestringisloaded with acoilof wirewrapped closelyround it.This hastheeffect ofincreasing theinertia withoutseriously impairingtheflexibility, which is anessentialpoint. Theinfluence oftension, again,isillustrated in.theprocessoftuning,which consists intightening upthe wireswhen these have stretched, orthepegshaveyielded,so thattheinstrument hasfallen inpitch,orbecome"flat." Inthenextnormal mode after thefundamental themiddle pointx=^lisatrest(Fig. 29).And inthesthmode, whose frequencyisby(3)stimes that ofthefundamental, there are s1internalpointsofrest, or"nodes," inaddition tothe ends. Midway between thesewehave thepointsofmaximum amplitude,or" loops."Eachsegmentintowhich thestringis divided bythenodes vibrates asinthefundamental mode ofa stringof1/sththelength. Asalreadystated(2)thesequenceofsimplevibrations withfrequencies proportionaltothenatural numbers 1,2,3, ..., which weheremeet with, hasimportant properties, musically *M.Mersenne(1588 1648), aFranciscan friar,was aschoolfellow and lifelongfriend ofDescartes, andmaintained anextensive correspondence with himandother ruen ofscience oftheday. STEINGS 71 andphysiologically.Ibsoccurrence invibrating systemsisof coursequite exceptional.Even inthepresent case,ifthe stringdeviateappreciablyfromuniformityorfromperfect flexibility,theabove scale offrequenciesisatoncedeparted from *. Wewere ledin16totheconclusion, onphysical grounds, that inanysystemoffinite extent the effect ofthemost generalinitial conditions consistent with itsconstitution may beobtained bysuperpositionoftheseveral normal modes, with suitable amplitudes andphase-constants. Weinfer thatthe mostgeneralmotion ofafinitestringcanberepresented by theformula srrct ,. (6) i \i providedtheconstants Ca,esbeproperly determined, the summation Sextendingover allintegralvalues ofs.An equivalentform is r,/ . STTCt .D.S7TCA .STT^,,.y-2[4,cosj-+jtfssinv-sinj- t (7) \i IJ I where As=Gscoses,B8=Cgs'm a (8) Ifthestringstart from rest inagiven positionatthe instant <=thecoefficients B8willvanish;ifitbestarted withgivenvelocities from theequilibrium position (y=0) the coefficients A8will vanish. Since thevalue ofeveryterm in(6)or(7)recurs whenever tisincreased by%1/c,thevibration isessentially periodic,as already provedin24.Inallotherrespectsthemotion ofthe string when started inanarbitrary manneris,from thepresent pointofview, ofacomplex character, being madeupofan endless series ofsimple-harmonicvibrations. Theresulting note isaccordinglymade upofaseries ofpure tones, consisting (ingeneral)ofafundamental, itsoctave, twelfth, double octave, andsoon. Ibisnotaltogether easytoexcite astringinsuch away *The factthat aparticular sequence ofnotes, musically related toone another, isassociated with lengths ofstring proportionaltothequantities 1)i.u>i> wasknown totheGreeks, andwastheorigin ofthename "harmonic" asappliedtothenumerical series. 72 DYNAMICAL THEORY OFSOUND that theresultingmotion shall bestrictly simple-harmonic, andthesensationaccordinglythat ofapuretone. But, as willbeshewn morefullyin 39,itispossibletosuppress allthetones belowanyassigned rank(s)bycheckingthe vibration atanode ofthe5thmode, as,forinstance, by contact with acamel-hairpencil. Theremainingnodalpoints ofthis constituent arethenpointsofrest, whilst half-way between them there isvigorous vibration. Theexperiment, which isvery striking,iseasily made with themonochord. Theenergyinanynormal mode iseasilycalculated. We find ,......(9) ...(10) The coefficients areequal,invirtue of22(3),andthetotal energyinthismode is Itisfurthereasily provedthat thewhole energyofthe stringisthesum oftheenergies correspondingtothevarious normal modes, viz. =2s*C* =Zs-(A* +B/)....... (12) This isageneral propertyofthenormal modes ofavibrating system.Theproof,inthepresent case,dependsonthefact that f Jl .STTX .S'TTX ,,. sinj-siny-dx=0, ............ (13) ol t ifs,sbeanytwounequal integers. See 32(4). 26. String excited byPlucking, orbyImpact. The relativeamplitudesofthevarious modes isofcourse amatter ofimportance,asonitthequalityofthenote depends (2).Usuallyastringisexcited inoneofthree ways,viz.byplucking (asintheharp, zither, &c.),bystriking with ahammer(pianoforte),orbybowing (violin, violon- cello, &c.). STRINGS73 Ifthestring bepulled asidethroughasmallspace /3,at adistance afrom theend oc=Q,andthen bereleased, the values ofthecoefficients in25(7)arefound tobe a) ,2/3Z- -1 .sira .STTX sirctwhence y=-^^ ^2- 2sin-j-am cos- ...(2) Themode ofcalculation willbeexplainedinthenextchapter (see 36).Wenotice that theharmonic oforder swillbe altogetherabsent ifsin(sirajl)=0,i.e.ifthepointofpluckingbeatoneofitsnodes; thiswasremarkedbyYoung (1841).Thus ifthestring bepluckedatthecentre, alltheharmonics ofeven order willbeabsent. Theformula(1)combined with 25(12)shews that, apartfrom atrigonometrical factor which liesbetween and 1,theintensities ofthesuccessive harmonics willvaryas1/s2 .Thehigher harmonics arethereforerelatively feebly representedintheactual vibration ofthestring. The effect oftheimpactofahammerdepends onthe manner andduration ofthecontact, and ismore difficult to estimate. Thequestionisindeed, strictly,oneofforced vibrations(28) ;butinthesomewhat fictitious casewhere the duration issosmall that theimpacthasceased before the disturbance(travelling with thevelocity c)hashadtime tospreadoverany appreciablefraction ofthelength, we maytreat theproblemasoneoffreemotion withgiveninitial velocityconcentrated onashortlength.Theresult is .A ,-> 2u, .STTCL ....4B=0,Bs= -sin-j-,...............(3) STrpcL where aisthedistance from theorigintothepoint struck, and/Arepresentsthetotalmomentum communicatedbythe impact.Hence 2a^1 .SirCl.STTX .STrct ,.. y_L2-sinj-sinr-smj-..........(4) J irpcsill^/ Asintheprevious problem,the 6-thmode isabsent ifthe originbeatoneofitsnodes.Aparfcfrom thetrigonometrical factor onwhich thiscircumstance depends,theintensities of thesuccessive modes are,accordingto25(12),nowofthe same order ofmagnitude. The unreal character ofthepre- ceding hypothesis betraysitself inthis result;butwemayat allevents infer that inthecase ofaverybriefimpact the higherharmonics arerelatively much more inevidence than intheformerproblem. Inrealit}'-theimpact,even intheease ofametallic hammer,isfarfrom instantaneous, thetime ofcontact, though veryshort asmeasured byordinary standards, beingatall events comparablewith theperiodofvibration ofthestring* The effect ofanimpulseoffinite duration hasbeen calculated byHelmholtz, towhom most ofthepresent theoryisdue,on thesuppositionthatthepressure beginsattheinstant t=0, and lasts foratime r,duringwhich itrises from 2ero toa maximum and fallstozeroagain, accordingtothelawsin(TT</T). Asomewhat simplerresult isobtained ifweimaginethelaw ofpressuretobe wherefirepresentsthetime-integraloftheforce from t= oo tot=+oo .This law,whosegraphical representationhasthe form ofthecurve inFig. 14,p.33,hasthedefect thatthere is nodefinite instant ofbeginningorending,butasthetruelaw isinanycaseunknown, itmayserve forpurposesofillustration. The interval oftimeduringwhich theforce issensible is comparablewith T,andcanbemade asnarrow asweplease bydiminishingT.The details ofthecalculation willmore convenientlyfind aplaceinthenextchapter (38).The result is As=0,^-^.^-'"''Bin^.......... (6) TT/JCS I^' When Tisinfinitesimal thisagreeswith(3). Inother cases theintensities ofthehigherharmonicsvaryase~8lrCT ' ,ifwe omit thetrigonometricalfactor. Althoughthepressureisthus rendered lessabruptas regardsitsvariation with thetime, itisstillassumed tobe *Kaui'muuu, Wied.Am:.,vol.LIV.(1895). STRINGS 75 concentrated atapoint.Ifwewere toimagineitdistributed continuously over ashortlengthofthestringthiswould further increase the relative weightofthelower harmonics (see 38). Accordingtoageneral principle,which ishereexemplified, andwhich will be,further referred tointhenextchapter, thehigher harmonics areexcited ingreater relativeintensity, themoreabrupt thecharacter oftheoriginatingdisturbance. From amusicalpointofview theharmonics after about the sixth are tobediscouraged,sincethey comesufficiently near tooneanother inthescale tobemutuallydiscordant. Inthepianoforte thehammers arecovered withlayersofsofter material, sothat thevariation ofpressure duringtheimpact isrendered moregradual. Thepointatwhich theblow isdelivered isalsoamatter ofimportance. Toobtain anote ofrichmusicalqualitythe lower harmonics should bepresentinconsiderable force, and themiddleregionsofthestringareonthisaccount tobe avoided. Ontheother hand, theharmonics ofhigherorder than thesixth areprejudicial,asalreadystated. Both re- quirementsaremetbyfixingthestriking pointatadistance ofabout one-seventh ofthelengthfromoneend. Thepartial tones which have nodes atornear thispointwillthen not beexcited atall,oronlywithcomparativelyfeebleintensity. 27. Vibrations ofaViolin String. Thetheoryofthevibrations ofastring when excitedby bowingissomewhat difficult, butthemain features havebeen elucidated byHelmholtz. Since thepitchisfound tobethat natural tothestring,thevibrations aretoberegardedasin asense" free," thefunction ofthebowbeingtomaintain the motion bysupplying energytomake upfor tlio losses by dissipation.Inthecase oftheviolin &c.,where thestrings areoflightmaterial andpassoverabridge restingonavery sensitive surface (oftheresonancecavity),these losses may berelativelyconsiderable. Themode ofaction ofthebow appearstobethat itdragsthestringwith itforatimeby friction, until atlengththelatterspringsback;after afurther 76 DYNAMICAL THEORY OFSOUND interval thestringiscarried forwardagain, and soon*, the complete cycle taking placeintheperiodofvibration. Inorder toobtain data formathematicalanalysis Helmholtz began byanexperimental studyofthecharacter ofthevibration atvariouspoints. Thedevice wasanoptical one, ofthekind employed byLissajous (18),bywhich therectilinear vibration ofthepoint examined iscompoundedwith anindependent vibration atright angles, whoseperiodiscommensurable, or nearly so,with that ofthestring.Amicroscopewhose axis is horizontal isdirected tothepointtobestudied, thestring itselfbeingvertical. Theeye-pieceofthemicroscopeisfixed, buttheobjectiveiscarried byoneoftheprongsofatuning forkandvibrates inavertical direction. When theforkalone vibrates theimageofabright pointonthestringisdrawn outintoavertical line;when thestringalone vibrates the appearanceisthat ofahorizontal line.When both vibrations coexist theresult would beaclosed curve iftheperiodswere exactly commensurable. Forexample,iftheperiodofthefork wereexactly commensurable with that ofthestring,and ifthe vibration ofthepoint examined weresimple-harmonic,the result would beone ofthecorrespondingseries ofLissajous figures (18);whilst iftherelation between theperiodswere inexact, thecurve wouldpassinsuccessionthroughthevarious phasesoftheseries. Intheactual circumstances theforms of thecurves aremodified, and itispossiblefrom theresult to make inferences astothetruenature ofthevibration studied. Fig.30. Theinterpretationisfacilitated bytheidealrepresentation ofthesuccessive phasesasorthogonal projectionsofacurve traced onarevolving cylinder.Itwasfound that thespace- *Inorder thatworkmaybedone itisnecessary tosupposethat the frictional force isgreater inthe first stage than inthesecond. This is consistent with theknown lawthat friction of(relative)rest isgreater than i'riulion ofmotion. Theremark isdue toLordIlayleigh. between forkandstring.Iftheportion ofthebroken lineinFig.30which lies between AandBbewrappedround a cylinderwhose circumference isequalto AB, itsprojections onplanes through theaxis willinclude such forms asare hereshewn(Fig. 31)*. Theperiodofvibration ofthepoint examined ismade upoftwo intervals, usuallyofunequal duration, during which thepoint moves backwards and forwards, respectively,with constant but (ingeneral) unequalvelocities. The ratio ofthetwo intervals isfurther ascertained tobeequaltothat ofthe twosegmentsintowhich thestringis divided bythepoint. These results have been confirmedbysubsequentobservers whohave obtained thespace-timedia- graminamore direct mannerf. In order thattheymaycome outclearly someprecautionsarenecessary. Some- thing dependsonthe skillwith which thebow isused,andapparentlyonthe qualityoftheinstrument. Inorder, also, that thediagramshould befreefrom minorirregularitiesthebowshould be *Intheactual experiments ofHelmholtz the frequencyofthestring wasfourtimes thatofthe fork. Thecircumference ofthecylinder intho above mode ofrepresentation then includes four periodsofthezig-zag line inFig. 30. iKrigar-Meuzel and llaps, Wicd. Ann., vol. .(1891). appliedatanode ofoneoftheharmonics, andthepoint observed should beatanother node ofthesame. Exceptatthetwoinstants ineach periodwhen thevelocity suddenly changes,theacceleration ofthepoint (P)examined is zero. Itfollows from 22(2)thatthecurvature ofthestring intheneighbourhoodofPvanishes, andthat theform ofthe stringatanyinstant isaccordinglymade upofstraight pieces. Pig.32. Itappearsthat alltheconditions oftheproblem canbesatisfied ifweassume thattheform isalwaysthat oftwosuchpiece's meetingatavariablepoint Q.InFig.32letAB(=I)bethe undisturbedpositionofthestring, and leta(=AN) and/3 (=NQ)bethecoordinates ofQreferred toAasorigin andAB asaxis ofabscissae. Theequationsofthetwoportions ofthe stringare ft=#B/a, ya=(Z- )/(*- ), .........(1) andthedifference ofthevelocities nearQonthetwo sides isaccordingly Inthetime 8talengthdtiofthestringistraversedbythe point Q,sothat amassp&Sthas itsvelocity increased bythe above amount. This istheeffect ofthetransverse force wherePisthetension, actingforthetime St.Equating the changeofmomentum totheimpulse oftheforcewefind STRINGS 79 Thepointofdiscontinuity Q(ofthegradient) must therefore travelrightorleftwith thevelocityc. Letussuppose thatQstarts fromAattheinstant t=0, andthat ftisatfirstpositive. Theobservations ofHelmholtz shew thatthevelocityatapoint x,viz. isduringanintervalosfcconstant, whence /8=Co(-), .....................(6) noadditive constantbeing admissible, sincej3must vanish with a.This istheequation ofaparabolicarcpassing through A,B. Theconditions oftheproblemaretherefore allfulfilled ifwe imagine Qtotravel backwards andforwardsalongtwosuch arcs,withvelocity c,inthemanner indicated inFig.32.In terms ofthemaximumdisplacement /3wehaveC 4/9 //I2 , andtheequationsofthetwoportionsofthestringatany instant aretherefore 2A=^r(*-Ky.=^(i-0....... 00 Itonly remains toresolve thismotion into itssimple- harmonic constituents. The details ofthecalculation are givenin 37.The result is 8/9 ^.1 -S7TX .STTCt , . y=-q2--sin-j-.sin-p-,............(8)7T- S* i L^' where thesummation embraces allintegralvalues ofs.Com- paringwith 25(7)wehave Aa=0,B.-yji...................(9)TTS3 x' These results, andindeed thewholeinvestigation,takeno account ofthepositionofthepointtowhich thebow isapplied. Itisplain, however, thatthepositionofthebowmust have some influence onthecharacter ofthevibration; and itis found infactthatthose normal modes areabsent which have a node atthepointinquestion.Itisforthisreason thatthe somewhat idealized vibration -form which isadoptedasabasis ofcalculation isonlyobtained initspurityatcorresponding nodes, Jjorceu. vj.uruiiu.uiis Thesimplestcase offorced vibration iswhere agiven simple-harmonicmotion y=/3cos(pt +a).................. (1) isimposedatapoint (x=a).Theportionsofthestringonthe two sides ofthispointaretobetreatedseparately. The results are ...(2) forthesesatisfythegeneraldifferential equation 22(2),they makey^=forx=0,andy2=forx=I,andtheyagree with (1)when x=a.Theamplitudeofy^ory2becomesvery great, owingtothesmallness ofthedenominator, whenever pa/cor p(Ia)/cisnearly equaltoamultipleofvr,i.e.when the imposed period 2ir/p approximatestoanaturalperiodofa stringoflength aorIa,respectively. Toobtain apractical result insuch casesweshould have totakeaccount ofdissipative forces. Thecase isillustrated bypressingthestem ofavibrating tuningforkonapiano string. Thesound swells outpowerfully whenever theportionofthestringbetween thepointofcontact andeither endhasanatural mode inunison with thefork. Thisplanisrecommended byHelmholtz asameans ofproducing pure tones, since thehigher modes ofthe fork, notbeing harmonic with thefundamental, arenotreinforced. When atransverse force ofamount Yperunitlengthacts onthestring,theequation (2)of22isreplaced by &y_tfy Ingeneral7willbeafunction both ofxand if. Thecaseofaperiodicforce JPcos(pt+a)concentrated onan infinitelyshortlengthofthewire atao=amaybededuced from STEINGS81 theformulae(2).Thevalue of/3interms ofFisfound from theconsideration thattheforcemustjust balance thepullof thestring onthispoint,i.e. Py 1'-Py 2' ...............(4) forxa.This leads to px.p(l a)sin*sm- - &=-T-^T- -pCosCpi+a)..........(5) -sin* o c Theformula foryzdiffersonlyinthat theletters xandaare interchanged; wehavehereaninstance ofthereciprocal theorem of 17,accordingtowhich thevibration atapointxduetoa periodicforce atamust bethesame asthevibration atthe pointaduetoanequalforce(ofthesameperiod)atx. Theamplitude becomes asarulegreatwhensm(pl/c)is small, i.e.when theimposed period approaches anaturalperiod ofthewholestring. Anindeterminate case occurs when sin(pafc)=andsin(pl/c)=simultaneously,thepointx=a beingthen anode. 29.Qualifications totheTheory ofStrings.Wehave in 26,27considered therelativeamplitudesof thedifferent harmonics when astringisexcited invariousways, butwemust notassume thatthecorrespondingrelative inten- sities areaccurately reproducedintheresulting sound-waves, which are starLedindirectly throughthesoundingboard. Ifweneglectthereaction onthestring, which mayfora considerable number ofvibrations beinsensible, wemayregard thestringasexertingoneachbridgeaforceproportionalto thevalue ofdy/dac there*, asgiven bytherespectiveformula. The differentiation introduces afactor sinthe coefficient of thesthharmonic, and soincreases theimportanceofthe highermodes. Ontheother hand, theamplitudeofvibra- tion ofthesoundingboard due toasimple-harmonicforce ofgiven amplitude,willvarysomewhat with thefrequency, *Thus inthecase oftheplucked stringitappears from Fig. 27that thepressureoneachendalternates between twoconstant values ofopposite sign. onthegeneral principleillustrated in 9*. This isprobably totherelative advantageofthelower modes. The effect of yieldingofthebridgesinmodifyingthenaturalfrequencies ofthestringhasbeen discussed byRayleighf;itisprobably inpractice very slight. Another cause which must bementioned asaffectingour results tosome extent istheimperfect flexibilityofthestring, orwire. Inthecaseofthehighernormal modes thesegments intowhich thestringisdivided maybesoshort that flexural couples come intoplay,andtend toraise thefrequency by increasingthepotential energyofagivendeformation. This willbereferred tolater(50).Afurtherpointisthat the abruptformspostulatedinthetheoryofpluckedorbowed stringsarenotexactly realized, andthatsuchinvestigationsas those of 26,27aretobeviewed asapproximations,which are howeverquite adequatesofarasthedetermination oftheampli- tudes ofthegraverandmore important harmonics isconcerned. 30. Vibrations ofaLoaded String. Weconclude thischapterwith thediscussion ofoneortwo problems which, besidesbeingofsome interest inthemselves, mayserve toremind usagainthattheharmonic scale offre- quenciesisafter allanexceptional phenomenon, even inthe case ofstrings. Take firstthecase ofastring, otherwise uniform, loaded withamassMatitscentre. Itisobvious that those normal modes oftheunloadedstringwhich have anode atthispoint areunaffected.Leavingthese onone side,weconsideronly those vibrations inwhich there isateveryinstantcomplete symmetrywithregardtothecentre. Ifthelateraldisplacement ofMbej3cos(nt+e),wehave, forthefirsthalf ofthestring, .nx sin ^1=^cos(ni+e) (1) sin^r-2c *Someinteresting experiments bearing onthese questions havebeenmade byBarton and Garrett, Phil.Mag. (6),vol.x.,1905. SeealsoBarton, Text- Book ofSound, London, 1908, 361. tTheory ofSound, 135. BT.B1JNU-B Theequationofmotion ofMis (2) where after thedifferentiations wemustsupposeas=JI.This gives nl nl_l. o7.ta/no^~~ A>' (*) where &iswritten forM/p,i.e.bisthelengthofstringwhose mass would beequaltothat oftheattachedparticle. The frequenciesaretherefore determined by nl/Zc a;^x2iacs,...,..................(4) where xl}#2,xa,...aretheroots ofthetranscendentalequation a;tanx=l/b.........................(5) Equationsmore orlessofthischaracter occur inmanybranches ofmathematicalphysics,andcanoftenbesolvedapproximately bygraphicalconstruction. Thus inthepresentinstance ifwe trace thecurves y=.ao, .....................(6) theabscissaegive the roots. Ifbberelativelysmall these fallalittle short of^vr, ITT, TT,...,respectively,andthe Fig. 83. 84DYNAMICAL THEOBY OFSOUND inagreement with 6(4). 31.Hanging Chain. rically*, , Emotive" ineP* f ition f .....................(i) op 8^=_?./a2/\ 9*2^a^ry......................(2) modesv a variable inplace of ThmrC ;llCeanewindependent (17S2)- totravel fromthelower endtothepoint sc,wehave T=| -77;=z./|-|, =^ra(4) Interms ofTasindependentvariable theequation (3)becomes Forthepresent purposewedonotrequirethecomplete solution, butonlythat solution which remains finitewhen r=0.This is where (7isarbitrary,asmaybeverified byactual differentia- tion,andsubstitution in(5).The function denned bythe series inbracketspresentsitself inmany physical problems; itiscalled the"Bessel's Function ofZero Order," and is denoted byJQ(nr)*. Hence, insertingthetime-factor, y=GJQ()ir) cos(?i.+e) (7) Thevalue ofTcorrespondingtotheupperend(x=I)is Tl=2V(%), (8) andthecondition that thisendshould befixed gives /o("T 1)=(9) This determines theadmissible values ofn.Tim firstfew roots aregiven by n-n/TT- -7(155, 17-571, 27546,..., (10) where thenumbers tend totheform s,&being integral.In themodes after thefirst, thevalues ofrcorrespondingtothe lower rootsgivethenodes. Thus inthesecond mode Uiero i.s anode atthepoint T/T,=P 7(J55/L-7571,or*//=T*/TI*='!{)(). Thegravest periodis2vr/?i=5'225V(^/ l(/)>whoreaw the,period ofoscillation ofarigidbarofthesamelengthisfr'l'M^(l/y). Thecomparisonverifies ageneral principlereferred t;oin 1(5, *Elaborate numerical tables oftheBcsscl'H PunctionH, calculated by Mcisscl andothorH, arcf^ivon byGrayandMathown, Treatise onllcKst'l.Fuin-tiona, London, 1895.Aconv(jnient abridgment isincluded inDalc'HFive.-L''iyitru Tables ofMathematical Functions, London, 1903. 86DYNAMICAL THEOEY OFSOUND accordingtowhich anyconstraint hastheeffect ofquickeningthe gravestoscillation. The firsttwomodes areshewn (ondifferent scales)inFig. 34,thetwonodal points repre- sentingthepointofsuspensioninthetwo cases. 3la.ApproximateDetermination of Free Periods. Wemayapplysome oftheresults ofthis Chaptertoillustrate further Rayleigh's ap- proximate method, ofwhich some account was givenin16.Itisassumed thatbyimaginary Motionless constraints thesystemisreduced toonedegreeoffreedom, sothat itsconfigura- tion atanyinstant dependsonasingle coordinate(q),thetypeofvibration being accordingly prescribed.Thefrequency-found onthishypothesis will, inthecase ofthe gravest mode, beanupperlimit tothetrue frequency,but willbeagoodapproximation toitiftheassumedtypeissuitablychosen. Takingfirstthecase ofauniformstring (25)oflength I,anddenoting byxdistance fromthemiddle point, letuswriteFig. 34. (1) sothat6rangesfrom\TTto+^TT.Thesimplest symmetrical assumptionwhich wecanmake forthedeflection is (2) theform atanyinstantbeing parabolic. Hence (3) |~(4) coefficients arerespectively 4 ,8P andthefrequency (n/2?r)isaccordingly given by 10P na==c/tt=7F......................(6) Wehave seen that thecorrect numerical factor ofPis 7r2(=9'87);thefrequencyistherefore inexcess bylessthan onepercent. Amuch closerapproximationisobtained ifinplaceof(2) weassume y=qcos2 (I+j3sin2 0),...............(7) where /3isaconstant tobechosen later. Thismakes -15(21+6 andtherefore ,35+14/3+11/326P(q.n~~ 21+6/3+/32' plz'""() Weknow thatwhatever valueweassignto/3theresult willbe inexcess; wetherefore choose @soastomake thefirst fraction aminimum. Itisshewn inbooks ontheCakwlus that the stationaryvalues ofthefunction _A+2H/3+B/33 U~ a+2h/3+bj3*' ^} aregiven bythequadratic (ab-A2 )u*-(aB+bA~ 2hff) u+(AB-#2 )=0.(11) Inthepresentcase thisreduces to 3w25Gu+84=(12) Wemust takethelower root,which is u=1-6444958. Hence (9)gives Since vr2=9'8696, theerror isveryminute. 88 DYNAMICAL THEORY OFSOUND Thesamemethod maybeappliedtothehangingchain (31).Assuming CO where asisnowmeasured from thetop,wefind ...(15) /32 )............. (16) Hence ,_5(3+4/3+2/3*)g .n"" 10+15/3+6/3* 'I................( } Theminimum value ofthis isfound tobeT4459. The con- sequent value oftheperiod (2-7r/?i)istherefore 5'226\f(l/ff),in almost exact agreementwith thecorrect result. 31b. Aeolian Tones. When acurrent ofairstrikes atensestringorwire atright angles, especiallyifthewind isintensified byhavingtopass throughanarrow slitalong which thewire lies,amusical note isoftenproduced.This isan.exampleofaforced vibration of asomewhat different kindfrom thosealreadyconsidered. The friction ofthewindoneach sideofthewiregivesrisetoaseries ofeddies which follow oneanother atregularintervals. Obser- vation insimilar cases onalargerscaleshew thatthetwoseries arenotsymmetrical, eddiesbeingshed offalternatelyonthe two sides. Theresult isanalternatingforceonthewire, atright anglestothewind. Iftheperiodisnotverydifferent from the naturalperiodofthewire,avigorousvibration ofthelattermay result. Therelation between thefrequency N,thevelocityVof thewind, andthediameter Dofthewind hasnotsofarbeen deduced fromtheory.Observation hasledtotheformula ~. CHAPTER III FOURIER'S THEOREM 32.The Sine-Series. The studyofthetransverse vibrations ofstringshas already suggestedaremarkable theorem ofpuremathematics, towhich some further attention mustnowbegiven. The theoryofthenormal modes hasledus(25)totheconclusion that the freemotion ofastringoflengthI,started inany arbitrary manner, canbeexpressed byaseries oftheform ,,/ . srrct ,D.S7rct\ .STTX71, y=2 fAscos-y-+ JBssmJsm....... (1) where s=l,2,3,...,providedtheconstants As,B8beproperly determined. Inparticularifthestringbesupposedtostart from, rest attheinstant t=inthearbitrary formy=f(x}}it should bepossibletodetermine thecoefficients Assothat , ..................(2) forvalues ofxrangingfromx=tox I.This isaparticular case of"Fourier's Theorem*." Since Iisatourdisposal we may conveniently replaceit(forgeneral purposes) by TT,and thestatement then isthatanarbitrary functionf(x) canbe expressed,forvalues ofxrangingfrom toTT,intheform f(x)=A1sinx+A.,sin2#+...+Assinsac+.......(3) *J.B.J.Fourier (17681830). The history ofthetheorem isclosely interwoven with that ofthetheoryofstrings, and ofthetheory ofheat- conduction. Fourier's own researches areexpounded inhisTlieorie dela Chaleur, Paris, 1822. Anoutline ofthehistoryisgiveninProf. Carslaw's book cited onp.9G.The subjectistreated mostfullybyH.Burkbardt inhis reportentitled Entwickelungen nach oscillirendenFunktionen..., Leipzig, 1908. moreover notreferred totherestrictions whichphysicalcon- siderations alone wouldimposeonthecharacter ofthearbitrary functionf(x). Leavingsuchpointsforthemoment, and as- sumingthetheoremprovisionally,weproceedtothedeter- mination ofthe coefficients. Ifwemultiply both sides of(3) bysinsx,andintegratefrom a;=tox=TT,wegetontheright hand aseries whosegeneralterm is Ar\sinrxsinsxdx Jo fir=^A rl{cos(r s)xcos(r4-s)x}dx....(4) jo When theintegers r,sareunequalthisvanishes, since each cosinegoesthroughitscycleofvalues, positive andnegative, once oroftener within therangeofintegration. Butwhen r=s,the first cosine isreplaced byunity,andtheresult is Hence A2f"==-I/(#)sinsxdx................ (5)TTjo Theprocess maybeillustrated byafewexamples. Take, first, thecaseof f(x)=x(Tr-x), .....................(6) which isrepresented byanarcofaparabola. We find, after aseries ofpartial integrations, 9f*" A.As=-\ x(-7r-x)$msxdx =-(1-cossTr). ...(7)TTJo vrs-J^/ V/ This isequalto or8/7rs3 ,accordingassiseven orodd. The theorem therefore becomes -^ sin3^+-sin5ic +...V...(8) Ifweputaj= TTinthisweobtain theformula 7T3_1 1 32 33+53 which isknown onothergrounds tobecorrect. Theequalityin(8)mayalsobetestedgraphically. Itisfound that the discrepancy between thegraph ofX(TT-X) and that ofthe FOURIEE'S THEOBEM 91 functionrepresented bythe firstthree terms ontherighthand issoslightthat itwould bebarely perceptible onascale suited tothepagesofthisbook. Inthenextexamplethegraphof/0) consists oftwostraight linesthroughthepoints x=0,x=TT,respectively, meetingat anangleatthepointas=a.Ifweassume theordinate atthe latterpointtobeunity, wehave /(*)= / [0<*<],, j(^) (Tr #)/(TT a) \<x.<no<ir\. j Wefind, aftersome reductions, <vsinsxda>+ (TT- of)sinsoodx \ / a(TT-a)' s2sinsa....(11) Eig. 35. 92 DYNAMICAL THEOEY OPSOUND Thus fsin a.sin#+~-sin2orsin2# (7r a)V ...). ...(12) Asacheck onthis result wemaypuba=^7r,a?=^7r;this gives =1+1+1+................... (13) which isknown toberight.Thisexampleisofinterest in connection with thetheoryoftheplucked string (26,36). Fig.35shews thegraphoff(x) togetherwith that ofthe functionrepresented bythe firsteightterms oftheseries on theright hand ofequation (12),inthecase of=|7r.The fourth andeighth terms contributenothingtotheresult in this case, sincethey correspondtomodeshavinganode atthe point plucked. Again,letf(x)=7r os. ..................... (14) 2f' 2Wefind As\(TT&')sins#c&c=-............. (15) Thetheorem therefore asserts that TTx=2(sinso-+-|sin2#+^sin3#+...)....(16) Ifweputx=\IT,weobtain 7T _11 which isEuler's formula forthequadratureofthe circle. Theformula(16)also verifiesobviouslyfor#=7r; but ifwe putsc=weseethat there issome limitation toitsvalidity. Thenecessary modification isstated in 34.The series is moreover much moreslowly convergent than inthepreceding case; this isillustrated byFig. 36,which shews thegraph ofTTxtogetherwith that ofthefunctionrepresented by the firsteight terms ofthe series. Foranyvalue ofxother than wecanobtain anapproximationasclose asweplease, provided wetake asufficient number ofterms, butthesmaller thevalue ofa?thearea fcerwillbethenumber ofterms reouired Fig. 36. Thepreceding illustrations, with thediagrams,afford at allevents apresumptioninfavour ofthetheorem inquestion, butshew atthesame time that itissubjecttosome restric- tions. Thetheorem admits ofindependentmathematicalproof under certain conditions astothenature ofthe" arbitrary" functionf(x).We shall, however, notenterupon this,but shall content ourselves with thefollowingformal statement: Ifweform thesumofthe firstmterms oftheseries(2); andwrite #)=^isinx+sn where.+Amsinmx, (18) 2f*"As=If(as)sin.sxdx,............ (19) 7T./0 itmaybeshewn that, foranyassignedvalue ofxintherange from toTT,thesumfm(x)willtendwithincreasing mtothe limitf(x), providedthefunctionf(od)iscontinuousthroughout theaboverange,hasonlyafinite number ofmaxima and minima, andvanishes for#=and#=TT. Itwillbenoticed that theconditions herepostulatedare 94 DYNAMICAL fulfilled asamatter ofcourse byanyfunction which itis natural toassume asrepresentingthe initial form, orthe initialvelocity,ofatense string. We also seethat the difficulty metwith inthecase of(16)canbeaccounted for bythe fact that thefunction does not vanish for ac=0. Anextension ofthestatement fcomeet such cases willbe given presently (34). 33.TheCosine- Series. Thetheoryofthelongitudinalvibrations ofrods, orof columns ofair,leads, inaddition, toasimilar theorem relating totheexpansionofanarbitraryfunction inascries ofcosines. Theformal statement isnow asfollows : Ifwewrite fm(#)=A+Alcosx+A2cos2# -f-...+Amcosmx, (1) where A=-If(x}dx, (2) whilst fors> 9 / (3) itmaybeshewn that asmincreases thesumfm(x)willtend to thelimit/(#), provided f(x)iscontinuous throughoutthe rangefrom toTT,andhas atmost afinite number of maxima andminima. There isnownorestriction astothe values of/(O)and/(TT). Ifthedetermination oftheeffect ofspecialinitial conditions inalongitudinally vibratingbarwhich isfree atboth ends were asinterestingaproblemasitisinthecaseofstrings weshould have recourse, tothecosine-series. 34.Complete Form ofFourier's Theorem. Discon- tinuities. Thequestionarises astowhat isrepresented bythe sine-series orthecosine-series, supposed continued toinfinity,when xliesoutside thelimits and TT.Theanswer issupplied bytheconsideration thatboth series areperiodic functions ofx,theperiod being ZTT,whilst theformer isanodd,the latber aneven function ofso* This isillustrated bythe annexed graphical representations,inwhichf(cc}isgiven primarily onlyfortherangeTT,but iscontinued inone case asanoddandintheother asanevenperiodicfunction ofx.Itwillbenoticed that intheformer casethestipulation that/(a?)istovanish forx=andx=TTisnecessaryif discontinuities aretobeavoided. -2K Since anyfunction /(>) given arbitrarilyforvalues ofx ranging (say)from-TTtoTTcanberesolved intothesum ofan evenandanoddfunction, viz. ...(2) =- *An"odd" function isonewhich issimplyreversed insignwith x, like aflorsiu. An"even" function isonewhich isunaltered invalue when thesignofxischanged,like x-orcosx.wederive themore generaltheorem thatthesum fm(x)=A+A1cosx+A2cos2a+...+Amcosmx +Blsina+JJasin20+...+-Bmsinmx, where 96 DYNAMICAL THEORY OFSOUND whilst fors> 1f71" 3f71" As~-{/(a?)+/(- a?)}cosstff&e=-f(x)cos TTJo 7Tj_ w If71" /( #)}sm&<&=-!/(#)s 7J"J TT tends with,increasing wtothelimit/(a?), provided /()is continuous from x= TTto00= vrand hasatmost afinite number ofmaxima andminima, andprovidedalso that /(7r)=/(7r). Forvalues ofasoutside thisrangethe limit; represents, under these conditions, aperiodicfunction oi period2?r. This isthecompleteform ofFourier's Theorem, andincludes theothers asspecialcases. Weshould beleddirectly, onphysical grounds,tothisform ofthetheorem ifwewere toinvestigatethe"longitudinal" vibrations ofthecolumn ofairinareentrant circular tube. Wehave sofarsupposedthefunction f(%)tobecontinuous, aswell asfinite, evenwhen continuedbeyondtheoriginal rangeasaperiodicfunction. But thetheorems hold, with amodification tobestatedimmediately,even if/(a?)have afinite number ofisolated discontinuities. In.such acase theseriesfm(ad)stillconverges, withincreasing m,tothevalue of/(*), exceptatthepointsofdiscontinuity.But ifabe apoint wheref(x) abruptly changesitsvalue, thesumfm(a) tends tothelimit where /(a 0)andf(a+0)representthevalues off(x)at infinitesimal distances tothe leftandright, respectively,ofthe pointa.Forexample,inthecase ofthesine-series 32(3), if/(#)doesnotvanish when #=orwhen XTT,there is discontinuityatthesepointsintheperiodic function, and the seriesfm(0),forexample,hasthe limit 0,which is thearithmetic mean ofthevalues ofthecontinued function onthetwo sides ofthepoint #=0. This isillustrated in Fig.36. 35.Law ofConvergence ofCoefficients. Itremains tosaysomethingastothelawofdecrease of thesuccessive terms. Itisevident atonce that under the FOUEIEB'S THEOBEM 97 conditions laiddown thevalues ofthe coefficients A,and J5g mustultimately diminishindefinitelyassincreases, owingto themore andmorerapid fluctuation insignofcoss# and sinso),andtheconsequent morecomplete cancellingofthe various elements inthe definiteintegralsof32(5)and 33(3). More definite results have been formulated byStolces. Thefollowing statement must beunderstood torefer tothe function ascontinued inthemanner aboveexplained; and care isnecessary,inparticular cases, toseewhether discon- tinuities of/(#)oritsderivatives areintroduced atthe terminalpointsofthevarioussegments: If/(%) have(inaperiod)afinite number ofisolated discontinuities, the coefficientsconverge ultimately towards zero likethemembers ofthesequence l This isexemplified by32(16)andFig.36. If/(.#)iseverywhere continuous, whilst itsfirst derivative /'(#)hasafinitenumber ofisolateddiscontinuities, thecon- vergenceisultimatelythat ofthesequenceill!'22'32'42'' This isillustrated by32(12)andFig.35. If/(#), f(pc)arecontinuous, whilst f"(x)isdiscontinuous atisolatedpoints,thesequenceofcomparisonis 1Ill '23'33'43'"" asinthecase of32(8).And, generally,iff(x) and its derivatives uptotheordern1inclusive arecontinuous, whilst thenthderivative has(inaperiod)afinite number ofisolated discontinuities, theconvergencyisultimatelyas Thenature oftheproof,which issimple, maybebriefly indicated forthecaseofthesine-series, wenave, oyapartial integration, 2r* As=-l/(oc)sinsxdx _![?. /(a-)cos cl+r/'<cossxdx,...(4) sL77" JS7rJ Avhere theintegratedterm istobecalculatedseparatelyfor each ofthesegments lyingbetween thepointsofdiscontinuity of/0)>#any,which occur intherange extendingfrom as=to flj=7rinclusively.Forexample,ifasin32(14)theonly discontinuityisat=0,itsvalue is2/(0)/W. Inanycase there is,forallvalues ofs,anupperlimit tothecoefficient of l/inthefirstpartof(4);wedenote this limitbyM.The definiteintegralinthesecond term tendsultimatelyto zero, assincreases, owingtothefluctuations insignofcossx. HenceAgisultimately comparablewithM/s.Ifthere isno discontinuityof/(#), even atthepointsoc=0,os=TT,thefirst term intheabove value ofAsvanishes, andcontinuingthe integration wefind 1f2 12 I"71" As=---/'(#)sin sx I/"(#)sinsxdx. ...(5)S"[_7T JS7TJo Inthe firstpart, regard must behadtothediscontinuities of /'(#),ifany. Denoting byMtheupperlimit ofthecoefficient of1/s2 ,weseethatAsisultimately comparablewithM/s2 ,the second term in(5)vanishingincomparison, bytheprinciple offluctuation. The further course oftheargumentisnow sufficiently apparent. 36. Physical Approximation. CaseofPlucked String. Ithasbeenthought worth while tostate Fourier's theorem withsome care,although wedonotenter intothedetails ofthe mathematicalproof, which isnecessarily somewhat intricate, owingtothevarious restrictions which areinvolved*. From aphysical pointofview thematter maybedealt with, andperhaps adequately,inamuchsimpler manner. Toexplain this, itisbest totakeadefiniteproblem, forinstance that of *Themost recentEnglish treatise onthesubjectisthat ofProfH.S.Carslaw, Fourier's Series andIntegrals, London, 1906. FOUBIEE'S THEOEEM 99 theplucked string (26).The differentialequation, andthe terminal conditions, aresatisfied bythe finite series A.TTX irct . .ZTTX 2-Trci T/= -diSin-j-cos-y-+.4.2sin T-cos j1- , . ._.+Amsm jcos T,...(1) each term ofwhichrepresentsanormal mode ofvibration. Thismakes theinitialvelocity zero, whilst theinitial form, is ..TTX . .27r# . .rmrx .n. y=A1sm -j-+-4asin-j-+...+Amam^-....(2) Thequestion wenowhave toconsider is,how todetermine the coefficients AltAz,...Amsothat(2)mayrepresent,asclosely asmay be,aprescribedinitial form y-/(*>.........................(3) There aremanyreasons why,from thephysical pointof view,wemaybecontent with anapproximatesolution ofthe problem. Leavingaside suchquestionsastheresistance ofthe airandtheyieldingofthesupportsattheends ofthestring, we have stilltoremember that insubstitutingamathematical line ofmatter, capable onlyofexerting tension, wehaveconsiderably over-idealized thecircumstances. Inthehighernormal modes, atallevents, theimperfect flexibility,andtheuncertaintyasto thetruenature oftheterminal conditions, render thisrepresenta- tionsomewhatinadequate,sothatasolution whichprofessesto determine these modesaccuratelyisopentothecriticism that itattemptstoomuch.Again,theassumed initial form in which twostraight pieces meet atapoint,isonewhich can onlybeapproximatelyrealized;ifwegotoofarinthisdirection weshouldproduceapermanent bend, orkink, inanactual wire. Thedetermination ofthe coefficients inthe finite series (2)willdependonthekind ofapproximation aimed at.For example, wemightdivide thelengthofthestringintom+1 equal parts, andchoose thecoefficients sothatthefunctions(2) and(3)should beequalatthemdividing points. Thecurves represented bytheseequationswillthen intersect inmpointsin addition totheends. Another method istomake thesum of thesquaresoftheerrors involved inthesubstitution of(2)for (3)assmall aspossible.Thus if,forshortness, wereplaceIby v,wehave tochoose thecoefficients soastomake theintegral J Hence thismethod ofleast squares, appliedtotheexpression (2)consistingofafinitenumber ofterms, gives preciselythe values ofthe coefficients which were obtained byFourier's process*.Each coefficient isdetermined byitself, andthe effect ofadding more terms to(2)istoimprove theapproxima- tion, withoutaffectingthevalues ofthe coefficients already found. Ifwereverb togeneral units, theformula (6)is replaced byrjy^)_(A 1sin as+42sin2as+...+Amsinww Jo aminimum. Ifwedifferentiate with respecttoAsweget Sf(x)-(Alsma;+Azsm2a;+...+A msinmx)}siiisxda6==:0, (5) o or,by32(4), 2 (6) 2fl= |Isnrx7 ,>_, --dx............. 00 Inthecase oftheplucked string,theform towhich we endeavour toapproximateis y=(3x/a [0<tf<a], y=/3(I-x}j(l-a)[a<x<V] (8) Theresult isobtained atoncefrom 32(11)ifwewriteTrccjl forx,andtherefore-rrajlfora,IforTT,andintroduce thefactor/9. Thus ./f..sin-,,............... (9)^s-TT- .6'V2 a(Z a) asstated in 26(1).Thenature oftheapproximationis illustrated inFig.35. 37.Application toViolinString. Toapplythemethod totheproblem ofthe violinstring (27)>wetake asorigin ofttheinstant when thepointQin *This theorem isduetoA.Toepler (1870). THKOKEM 101 Fig.32starts fromAtodescribe theupper parabolic arc.At thisinstant wehavey=Q,everywhere, whilst 4/9 nc,.y=--(*-*) =0]..........(1) Wethereforebegin with thefinite series A.irac .-rrct . <7r .?rcy=AIsin-y-sm-j-+A2sin-sin^-+... (ibLi A.mirx .m-jrct,_.+Amsmj-smy. ...(2) This satisfies thedifferentialequation, andmakesyfort0. Itonlyremains sotodetermine theconstants thattheseries . D../rt.sm-+JBasin--+ ...+5msm-y-,...(3) S7TC , T^.whereB^^A,,..................... (4) mayrepresent theinitial distribution(1)ofvelocity,asnearly aspossible. Thedetermination ofBshasvirtually beenmade in|32(15). With thenecessarymodifications ofnotation we find asstated in 27.Thegraphofthe initialvelocity,andthe approximation attained bytakingthe firsteightterms ofthe series(3),areshewn inFig.36. Itwillbenoticed that ourapproximationhaseven an advantageover theresult obtained bycarryingtheseries to infinity. Inthelatter case, the initialvelocity,asrepresented by(1),isdiscontinuous when x=0,beingzero for=0,but equalto4/3c/when xdiffers ever solittle from 0.The idealized representationofthemotion in 27 isinthis respect imperfect ;theparabolainFig.32should beslightly modified soastotouch the lineABatitsextremities. 38. String Excited byImpact. Asafinal example wetakethecaseofastringstartedby animpact,asin 26.Webeginwith thecase ofaforce 102 DYNAMICAL THEORY OFSOUND distributed continuouslyinspaceand intime, thedifferential equation being 92y__ 2&y,Y /i\ W 9#2p' asin28.Suppose,inthe firstplace,that -=/ 1(*)sin^+/a(Osin- 7-+...+/m(Osin-T- >(2) pt v & thecoefficients beingknown functions of t.Theequation (1) isthen satisfied by .7TSD .2-TttE Tn/JTOD /ON .-y+97 2sinr-+ ...+^,,iSin j,...(.o; L b v ij provided/.,A=/ s(0/d\ (4) Thesolution ofthisequationhasbeengivenin 8.Ifwe assume that%=0,%=for t= oo,andthat/ s(i)issensible onlyforafiniterangeoft,theresultingvalue ofv)sis I .STTCt riB sm Tls S-JTC LSTTCtfg(t)cos u I STTCt STTCt ifasaparticular caseweput 7T.(6) wehave 57TCsinSTTCt by8(18). Asafunction ofi,Fnow follows thespecial law indicatedbythelastfactor in(6),atevery pointofthestring, butwehave notyetmadeanyspecial assumptionastothe distribution oftheforce over thelength.Itstime-integralis given by Ydt_ sin+0.sin*f+...+0,sin?! .(8) Wemaynowseek todetermine thevalues ofthecoefficients sothat thisexpression maybesensibleonlyintheneighbour- hood ofthepointx-a.Weassume, then, that 1f00 .., FOUBIEB'S THEOREM 103 where</>(a;)vanishesexcept between thelimits a-eanda+e, say.Theformula(7)of36thengives (,)sin^(10) Ifebesmall, the series thus obtainedconvergesatfirstvery slowly, andagreatmany termsmight have tobetaken to secure areasonableapproximation. Intheterms oflower orderwehave sin & &Ja- piU nearly, where ^=pJ' ^(n^dx,..................(12)Jae i.e./Arepresents thetotalimpulse. Thecorrespondingterm, in thevalue(3)ofyis 2yW1_<,-p-n STTCL .STTX .sirct /10. ~r .-eSVC7/i .sin-j-.sm =-.sm = . ...(13)pvrcs III' Buthowever small emay be,solongasitisnotevanescent, thevalue ofG8given by(10)willultimatelytend tozerowith increasing s,owingtothemore andmorecomplete mutual destruction ofpositive andnegativeelements under theintegral sign. This shews theeffect ofdiffusingtheimpulseover a small butfiniteportionofthestring. The case ofaninstantaneous localimpulseisobtained by puttingT=(cf. 26). 39.General Theory ofNormal Functions. Har- monic Analysis. Thespace which hasbeen devoted toFourier's theorem is nomorethan iswarrantedbyitsimportance, especiallyinrela- tion tothetheoryofstrings,but itiswell toremember that from thestandpointofthetheoryofvibrations thetheorem isonly oneoutofaninfinite number which canbebased onthesame kind ofphysicalconsiderations.Every vibratory systemhas its own series of"normal functions," astheyare called, which expresstheconfigurationofthesysteminthevarious normal modes. Inthecase ofauniformstring,orofthedoubly-open organ pipe,these functions happentohave thesimpleform 104 DYNAMICAL THEOEY OPSOUND sin(sTTX/l),orcos(sirsc/l), respectively. Morecomplicatedforms willbemetwithwhen wecome tothetheoryoftransverse vibrations ofbars,and tothat ofmembranes;andeven inthe casesjustmentioned thesimplicityoftypewould atonce disappeariftheuniformityofline-density,orofcross-section, respectively,weredepartedfrom. Insomeproblems, indeed, ofconsiderable interest, e.g.thatofthevibrations ofarectangular plate,thepreciseform ofthefunctions has still tobediscovered. But inanycase thefunctionstheoreticallyexist;andonthe principlethatanyfreemotion whatever ofthesystemconsists ofsome combination orother ofthevarious normal modes, it must bepossibletoexpress anyarbitraryinitial state, and thereforeanyarbitraryfunction ofpositioninthesystem, bya series ofnormal functions. Such preeminence asattaches to Fourier's theoremis,from thepresent pointofview,duemerely tothefactthat initwehave thesimplest exemplificationof thisprincipleinthecase ofacontinuoussystem, andtheone where thephysical induction hasbeenmostfullycorroborated byindependent mathematicalproof.Itmayalsobeadded that itisonlyinthecase ofstringsthat thecalculation ofthe effect ofparticularinitial conditions hasanygreatinterest. There ishowever anotherpointofview from which the resolution ofafunction intoaseries ofsines orcosines ofthe variable isofpeculiar importance,viz.whenwearedealingwith functions ofthe time. Thedynamical reason for thishas already been dweltupon (19). When afunctionf(t)isknown tobeperiodic, ofperiod r, itsresolutionbyFourier's theorem is /()=A.+A,cos +A.2cos +A,cos?I* +>sn + zsin +5,ain +...,(1) T T 7-t \/ where At=j*f(t)dt, .....................(2) whilst fors>0, FOURIER'S THEOREM 105 This isofcoursemerelyarestatement ofthetheorem of 34,with thenecessary changesofnotation. Itwillbenoticed thatAQrepresents themean value ofthefunction. Wehavealready been ledtoformulae ofthetype (1)as expressingthemotion atanyassigned pointofafreelyvibrat- ingstring,theperiod rbeing equalto2Z/c. Anotherimportant acoustical application istotheanalysisofaperiodic current of air,asinthesiren, orthereed-stopsofanorgan (90). Again, inthecaseofeloctromagneticallydriventuning forks, aperiodic current canpowerfully excite, notonlyafork inunison with itself,butalsoothers whose naturalfrequenciesarerespectively, twice, three times, ...asgreat.This isduetothefactthatthe disturbingforce isofthetype (1),theselective resonancetaking place accordingtotheprinciplesof 9. Various mechanical contrivances forresolvingagiven periodiccurve into itssimple-harmonic constituents, and converselyforcompoundinganumber ofindependentsine- aridcosine-curves whoseperiodsareas1,%,$, ...,have been devised byLord Jelvin and others. From thestandpointof thepresent subject themost remarkable ofthese isperhaps themachine constructed byProf. A.A.Michelson, inwhich provisioniamade forasmanyas80constituents*. Itishardly necessarytosayexplicitlythat theresolution ofaperiodicfunction oftintheform(1)canonlybeeffected inoneway,thevalues ofthecoefficients asgiven by(2)and(3) beingdeterminate. Inparticular,aseries oftheabovetype cannot vanish for allvalues oftunless itscoefficientsseverally vanish. Thus hiafreely vibrating string,ifthemotion atany given pointxbeprevented,asbytouchingwith acamel-hair pencil,thecoefficients ofcos(s7rc/J)and sin(sTret/I}inthe generalformula (7)of25must bezero,i.e.wemust have . .STTX -D.STTXf., A.esu\-j-=Q,J3asm-y-=(4) forallvalues of s.Unless scbecommensurable with Ithis requiresthatA8=0,Ba=0,and thewholestringwillbe *Phil. May. (5),vol.XLV.(1898);thispapercontains anumber ofmost interesting examplesofresults obtained. Tlieconstruction isalsoexplainedin hisbookOuLight Waves and their Uses, Chicago, 1903. 106 DYNAMICAL THEOEY OFSOUND reduced torest. Intheexcepted case theconditions(4)are satisfiedindependentlyofthevalues ofAsandBgwhenever sin(STTX/I)=0,i.e.those normal modes remain unaffected which have anode atthepoint touched. Aquestion arises astothe effect ofnon-periodicforces on adynamical system. Forthereasonalreadysooften insisted upon,itisconvenient, wheneverpossible,toresolve theforce intoaseries ofterms ofthetype Acospt+Bsinpt ...................(5) Each element AcosptorBsinptthenproduces throughout thesystemitsown effect, viz.anoscillation ofthesametype andperiod, theconfiguration and itsamplitude dependingon thespeed p.Insome cases theresolutionpresentsitselfquite naturally,asforexampleinthetheoryofthe tides. The disturbingeffect ofthesunandmoon, when account istaken oftheirvarying declinations, andoftheinequalitiesintheir orbital motions, canbesufficiently represented byaseries of terms ofthetype (5). Itfollows thatthetide-heightatany particular placemust beexpressed byaseries oflikecharacter, inwhich thevalues ofpareknown. The theoretical determina- tion ofthecoefficients isoutofthequestionfortheactual ocean, with itsvariabledepth andirregular boundaries, but their values canbeinferred aposteriori withmore orless accuracy from acomparisonoftheformula with observation, andwhen once ascertained canbeused forprediction*. When thedisturbingforce isperfectly arbitraryincharacter withoutanyobviousperiodic elements, thequestionismore complicated. There isaform ofFourier's theoremspecially appropriatetothis case,but itsapplicationisusually difficult, and itissimplertohave recourse, asin38,totheformula(12) of 8.Theobjection that thisimplies aknowledgeofthe wholeprevious historyofthesystemismet ifweintroduce theconsideration ofdamping, which isinreality always present. Theequation *Foranelementary account ofthematter seeSirG.H.Darwin, TheTides, London, 1898. maybewritten '*= '/(*), providedn'2=w2-^2...................... (8) Hence, bytheformula referred to, 'tfe J=-,e~^ktsnnt e cosnnJx' i r ,e*cosn'ile* f(t*}sinn'tdt. -.-(9)n JJxy v' Ifwehavex=0,cc=for t= oo,thelimits ofintegration are oand t.For instance, thevalue of acwhen it= becomes .=_!Pe^fsinn'tdt(10) 00 Owingtothepresenceoftheexponentialfactor itisonlyfor acertain rangeofnegativevalues oftthat thefunction under theintegral signhasasaruleanappreciablevalue. Inother words, theeffects oftheaction oftheforcepriortoacertain antecedent epochhavepractically disappeared. CHAPTER iy BABS 40.*+*,*!.vwy "j.7ieory 'arynotions from ors"i , ionofabody in'astLT ?^WM >the- aThis issuffieLl dLe/bfr""hmo'" two lines inthesubah,J.7- iyProperty thatany P-He, regain6 ,;;ta^plt, w" ,H%"^^^ d relative toother line? in ti? v'Ugh their directi Aparallel^ ttrefoet"brt"'Be"USUa"^alterei eas% folj, that thXr^y/T11610^".^ it lines are alfcered in,,,!nlteParallel'stght -ually bedifee^fed^^:*his -"howefer Itcanbeshethat tldlrect 'mthesubstance. directions inthesuttnceT ,^""""^ PerPdicuIar larafter thedforn" onWT"""" mUtUa"^*"** ofthestrai? Ttt"'""caUed the"P^ipJ the formal^^'^-cessary, forOUrpurposes/to inevidence inSeX1Ste "Ce fsuch **be fromS' tportion ofthesubstance LTf/Dy rigina"y areinthedfrecti of theoriginallength, viz.increase oflengt BAES 109 iscalled the"extension"; itwill ingeneral bedifferent for different directions ofPQ. Inthetheoryofelastic solids, eis alwaysaveryminute fraction. Wedenotebye1}e2,e3,the extensions inthedirections oftheprincipalaxes. Theratio oftheincrease ofvolume totheoriginal volume iscalled the"dilatation." DenotingitbyA,andconsidering thechangeofvolume ofacubical block whoseedgesarealong theprincipal axes,wehave or(2) theproductsofsmallquantities being neglected. There aretwospecial typesofhomogeneousstrain which requirenotice. First, supposeex=e2=e3,=e,say.Any origin- ally spherical portionofthesub- stance then remainsspherical, and theextension istherefore thesame inalldirections. The strain may accordinglybedescribed asoneof uniform extension; andwenote that e=A. Again,takethecaseofej= e2, =e,say,whilst es=0,andtherefore A=0.Asquare whosediagonals AOC,BOD areparalleltotheaxes 1,2isconverted intoarhombus A'B'G'D', andsince (l+e)2-f-(l-e)2=2, tothe first order, thelengthsofthesides areunaltered. Also tanA'B'O= =tan(TT+e), .(3) sothattheanglesoftherhombus are vr+2e.Another view ofthisstate ofstrain isobtained ifweimaginetherhombus A'B'G'D' tobemoved initsownplanesothatA'B' coincides withAB. This islegitimate,since nodisplacementofthe bodyasawhole affects thequestion. Wethen seethatany twoplanesofthesubstanceparalleltoABandtheaxis3are displaced relativelytooneanother, withoutchangeofmutual noDYNAMICAI, THEORY OPSOUND ~" ' axis3,inthe thesecond halfofthefigure. ccThis ,' D'Om Kg. 39. 41.Stresses. theportions ofmatterWearehereconcerned anexceeding!,^ question areconfined tr, t ^P common bounda islfVh"f88*"^ ofeithor stratunl mfy^~"^^". portional to itsOTeafJ"the ntensi aeoordmgly specified bytheforco ^. maybeofthenature either Ofanormal oroblioue, oreven*>.* -?^m&^J'between adjacent *S ther side. ofmatter in trata, whose-portion 8treSS ""' sassumed tobeuniform inthedirection ofthenormal;butforareasonalreadyindicated weneed notstoptoprovethistheorem. Theplanesinquestion arecalled the" principal planes"ofthestress, andthecorre- spondingstress-intensities arecalled the"principalstresses." Theyareusuallyreckoned aspositive when ofthenature of tensions; wedenote them bypi,p t,pa. There arecertainspecial typesofstress tobenoticed. First letpi=p2=p3.The stress acrossevery planeisthen in thedirection ofthenormal, andofuniformintensity,asin hydrostatics. Next,letpi*=pa,=p, say,whilst p3=Q.Consider aunit cubewhose faces areparalleltotheprincipal planes. The portionincluded between thefacesrepresented byAB,DA, in thefigure,andthediagonal plane represented byBD, isin equilibriumunder three forces. Two ofthese forces are parallelandproportionaltoDAandAB, viz.theforces onAB andDA, respectively. The third force isthereforealongand Fig.40. Fig. 41. proportionaltoBD\ arid itsamount(-DT)perunit area isp. Asimilar result holds withrespecttothediagonal plane AC. Acube fourofwhoso faces arcparalleltothesediagonal planes isinequilibriumundertangential stresses, inthemanner shewn. Thistype-,isaccordinglycalled a"shearingstress." Itsamount(CT)inspecified bythetangentialforceperunit areaontheplanesinquestion. 112 DYNAMICAL THUUlti ui<KVU^IJ Ingeneralthestates ofstrain and stress inabodyarenot uniform, or"homogeneous,"butvary continuously frompoint topoint;buttheabove notions are stillapplicabletothe infinitelysmall elements intowhich thebodymaybeconceived tobedivided. 42. Elastic Constants. Potential Energy ofDefor- mation. Thetheoryofstrains isamatter ofpuregeometry;that of stresses oneofpurestatics. Whenwecome toconnect thetwo werequiresomephysical assumption.The usualhypothesis, known as"Hooke's law,"*isthat the stresses arelinear functions ofthestrains. Thislawceases tohold, evenapproxi- mately, when the strains exceed certain values called the "elastic limits"; but forthepurposesofacoustics itmaybe adoptedwithout hesitation, onaccount oftheexcessive minute- ness ofthevaryingstrains withwhich weareconcerned"f. Inan" isotropic"substance, i.e.oneinwhich there isno distinction ofpropertiesbetween onedirection andanother, the principalaxes ofstrain mustevidentlycoincide with those of stress. Moreover theprincipalstress pzmust involve the principalstrains ez, 3symmetrically,andsoon.Themost general assumptionconsistent with thisrequirement, andwith Hooke's law, isoftheform Pi-X(ej.+ea+3)+2/tiej,' p2=A,(ei+e2+e,)+2/A6 2,-(1) ps=X(e!+eaHea)+2pe a, , whereA,,/*areconstantsdependingonthenature ofthe material^. Itwillbenoticed that ej,e2)esarepureratios and thatthedimensions of\, /j,aretherefore those ofstress, orforce *Eobert Hooke(16351703), professor ofgeometry atGresham College 16651703. tIfHooke's lawweresensibly departed from, thefrequencies ofthenormal modes ofavibrating barwould nolonger beindependent oftheamplitude. Since theearisvery sensitive tovariation ofpitch, thiswouldeasily bedetected. Thisremark isduetoStokes. tThere isagreat diversity ofnotation asregards these constants. The above symbols arethose introduced byG.Lame"(17951870), professor of physics attheEcolepolytechnique 1832 44. There arevarious combinations oftheconstants X,pwhich important inphysics, aswell asintechnical mechanics. In liform dilatation wehavepl=pa=p3(=p, say),e1=e2=es J-A),whence p=(X+^)A...................... (2) aceifwewrite/c=X+f^)........................ (3) illdenote the" volume-elasticity"or"cubicalelasticity"of substance, i.e.theratio oftheuniform stress tothedilatation ichitinvolves. Next suppose that ej=-e2=e,es=0,and therefore =-pz=p, ps Q,which isthe case ofapure shear, olvingashearingstress. Accordingtotheinvestigations 40,41theshearingstress is-ar=p,andtheshear is 2e.Hence, from(1),&pi),........................... (4) fjudenotes the ratio ofthestress tothestrain (appropriately :asured) inapureshear. Itiscalled the"rigidity"ofthe Dstance. Again, suppose wehaveabarstretched lengthways,butfree mlateral stress. Weput, then, in(1),p2-0,ps=0.This ,dsto =J&a,........................... (5) (6)X+JU. +//. iisratio ofthelongitudinalstress tothecorresponding tension iscalled "Young'smodulus" ofelasticity;its clinical importanceisobvious. We also find 2=63=-0-eu ........................ 0) Xa= (8) .iisfraction accordinglymeasures theratio oflateral con- action tolongitudinalextension under thecircumstances pposcd;itisknown as"Poisson's ratio."* *S.D.Poissou (17811840).Hischief contributions toacoustics relate to ovibrations ofmembranes and plates, and tothegeneral theory ofsound- ivosinair. 114 DYNAMICAL THBOEY OPSOUND Bysolvingtheequations (1)wecanexpress e,,ez,esasline functions ofp1}p2,p,.Itisobvious, however, thattheformu for e,must involve p*andpssymmetrically;andfrom th consideration, andfromthephysical meaningsoftheconstan Eand <r,itfollows immediatelythat theresult must 1 equivalentto .(9) Ofthevarious elastic constants and theircombination oneorothermayappear specially important, accordingtoth nature ofthequestioninview,and thismayaccount forth great diversityofnotations which hasarisen. Inanycasetw independent quantitiesarenecessaryand sufficient todefine th elastic behaviour ofanisotropicsubstance. From aphysics standpointKandpmight appeartobethemostfundamental whilst asregards facilityofdirect measurement preference ma; begiventoEandp,whence Kand <rcanbederived byth formulae which followeasily from(3),(6)and(7).Onaparticula hypothesisastotheultimate structure ofanelastic solic Poisson wasledtotheconclusion thatthetwo elastic constant; arenotindependent, butareconnected byaninvariable relation which inournotation isexpressed byA.= //..Thismakes *=$A*E=%n ><r=J................ (11) Onexperimental grounds Wertheim (1848) proposedthe relation X=2 yu,,which makes *= /*,E=K, <r=!................(12) More accurate methods ofmeasurement, introduced bj Kirchhoff*andothers, support theview,which hasbeen con- sistently heldbyEnglish physicistsf that there isnonecessary *^*?'^JS"***(18M-87)'Professor ofphysics atHeidelberg 1854-75,atBerlin 1875-87; famous forhisshare inthediscovery ofspectrum analyeis,battheauthor also ofimportant memoirs onthetheory ofelasticity and itiapplications tothevibrations ofbarsandplateetNotably byGreen(1798-1841), StokeB, andLord Kelvin. Definite relation "between Xandp,andconsequentlynouniversal value of cr.Wemaynote that inanabsolutely incompressible Medium weshould have *=oo, E=3fj,,o-=i(13) Thefollowingtablegivestheresults ofafewdeterminations "byEverett(1867). The second columngivesthevolume- densityingrammes percubic centimetre. Thenext three columnsgivetherespectiveelastic constants, indynes per square centimetre. These arefollowed inthelastcolumn by thecorrespondingvalues ofcr.The lasttworows illustrate the fact thattheelastic constants mayvaryappreciablyindifferent specimens ofnominallythesame substance. For technicalpurposesthe elastic constants E,K,/*are oftenexpressedingravitation measure, e.g.ingrammes per squarecentimetre. Thecorresponding numbers intheabove table arethen divided byg.Another mode ofspecification, employed byYoung,isinterms ofthelengthofabarofthe particular substance, whoseweight perunitareaofcross-section would beequaltothemodulus inquestion whenexpressedin gravitation measure; this iscalled the"length-modulus." Thus ifLbethelength-modulusofextension ofabarfreetocontract laterally wehave E=gpL (14) Taking g=981,theabove tablegives,inthecase ofsteel,L=278x106centimetres, 116 DYNAMICAL THEOliY OFSOUND Thepotential energy (W)perunitvolume ofastrained isotropicsubstance maybefound bycalculatingthework done bythestresses onthefaces ofaunit cube, onthehypothesis thatthestrains increase from zero totheir final valueskeeping theirmutual ratiosunchanged.Theaveragestresses arethen one-half thefinal stresses. Thus inthecase ofauniform dilatation Awehave Inthecase ofapureshearvj,(15) Intheextension ofabar,with freedom oflateral con- traction, W=lp&=W................... (17) Inthegeneralcasewehave W=i(PA+p.2ez4-ps3) f-|fji{(e2-e3)2+(e,-e,)+(e,-ea)*J....(18) This shews that inorder that thepotential energy maybe aminimum intheunstrained state Kand/j,must bepositive. Itisotherwise obvious from themeaningofthesymbolsthat ifeither ofthese werenegativetheunstrained statewould be unstable. 43. Longitudinal Vibrations ofBars. Wetake theaxis ofxalongthebar,anddenote byx+% thepositionattime tofthat cross-section whose undisturbed positionisx,sothat denotes thedisplacement. Anelement of lengthisthen altered from &xtoB(sc+),or(1+')Sx,where theaccent denotes differentiation withrespecttoao.Equating thisto(1+e)Sac,wehave -|a) The tension across thesectional area(&>)istherefore Eew. The acceleration ofmomentum ofthemass included between thetwocross-sections corresDondinsr toocandx+8aisDU>X . . "- Ifthesection beuniform, thisreduces to where d*=E]p........................... (4) Itwillbenoticed that inthisinvestigationitisnot necessarytoassume thesubstance ofthebartobeisotropic, providedthepropervalue oftheYoung'smodulus betaken*. The result isalsounaffected ifthebar,orwire,besubject toapermanent longitudinal tension, since byHooke's law thestress due totheextension (1)maybesuperposedonthe permanent tension, solongasthelimits ofperfect elasticity arenottransgressed. Asin23thegeneralsolution of(3)is %=f(ct-a:) +F(ct+x))............... (5) representingtwowave-systems travelling unchangedinopposite directions with thevelocitycgiven by(4).Interms ofthe length-modulus,wehaveby42(14) this isthevelocity duetoafallfrom restthroughaheight |L. Some numerical values ofcaregiveninthelastcolumn ofthe table onp.119. Theapplicationtoparticular problems maybetreated very briefly. The various cases that arise presentthemselves in amore interestingformwhenwecome tothevibrations of columns ofair. Inthecase ofarodorwire fixed atboth ends,wehave = forx-andx=I(say) ;andthemathematical theory *Inan'aeolotropic"orcrystallinesolid thevalues ofEwillbedifferent forbars cutindifferent directions from thesubstance. H8 DYNAMICAL! TiiliUltY O*'HUUJND isexactlythesame asinthecaseofthetransverse vibrations of astring. Thefrequenciesofthevarious modes aregiven byN=sc/2l,........................ (7) where s=l, 2,3,....The result isunaffectedbypermanent tension inthewire. When therod isfree,thecondition ofzero stress atthe endsgivesf=for ocandx=l.Introducingthiscondition in(5)wefind F'(ct)=f'(c), F'(ct+l)=f(ct-l),......(8) forallvalues oft.Theformer ofthesegivesonintegration F(ct)=f(ct),..................... (9) noexplicitadditive constantbeing necessarysince itmaybe supposedincluded inthevalue off(ct}. Thesecond relation thengives f(ct+l)=f(ct-l) +C................ (10) Theconstant isconnected with thetotalmomentum ofthe barinthedirection ofitslength. Wehave, from(9)and(10), fdx=c ({f(ct-x)+f\ct +x)}dx=cG....(11)JO JO Sincenothingessential isaltered ifwesuperpose anyuniform velocityinthedirection ofthelength, wemayassume the mass-centre tobeatrest, inwhich caseC=0.Theformula (10)then shews that theresidual motion isperiodic,since everythingrecurs when tincreases by21/& Intheanalytical processforascertainingthenormal modes weassume thatfvaries ascos(nt+e),whence and %=(Acos~ +Bsm}cos(nt +).......... (13)V o G)' ^ ' Theconditions thatd%/dx~for a?=andx-IrequireB=0. sin(nljc)=0,whence nl/c-STr, ........................(14) where 5=0,1,2,3,...,thescale ofperiods being harmonic. Thenodes(=0)aregiven bycos(STTX/I)=0,andtheloops, orplacesofzero stress, bysin(s-rrx/l)=0.Inthegravestmode (s1)wehave anode atthecentre*. Ontheprinciples explainedin 16,32themost general freemotion ofthebar,under thepresentconditions, may be expressed byaseries * "? A7rT> S7Tc rr ,-,,-\ f=S (.4.cos --+5 ssm-^ Jcos-j-,...... (15) where s=0,1,2,3,....Thus ifthebarbestarted from rest inthestate ofstrain defined by< -/(*) [<-0],.................. (16) wehaves=0;andweinfer that itmust bepossibleto determine thecoefficient A8sothat .(17) forvalues ofxrangingfrom to I.This istheresult referred tobyanticipationin33. Thelongitudinal vibrations ofbars orwires have hardly anypractical applicationofimportance, exceptinsome primitive forms oftelephone. Asregards bars, thepitchisvery high compared with that ofthetransverse vibrations, which ibis difficult toavoidexciting simultaneously. Again,ifwecompare thefrequenciesoflongitudinal vibration ofatense wire wibh those ofthecorresponding transverse modes, theratio willbo that ofthewave-velocities, i.e.of>J(Elp)to^(P/pco), whereP denotes thepermanent tension. Ifebetheextension duo toP,wehaveP=^.&>,andtheratio isl/Ve,which is usually very great f.Longitudinal vibrations maybeelicited onthemonochordbyrubbingthewirelengthwise withapiece ofleathersprinkled with resin; theresulting note isvery shrill. Itisassumed inthepreceding theory that theextension andtheaccompanyingstress areatanyinstant uniform over thecross-section; inother words, wehaveassumed that the *The case s=needs, instrictness, separate examination. Itloads to=/l(l+ai),which may beinterpreted asanoscillation ofinfiuik-ly long period.Ifthemass-centre beatrestwehaveAoQ. +ThiscomparisonisduetoPoisson(1828). 120 DYNAMICAL THEORY OFSOUND lateral contractionadjustsitself instantaneously throughthe thickness. This isnotquite exact, asthere isacertaindegree oflateral inertia, buttheerror isinsignificantsolongasthe wave-lengthislarge comparedwith thediameter. Inthe modes ofveryhighorder itmightbecome sensible, but these areinanycase ofnoimportancefrom thepointofview ofacoustics. Acorrection hasbeeninvestigated byLord Rayleigh. 44.Plane Waves inanElastic Medium. Thetheoryofplane waves inanunlimitedisotropicelastic medium issoclosely analogoustothat oflongitudinalwaves inarodthat itmaybebrieflynoticed here. Itisassumed that thestate ofthingsisatanyinstant uniform overany plane perpendiculartothedirection ofpropagation (#). Such wavesmaybeoftwotypes, which aredistinguished as"dilatational"or" longitudinal,"and"distortional"or "transversal," respectively. Intheformer class thedisplace- ment iswhollyinthedirection ofpropagation. Denotingit byf,wehave, inthenotation of42, ea=df-fdac,e2=0, e3=0, andthereforep^=(X+2/i) e^(K+f^)9/9#.......... (1) Consideringtheportionofmattercorrespondingtounit area ofastratum ofthickness Bx,wehave whence =or ......................... (2) if d?=(K+$(i)fp (3) Some numerical values ofthewave-velocity aaregivenon thenextpage, and itwillbeobserved thattheyareinall cases greaterthan thecorrespondingvalues ofc,aswastobe expected,since thepotential energy duetoagivenextension 9f/9#isnowgreater owingtotheabsence oflateralyielding. Inthesecondtypeofplane waves thedisplacementis everywhereatright anglestothedirection ofpropagation. Itmayberesolved intotwocomponents paralleltoyandz respectively,which maybetreatedseparately. Considering theformer component (77)alone, weseethat thestrain at anypointconsists inashear ofamountdijjdx. Theconsequent stress across anyplane perpendiculartoOx isparalleltoOy, and itsintensityisfjidrj/das. Henceformingtheequationof motion ofaportionofmatter defined asbefore wehave or.(4) .(5) Some values ofthewave-velocitybaretabulated below. Wave-velocities(metres per second). Itmaybeshewn thatanylocal disturbance inanunlimited elastic medium breaks upintotwowaves, divergingwith the velocities aand I,which tend ultimatelytoassume the " longitudinal"and"transverse" characters, respectively. The theoryishistorically importantinrelation toOptics, but inourpresent subject greatcautioi] isnecessaryin drawinginferences astothepropagationofwaves inlimited solids. Wehave alreadyseenthat inacylindricalorprismatic rodthevelocityoflongitudinalwaves isquitedistinct from a, andthetheorybecomes altogetherdifferent inthecase of flexural vibrations, tobereferred topresently.Inthese cases amodification wasofcourse tobeexpected,since thewave- lengthisunderstood tobelarge comparedwith thedimensions 122 DYNAMICAL THEOEY OFSOUND ofthecross-section. Buteven intheother extreme, when all thedimensions ofthebodyarelargecomparedwith thewave- length,thecircumstances maybeprofoundlymodified bythe existence ofafreeboundary. Anewtypeofwaves, called after thediscoverer the"Rayleigh waves"(1885), make their appearance, andunder some conditions maybecome, from the observationalpointofview, predominant. These aresurface waves inwhich theagitation penetrates onlytoarelatively smalldepth. Theirvelocityissomewhat lessthan that of the distortional waves; thus foranincompressiblesolid it is'9554b,whilst onPoisson'shypothesis (o-=i)itis'91946. Inmodern observations ofthetremors due todistant earth- quakes threephasesofthedisturbance areoftenrecognized The first isinterpretedasdue tothearrival ofthedilatational waves, propagated directly throughthesubstance oftheearth, thesecond asdue tothat ofthe distortional waves, also propagated directly, andthethird tothat oftheRayleigh waves, which have travelled overthesurface andaretherefore delayed more than inproportiontothedifference ofwave- velocity*. The latter waves asthey spreadover thesurface arelessattenuated than theformer, whichdivergeinthree dimensions. Ithasevenbeenattemptedtodeduce estimates ofthevolume-elasticity andrigidityofthematerials ofthe earth from thevarious wave-velocities, asinferred from the seismic records"J". 45. Flexural Vibrations ofaBar. Weproceedtothe transverse vibrations ofabarnaturally straight. Toavoidunnecessary complications wewillsupposethat thebarhasa^-^ longitudinal planeofsymmetry, andM } that theflexure takesplace parallel-t-S tothisplane. We will alsoassume forthepresentthat thetotallongi- tudinal stress onanysection iszero.j Theresultant stress atasection there-Fig. 42. *R.D.Oldham, Phil. TYcms. A,1900. tProf. A.E.H.Love, Phil. 2Vans. A,vol.ccvn., p.215(1908). forereduces toatransverse" shearingforce" F>andacouple or"bending moment"M.These willbefunctions ofas,the longitudinalcoordinate. Iftjdenote thelateraldisplacement, paralleltotheplaneofsymmetry, then, resolving transversally theforcesactingonanelement 8%ofthelength, wehave pco8# .tf=8F, Again,if K,denote theradius ofgyrationofthearea ofthe cross-section GOabout anaxisthroughitscentre ofgravity, normal totheplaneofflexure, theelement ofmass is ultimatelyadisk ofareato,thicknessSac,andmoment of inertia pwBx.K?*.Since theaxisofthisdiskhasbeenturned throughasmallangle dij/dx from thepositionofequilibrium, theequationofangular motion is , , ._,.whence,w-_L=_+jr. ..................(2) Tfweeliminate Fbetween(1)and(2)wehave providedthesectional area cobeuniform. Wehave next toexpressMinterms ofthedeformation ofthe bar. Consider inthe first instance thecase ofabar uniformly bent, sothat itsaxisbecomes anarcofacircle. Ibisevident fromsymmetrythat theshearingforceFnow vanishes, and ithardly needs calculation toshew that the strain inanypartofthecross-section willbeproportional to thecurvature. Hence byHooke's lawtheresultantcouple717 willalsovaryasthecurvature, or M=3&/R,........................(4) whereRistheradius ofcurvature, and 33isaconstant dependingontheshapeand sizeofthecross-section, andon the elasticpropertiesofthematerial. *Thesymbol Kisnotrequiredatpresent initsformer sense asnnclastic constant. 124 DYNAMICAL THEOEY OFSOTJNX> Thevalue of33isfound asfollows.Wetake rectangula axes Gy,Gzintheplaneofa section, theorigin beingatthe centre(i.e.thecentre ofgravity ofthearea),andtheaxis of* normal totheplaneofflexure. Assumingtheaxis ofthebar, i.e.thelinethroughthecentres ofthe sections, tobeunex- tended, weseethat ifRdenote theradius ofthe circle into which itisbent, thelength ofalongitudinallinear element whose distance from theplane xzisyisaltered intheratio ofR+ytoR,and that the extension isaccordingly yJR. Thecorrespondingstressper unitareaofthesection isEyjR, whereEistheappropriate Young's modulus- longitudinaltension istherefore Eff=-~\\ydydz~Q.-LitJJ Thisjustifies theprovisional assumption that the axis (as abovedenned)isonthepresent hypothesis unextended. For thebending moment wehave, taking moments about Gz, (5) Except inthespecial casejustconsidered, viz.that) ofabar bentstaticallyintoanarcofuniform curvature, there willbo ashearingofcross-sections relative tooneanother, and also awarpingofthesections sothatthese donotremainaccurately plane. Anexactinvestigation isoutofthequestion,but enoughisunderstood ofthematter towarrant thestatement thattheadditional strains thus introduced areasarule small compared with those taken account ofinthepreceding calcula- tion. \\ethereforeadopt theformula(5)assufficiently ......................(6) Substitutingin(3)weobtain P^ Formostpurposesthisequation maybesimplified bythe omission ofthesecond term, asweshall seeimmediately. Thekineticenergyofthebar is <fe ......(8) Thesecond term, whichrepresentstheenergyofrotation ofthe elements,isusually negligible. Thepotential energyisfound, inaccordance with 42(16), byintegratingtheexpression ^Ef?,=$Eyz/R2 ,firstover the area ofthecross-section, andthen over thelength;thus (9) Consider foramoment thepropagationofasystemofwaves ofsimple-harmonic profile alonganunlimited rod,assuming 77=Ccosk(ct os) (10) Since everythinghere recurs whenever ccisincreased by2-7T/&, theconstant kisconnected with thewave-lengthA,bythe relation &=27T/X (11) Onsubstitution theequation (7)isfound tobesatisfied provided I,"2p1K-Ka ^1+k^K p Thisgivesthewave-velocity c,which isseennottobeadefinite quantityfixedbytheconstitution oftherod,buttodependalso onthewave-length.Totrace theprogressofawave ofany typeother than (10),itwould benecessarytoresolve thewave- form intosimple-harmonicfunctions ofx.Each ofthese would travel with itsownvelocity,sothat theresultantwave-profile wouldcontinuallyalter. For thisreason itwould behopeless tolook forageneralsolution of(7),oreven ofthemodified form(13)below, ofthesamesimplecharacter thatwemet with inthetheoryofstrings (23),andagaininthat ofthe longitudinalvibrations ofrods. Afurther remark isthatwhen wesubstitute from(10) in(7),thesecond term isoftheorder JC-K*ascomparedwith the first.When thewave-lengthislarge comparedwith the dimensions ofthecross-section this isaverysmallquantity, andtheterm inquestion,which arosethrough takingaccount oftherotatoryinertia oftheelements ofthe bar, viz. in equation (2),maybeneglected.Itiseasytosee,and itmay beverified aposteriori, thatthesamesimplificationislegitimate indiscussingthevibrations ofafinite bar,atallevents solong asthedistance between successive nodes islarge compared with K..Weaccordinglytake theequation * - asthebasis ofoursubsequent work, togetherwith theformulae 27?-dM 46.Free-Free Bar. Toascertain thenormal modes ofafinite barweassume as usual that??varies ascos(?i+e). Theequation (13)ofthe precedingsection then reduces to where m4=nsp/KiE.........................(2) Itistobenoted thatmisofthenature ofthereciprocalof aline. Thesolution of(1)is t]=Acoshmx+Bsinhmx+Gcosiiix+Dsinmx, (3) thetime-factorbeingforthepresentomitted. Thethree ratios A:JB:0:D, andtheadmissible values ofm,andthence ofn2 , arefixedbythefourterminal conditions, viz.two foreach end. Take firstthecase ofaperfectlyfree bar,oflengthI,say. Ifwetake theoriginatthemiddle*, these conditions are,by 45(14), V'=0, 17"'=[>=i/] (4) Thenormal modes fallnaturallyintotwoclasses; inoneof theseV)isaneven, intheother anoddfunction ofx.Forthe symmetrical vibrations wehave r)=Acoshmx+cosmac, (5) with theterminal conditions Acosh\ml~G cos\ml=0,] Asuih^ml +Csin^ml = whence tanhfanl=-tan|ml(7) Fig. 44. The roots ofthisequationareeasily foundapproximately by graphical construction, viz.astheabscissae oftheintersections ofthecurvesy=tana>,y=-tanh, the latter ofwhich is Thisimprovement ontheordinary procedure isduetoSirA.G.Greeuhill,Mess, ofMath. vol. xvi., p.115(1886). 128DYNAMICAL THEOBY OPSOUND asymptotic tothelinev=-1Th* fi aroximatelllie%u^shews thatwehareapproximately where5=1 9Q j thefrequent ViLaS1S8maIL Jtfollows **>(2)thati^iy~^rf,y 6Xacfc co^tation oftherootswehave"" & ,(9) where~e fr~ ......................(10)Hence,=tan^(^ e-28==.-2a3__ 3_6a lntheymmetric modes wehave ...A,^^^sinh withtheterminalconditions whence m=s+1) where s=l33 ri /o frequencies are'an..... COTOSP to tanA=r^ _1-tanh\ml _ml ~ m s(16) Where Ss=e~^-^ TT......................(17)Hesoe A.^(t^. fc^ Sincef,=-OOO^Q ti,......^^600039, theapproamation BARS129 Combiningtheresults forthetwoclasses itisfound that mll-n-=1-50562, 2-49975, 3'50001, ..., ...(19) where thevalues forthesymmetric andasymmetric types alternate. Thesubsequent numbers areadequately represented bys+$.The fact that thefrequencies areapproximately proportionalto32 ,53 ,72 ,...wasascertained, from observation alone, byChladni*. Toexamine theformassumed bythebarinanynormal mode, \verequire theratio ofthearbitrary constants, asdeter- minedby(6)or(13). Thus inthecase ofsymmetry wehave vf=C(cosmlcoshmcc+coshmlcosmx)cos(nt+e),(20) wheremisaroot of(7).Thecurvemaybetraced with the helpofatable ofhyperbolic functions, andthepositions ofthe nodes foundbyinterpolation. Theform assumed inthe gravest mode isshewn inFig.45.Thenodes here areata distance of-224ofthelengthfrom theends. Fig. 45. Thecorrespondingformula fortheasymmetric modes is 17=C(sin|mlsinhmoo+sinh|mlsinmai)cos(nt+e), (21) wheremisdetermined by(14). 47.Clamped-free Bar. Thenextmostinterestingcase isthat ofabarclampedat oneendandfreeattheother. Here alsothere isanadvantage intakingtheoriginatthemiddlepointofthelength f.The terminal conditions then are (i) *E.F.F.Chladni, born atWittenberg 1756, died atBreslau 1827. Distinguished byhisexperimental researches inacoustics. These arerecorded inhisbookDieAkustik, Leipzig, 1802. +Greenhill, I.c. 130 DYNAMICAL THEOBY OPSOUNDattheclamped end,and thecondition!=Sn'-'...............(3) whence~s -S1n, (II) roots of(12)aregi;enbyCOtl1*aiealso ^=(*""i)?r~^ ........ ..Hence whereo.................(14) andtherefore^e~ Thefrequenciesofthewhole series ofnormal modes, after the first, areapproximately proportionalto32 ,52 ,72 ,...,asfound experimentally byChladni. The accurate solutiongives,to fiveplaces, ml/7r=-59686, 1-49418, 2*50025, (17) Inthemodes which follow the firstwehaverespectively one, two, three,...internal nodes. Theannexedfigure shews the gravestmode. Fig. 46. Other problems,which arehowever ofloss interest, maybe obtained byvaryingtheterminal conditions. We willonly notice thecasewhere both ends are" supported,"i.e.fixed in positionbutfreefrom terminalcouples.Theconditions then are,by45(14), ^=0, V'=0=id (18) Inthesymmetricalclasswehave 77=cosmx .cos(nt4-e), (19) with cos^nd 0,whence ml/7r=l,3,5,..., (20) latheasymmetricclass /;C*sinmas.cos(nt+e), (21) with?7i//7r=2,4,6, (22) Thefrequencies arc,by46(2),proportionaltothevalues of in2 ,andsotothesquaresofthenatural numbers. Thefoundations ofthetheoryofthetransverse vibrations were laidbyD.Bernoulli(1735) andEulcr(1740). The latter alsogavethenumerical solution oftheperiod equation inafewofthemoreimportantcases. Inmore recent times thecalculations, includingthedetermination ofthenodes &c., havebeengreatlyextended byLissajous (1850), Seebeck*(1848) andLordRayleigh. 48.Summary ofBesults. Forced Vibrations. Inanyoneofthepreceding cases, and inanyparticular mode,mvariesinverselyasI,and therefore, by46(2),the period %ir/nwill forbars ofthesame materialvaryas 12 /K. Hence forbarswhich areinallrespectssimilar tooneanother (geometrically)theperiodwillvaryasthelinear scale. For bars ofthesame section theperiodisasthesquareofthe length. Asregardstheshapeand size ofthe cross- section, everything dependsontheradius ofgyration K;thus forbarsof rectangularsection thefrequencyvaries asthethickness inthe planeofvibration, and isindependentofthelateral dimension. This latter statement needs, however, somequalification ;itis impliedthatthebreadth issmall comparedwith thelengthof the bar, or(more precisely)with thedistance between con- secutive nodes. When thiscondition isviolated theproblem comes under themorecomplex theoryofplates (55). Itisofinterest tocomparethefrequenciesoftransverse and longitudinalvibration ofabarincorrespondingcases. Fora barfreeatbothendswehave, inthegravesttransverse mode, jy-'~fi, TT^W" Hirf-W-x (1-50562)*,.........(1) whilst inthegravest longitudinal mode Hence ~=7"122 ......................... (3) Thisexplainstherelative slowness ofthetransversal modes. ThecomparisonisduetoPoisson. Wepassover thequestionofdeterminingthemotion consequentonarbitraryinitial conditions, bymeans ofthe normal functions. Inthecase ofthefree-free bar, forexample, these aregiven bytheexpressionsinbrackets inequations (20) and(21)of46. *L.F.W.A.Seebeck (180549), professorofphysics atLeipzig. Thetheoryofforced vibrationsagain,isoflittle acoustical interest, althoughithassome technicalimportance. Asimple exampleisfurnished bythecouplingrodwhich connects the wheels ofalocomotive.Attending onlytothevertical com- ponentofthemotion, andtreatingthebarasuniform, wehave tosolve theequation (13)of45subjecttotheconditions i)={3cos(pt+a), 37"=[a?=$Z],.........(4) where nistheangular velocityofthewheels, and /3isthe vertical amplitude. The forced oscillation isevidentlyof symmetrical type, andwetherefore assume v]=(Acosh 'mac+Ccosmx)cos(pt4-a).......(5) This satisfies thedifferential equation, provided m'=p*p/K*J;........................(6) whilst theterminal conditionsgive Acosh^ml+Ccos\vnl=/?,.^.Acosh^mlCcos\ml=0,J............ thelatter equation expressingtheabsence ofterminalcouples 0?"=0).Hence (8) The oscillations would becomedangerously largeifcos^ml were small, i.e.iftheimposed frequency (jj/2?r)were toap- proximatetothat ofoneofthesymmetricalfreemodes ofthe barwhen"supported"attheends(47(20). 49. Applications. Theuseoftransverse vibrations ofbars inmusic isre- stricted bythefactthattheovertones arenotharmonic tothe fundamental. Ifaflatbur,otherwise free,besupportedatthe nodes ofthefuridumontal(Fig. 45),and .struck with asoft hammer, theproductionofovertonesis,however, insome measurediscouraged, andmusical instruments ofakind (such asthe"glass harmonica") havebeen constructed onthisplan. Themust important applicationisin the-tuningfork. J o abaristolower thepitchofthegravest mode andtomake the nodesapproachthe centre. Itwasfound byChladni that when thebartakes theform ofanelongated U,thenodes are veryclose tothebend. Theamplitudeofvibration atthe centre ofthebend willtherefore besmallcomparedwith that attheendoftheprongs. Thecircumstances aresomewhat modified bytheattachment ofthestem, butthetransmission ofenergyiscomparatively slow,andthevibrations have con- siderablepersistence. Aforkmayalsobecomparedtoacouple ofbarseachclampedatoneend,andtheformula(2)of 46, withmlfir='59686, maybeused toestimate thefrequency theoretically.Ifthisanalogy were exact there would ofcourse benolossofenergyofthekindjustreferred to. Massive forks areusuallysetinto vibrationbymeans of avioloncello bowappliedtooneprongnear thefreeend.The productionofovertoneshavingnodes inthisneighbourhoodis thusdiscouraged. Thefundamental isfurther reinforced re- lativelytotheother modes ifthestem bescrewed intothe upperfaceofaresonance box ofsuitable dimensions. When afork isexcited inthis orinotherways,itoften happensthatthemotion isnotinthefirst instance,symmetrical withrespecttothemedialplane. Inthatevent thevibration mayberegardedasmadeupofasymmetricalandanunsym- metricalcomponent. These will ingeneralhaveslightly differentfrequencies,andbeatsmaybeproduced. Butunless thestem bevery firmlyfixed thevibrations ofthelatter class arerapidly dissipated bybeing communicated tothesupport, sincetheyinvolve anoscillation ofthecentre ofmass ofthe fork. The firstovertone ofaforkmaybeelicited inconsiderable intensity bybowingoneoftheprongsnear thebend ;thenote producedisveryshrill. 50. Effect ofPermanent Tension. Inthetheory developedin45itwasassumed that the longitudinal tension, whenintegratedover thearea ofthe cross-section, vanishes. Itiseasytoseethat the effect of uension JTismerelytoaaaaterm rvf totne equation (13)of45,sothat where c2=P/po> .........................(2) Thisequation hasbeenemployedtoestimate theeffect of stiffness ofapiano-wire onthesequenceofproper tones, but thematter iscomplicated bytheuncertaintyastothenature oftheterminal conditions. Awire, where itpassesovera bridge, cannot bequite accurately regardedeither asmerely"supported"oras" clamped." Thequestionwillperhapsbe sufficientlyillustrated if\veconsider awave-system v)=Ccosk(ct-x')..................(3) onanunlimited wire.Wefind,onsubstitution in(1) C*=c2+c 12 )........................(4) Ewhere da=.fcV, ........................(5) i.e.c-iisthevelocityoftransverse waves oflength 2vr/A;onabar freefrom tension. Wehave seenthat inthecaseofapiano- string Efpislarge comparedwith c<>2 ;ontheother hand Kis usually anexceedingly minute fraction ofthewave-length.In thegravermodes ofapiano-stringthissecond influencepre- dominates, and(Cj/c,,)2issmall; thewave-velocityispractically unaffected bystiffness, andtheharmonic sequenceisnot disturbed. Itisonlyinthecase ofthemodes ofveryhigh order, where thelengthisdivided intoalargenumber of vibrating segments,thatasensible effect could belooked for. Ithasalready been stated that inthepianofortesuchmodes are, sofarasmay be,discouraged onindependent grounds.Inany case itappearsfrom(4)that theeffect ofstiffness isrelatively lessimportant,thegreaterthevalue ofc,i.e.thetighterthe wires arestrung. 51. Vibrations ofaRing. Flexural andExtensional Modes. Thetheoryofthevibrations ofacircularringisimportant asthrowing lightonsome laterquestionswhich canonlybe assumed tobesymmetrical withrespecttoaplane perpen- dicular tothe axis.We further consideronlyvibrations paralleltothisplane. Let u,vbethedisplacementsofan element oftheringalong andatright anglestotheoriginal radius vector, sothat thepolarcoordinates oftheelement are changedfrom(a,0)to(a4-u,6+v/a).Werequire expressions fortheextension, and forthechangeofcurvature. Incon- sequenceoftheassumed smallness ofthedisplacements, we maycalculate theinstalments ofthesequantitieswhich aredue touandvseparately,andaddtheresults. The radialdisplace- mentbyitselfchangesthelengthofanelement froma$d to (a+u)<b0, and socauses anextensionu/a.The transverse displacement obviouslycontributesdv/ad6. The total extension istherefore (I) Again,inconsequenceoftheradialdisplacementalone the normal tothecurve isrotated backwards soastomake an angle du/ad6with theradius, andthemutual inclination ofthe normals attheends ofanelement a&Q isaccordinglydiminished byffu/adfi.B0.Dividingtheangle between thenormalsby thealteredlength (a+u)86wegetthealtered curvature, thus Since the transversedisplacementvbyitself contributes nothing,theincrease ofcurvature is Theresultant stress across anysection mayberesolved into aradialshearingforceP,atangential tensionQ,andabending moment M.Ontheprinciplesof 43,45wehave Eat BABS 137 thebending momentbeingnowproportionaltotheincrease of curvature. Resolving alongandperpendiculartotheradius vector the forces onamass-elementpwaSO, wehave(seeFig.47) and,taking moments about anormal totheplaneofthering, therotational inertiabeing neglectedasinthecaseofastraight bar(45).Thus 32wdP d-vdQ and.(4) .(5) These, togetherwith(3),are theequationsofourproblem. Itiseasilyseenthattheycannot besatisfied ontheassumption thatthetension Qvanishes, and thataccordingly somedegreeof extension isinvolved inany mode ofvibration. This is readilyaccountedfor,astress ofthis kindbeing necessarily called intoplaybytheinertia of thedifferentportions swinging inoppositedirections. Itmaybeshewn however that inthe "flexural" modes tobereferred topresentlythecorresponding strains aresmallcomparedwith those involved inthechangeof curvature. Eliminating P,Q,Mbetween(3), (4),and(5),wefind E (dv ,Fig. 47. du K?/du(G)v-u K-iuu, to"u\(_A I paa\d8 Toascertain thenormal modes weassume thatuandvvary ascos(nt+e).Again,theringbeing complete, uand vare necessarily periodicfunctions of0,theperiod being %TT,and canaccordingly beexpanded byFourier's theorem inseries of sines and cosines ofmultiplesof6;moreover itiseasily provedthattheterms ofanygivenrank intheexpansion must satisfytheequations separately. We find, infact, that a sufficientassumptionforourpurposeis uAcossd .cos(nt+e),v=Bsins8 .cos(nt+e),(7) where sisintegralorzero. This leads to ......(8) where /3=nWp/E......................... (9) Hence +B(fi._l).s=0.(10) SinceK/aissmall, thesum oftheroots ofthisquadraticin@is s2+1,approximately,whilst theproducts2 (s2I)2KZ JO?issmall. Thetworoots aretherefore fft(c.2_1\2.,20-e+i.ft-'^-l............. (ID approximately. Onreference to(8)weseethat theformer rootmakes B sAnearly. Thecorresponding modes areclosely analogous tothelongitudinalmodes ofastraight bar,thepotential energy being mainly duetotheextension;andthefrequencies, which aregiven by are,forsimilar dimensions, ofthelikeorder ofmagnitude.The case s= isthat ofpurelyradial vibrations. The vibrationscorrespondingtothesecond root aremore important. Wethen have, from(8),A+sB=0,nearly;thus A u=Acossd .cos(nt+e),v=--sinsd .cos(nt+e),(13) .,, s-(,v21)-EK- .....with ri>=\T2--................ (140&--Mpa*^' Itfollows from(1)that theextension isnegligible,andthe energy mainlyflexural. Thefrequenciesareinfactcomparable with those oftransverse vibration ofabar. Inthemode of order sthere are2snodes, orplacesofvanishingradial motion, butthese arenotpointsofrest, thetangentialmotion being there amaximum*. Inthecases=l the circle ismerely displacedasawhole, without deformation, andtheperiodis Fig. 48. accordinglyinfinite. Themostimportantcase isthat ofs=2, where theringoscillates between twoslightly elliptical extreme forms. Thearrows intheannexedfigure shew thedirections of motion atvariouspartsofthecircumference attwoepochs, separated byhalfaperiod, when thering passes throughits equilibrium position. Thedotted linespassthroughthenodes oftheradial vibration. Onefurtherpointistobenoticed. Owingtotheassumed uniformityoftheringtheoriginof6isarbitrary, andother modes, with thesamefrequencies,areobtained byaddinga constant to0.Inparticular wehave theflexural mode u=AsinsO.cos(nt+e),v=coss$ . Se),(15) with thesame value ofwaasin(14).Wehave hereaninstance ofthekind referred toin 16,where twodistinct normal modes *This pointisillustrated bythevibrations ofafinger-bowl when exc.ited by drawing awettedfinger along theedge. Thepointofrubbingisanode as regards theradial vibration, andthecrispations onthecontained water are accordingly most conspicuous atdistances of45:joueither side,whore t.huradial motion isamaximum. 140 DYNAMICAL THEORY OFSOUND have thesamefrequency,andthemodes themselvesaccordingly become tosome extent indeterminate. Thecasewould bealtered atonce iftheringwere notquite uniform, e.g.ifitwereslightly thicker atonepoint. Thenormal modes inwhich there isa node oraloop respectively,ofradial vibration, atthispoint would differsomewhat incharacter, andhaveslightlydifferent frequencies. Accordingly when both modes areexcited we should have beats between thecorrespondingtones. This isa phenomenonoften noticeable inthecase ofbells (and finger- bowls),theinequality beingduetoaslightdefect ofsymmetry. Theformula(14)isobtained more simplyifweassume ab inilio thattheextension maybeneglected,sothat by(1).Assumingthen w=gsms$,y=-coss#, ............ (17)s thekineticenergyis " q*.......(18) o Thepotential energy is,inanalogywith Art.46(9), 7rJW (19)^ ' Hence, putting g=<7cos(ii+e),..................(20) andexpressingthatthetotalenergyisconstant, wereproduce (14).ItappearsfromKayleigh's principle (16)thattheerror involved inthisneglectoftheextension willbeinexcess. Thevibrations ofaringinitsownpianowere first investi- gated byR.Hoppe (1871); theabovesimplified treatment of the fle-xural modes wassubsequently given byLordRayleigh. Thetheoryofvibrations normal totheplaneismore intricate, since torsion isinvolved aswell asflexure. Theproblemhas been solved byJ.H.Michcll(18Sf)), who finds, inthecase of circular cross-section, =.... &J+1+apa-1 where crisPoisson's ratio. CHAPTER Y MEMBRANES ANDPLATES 52.Equation ofMotion ofaMembrane. Energy. The vibrations ofmembranes arenotveryimportantin themselves, andtheconditions assumed forthesake ofmathe- maticalsimplicity are,moreover, noteasilyrealizedexperi- mentally. Thetheoryishowever, foratwo-dimensionalsystem, comparatively simple,andtheresultshelpustounderstand in ageneral waythecharacter ofthenormal modes inother cases where the difficulties ofcalculation aremuchgreater,and indeed ofteninsuperable. Theidealmembrane oftheoryisamaterial surface such that thestress acrossanyline-element drawn onitisalwaysinthe tangent plane. Weshall consideronlycaseswhere thesurface initsundisturbed state isplane,and isinastate ofuniform, or "homogeneous," stress;i.e.itisassumed thatthestresses across anytwoparallelandequallines arethesame indirection and magnitude. Wefurthersuppose,forsimplicity,thatthestress across anyline-element isperpendiculartothatelement. It follows, exactlyasinhydrostatics,from aconsideration ofthe forcesactingonthecontour ofatriangular area, thatthestress (perunitlength)isthesame foralldirections ofaline-element. This uniform stress iscalled the"tension"ofthemembrane; wedenote itbyP. Itsdimensions arethose ofaforce divided byaline, or[MT~*].Wetake rectangularaxes ofx,yintheplaneofthe undisturbed membrane, anddenote by thedisplacement normal tothisplane. Thesurface-density (i.e.themassper form theequationsofmotion wecalculate the forces onthe sides ofarectangularelement SxSyhavingitscentre at(&-,y\ Inthedisplaced position,thegradientofalineparalleltoxis 9/cte,andthat ofalineparalleltoyisd/dy. Hence thestress across alinethroughthecentre oftheelementparalleltoBy, when resolved inthedirection ofthenormal totheplane ay,is Pd%fdx.8y. Thecorresponding componentsofforce onthetwo edges Syoftherectangleare I-\'/-,I' [doc oac\( where theupper signsrelate totheedgewhose abscissa is x+ #,andthelower totheedgex\x. Thesum ofthese givesPd^/dx3 .SxSy,Asimilar calculation forthetwoedgesSas gives Pd^/dy^.Sx8y. The resultant force ontherectangleis therefore Theabove maybecomparedwith theinvestigation by which, inthetheoryofCapillarity,the tensions across the boundaryofanelement SSofasoap-filmareshewn tobe equivalenttoanormal force ,r+-~ where 7?},R2aretheprincipalradii ofcurvature ofthesurface. Itisshewn inbooks onsolidgeometry that, ifdenote distance from theplane xy,wehave atpointswhere theinclination ofthetangent planetoxyis small. Equatingtheexpression (1)totheacceleration ofmomentum oftheelement, viz.pSxSy.,weobtain theequationofmotion p^=P^+B (:3)n't/)//> Ati& i ^* This isduetoEuler(1766). MEMBRANES ANDPLATES 143 Thekineticenergyisgiven by taken over theareaofthemembrane. Thepotential energyisfound mosteasilyasthework requiredtostretch themembrane. Asinthetheoryof capillaritythis isequaltothetensionPmultiplied bythe increase ofarea.Now ifaprismbeconstructed onarectangular element SxSyoftheplane ocyasbase, this willcutoutfrom the displaced membrane anearly rectangular portion whose sides are andwhose area istherefore, tothesecond order, Hence Thesameexpressionisobtained bycalculating,from the expression (1),thework done bynormalpressures applied (asin22)todeform themembrane into itsactualshape^the ratio of toitsfinal valuebeing,atanystageoftheprocess, thesame alloverthemembrane. Theresult is The reader who isfamiliar with thetheoryofattractions will recognizethat this isequalto where inthe firstterm theintegrationextends over allthe elements Ssofthecontour, andSnisanelement ofthenormal toSsdrawn inwards, intheplaneofthemembrane. Since atafixededge f= 0",theformulaagreeswith(5). 144 DYNAMICAL THEOEY OFSOUND 53.Square Membrane. Normal Modes. Toascertain thenormal modes ofalimited membrane we assume asusual that varies ascos(nt+e),sothat E++-u, .................. (i) da? 92 wherek*=n*p/P......................... (2) Atafixed "boundarywemust have =0. Itisfound that thesolution of(1)subjecttothiscondition ispossible only foraseries ofdefinite values ofk,which determine, by(2),the corresponding frequencies. Inthecaseofarectangular membrane, wetake theorigin. atacorner, andtheaxes ofx,yalongtheedgeswhich meet there. The equationsoftheremaining edges being, say, 01=0,,y=b,theequation (1)andtheboundarycondition aresatisfied by ~ .S7TX .sVi//. .\ ,o\=(7sm sin?- cos(w+e),......... (3) Ch where s,sareintegers, provided Itmaybeshewn, byaneasyextension ofFourier's theorem, that(3)istheonlyadmissibletypeofsolution inthepresent case;itwasgiven byPoisson in1829. Inanynormal mode forwhich sors'>1,wehave nodal linesparalleltotheedges.Itappearsfrom(4)that ifthe ratio a- :62isnotequaltothatoftwointegers,thefrequencies are alldistinct, andthenodal lines arerestricted tothese forms. But ifa2 :&2iscommensurable, some oftheperiods coincide, and thecorresponding modes maybesuperposed inarbitrary proportions (16).The nodal linesmaythen assume agreat varietyofforms. Thesimplestinstance is that ofthesquare membrane(a 6),when &=(*+s*) (5) MEMBBANES ANDPLATES 145 Thusbysuperpositionofthemodes forwhich s=2,s=1and s=1,s'=2,respectively, weget .2-TttC .TH/ .TTX .27TV csm- - -. .sm-+Xsinsma a a a .TTX .Tryf TTOC ,rrry\ ,.sm sin-2cos hXcos 1, (6)a a\a a/x/ whereXmayhaveanyvalue. Forexample,inthecases X.=1 thediagonalsSD+y a,xy=0,respectively,arenodal lines. Thefigureshews thecasesX=0,X=\,X=1,which give asufficient indication ofthevarious forms thatmay arise. A-i Fig. 49.A--1 Again, bysuperpositionofthecases s=3,s'=1and s=1, s'=3,weget try.STTX.Try ^,irscsm sin^+\sin sma a a a sm ....(7)a^Ja a[c The cases X=0,X+J,X=1areshewn inFig. 50; intermediate forms arereadily suppliedinimagination.Astillgreater varietyisintroducedbythe factthat a number which isthesum oftwosquarescansometimes be soresolved inmore than oneway.Forexample,themodes forwhich=4,7,1,8,1 s'=7,4,8,1,j respectively,have allthesamefrequency. 146 DYNAMICAL TEEOBY. OFSOUND A-l X-o A-l Fig. 50. 64. Circular Membrane. Normal Modes. Inthecase ofthecircular membrane wenaturally have recourse topolar coordinates, with theoriginatthecentre. The differentialequation maybeobtained bytransformation of52(3),butamore directprocessispreferable. Take firstthecase ofthesymmetricalvibrations where isafunction ofr,thedistance from 0,only. The stress across acircle ofradius rhasaresultant P .^TTT.d^/dr normal tothe planeoftheundisturbed membrane, andthedifference ofthe stresses ontheedgesoftheannulus whose inner andouter radii arerandr+Brgivesaforce P.27TT.? or Equatingthis top.2-rrrBr .,which isthe acceleration of momentum oftheannulus, weget 32P9 I7*=- r3rV 3?- Ifvaries ascos(nt+e),thisreduces to f+13t+4,f.3r2rdr where I?n^pjP,asbefore.(2) MEMBRANES ANDPLATES 147 Ifweassume, asisnecessarilythecasewhen theoriginis included within theregiontowhich(2)applies,that %can "be expandedinaseries ofascending powersofr,thecoefficients (after thefirst)maybefoundbysubstitution in(2),andwe obtain e), (3) providedz~ "W 22.42 .(4) This.is theBessel's Function* ofzero order, "ofthe firstkind," which wehavealready metwith in31;itisrepresented graphicallyinFig.51. Ifabetheradius oftheboundary, 10 10 Fig. 51. supposed fixed, theadmissible values ofkandthence ofnare determined bytheequation J(ka}=0, (5) viz.wehave &a/7r=7655, 1-7571, 27546, 37534, (6) thenumberstendingtotheformm~ ,wheremisintegral. The first ofthese rootscorrespondstothegravestofallthe normal modes ofthemembrane. Inthewithmode there are m~I nodalcircles, inaddition totheedge, whose radii are given bytheroots oflower order. Thus inthecase ofthe second rootwehave forthenodal circle7tr/7r=7655, whence r/a='4356. Thecharacters ofthevarious normal modes will beunderstood fromFig. 51,which maybetaken torepresent asectionthroughthe centre, normal totheplaneofthe membrane. P.W.Bessel(1784 1846),director oftheobservatory atKonigsberg 148 DYNAMICAL THEOBY OF Thecompletesolution ofthe difi which isofthesecond order, would < two definite functions ofkr,each mull constant; butthesecond solution, whi Function"ofthesecond kind/' becomes isthereforeinapplicabletoacomplete case ofanannular membrane, however, circles, both solutions would beadmissil requiredinorder tosatisfytheconclitio Thetheoryofthesymmetricalvil membrane wasgiven byPoisson (1829 approximatelyafewoftheroots ofth When thevibrations arenotsymmei calculatingtheforces onaquasi-rectan bounded bytworadii vectores andtwo sidesbeing accordinglySrand r&0. curved sidesgivearesultant normal totheplane,whilst thestresses produce Equatingthesumoftheseexpressionsto^P{il/ani MEMBBANES ANDPLATES Since fisaperiodicfunction of$,ofperiod expanded (foranyparticular value ofr)inaserie cosines ofmultiplesof0,thus f=R4-Rlcos9+S,.sin -f... 4--Rscoss#+Sssins#4 byFourier's theorem;and this formula will whole membrane ifthe coefficients beregarded ofr.Moreover onsubstitution in(8)itappe; termmustsatisfytheequation separately.T atypicalsolution =Rscossd .cos(nt-he), .,,, provided-=-r= H---^ rgr2r^r The solution ofthis,which isfinite forr=0,ca: theform ofanascendingseries. Intheaccepts haveR8=AsJs(kr),where thefunction J8isdefii This isknown astheBessel's Function ofthe stl first kind. Asinthecase of(2)there isas^ which becomes infinite forr=0,butinthecase o: circular membrane thisofcourse isinadmissible. thenormal modes 150 DYNAMICAL THEOBY OF ! Wehavehere snodal diameters, given bj S0-h= |TT, fTT,. andaccordingly arrangedatintervals of value ofkafter thelowest wehave one whose radii aregiven bytheroots oflow s=l, where there isonenodal diameter, &a/7r= 1-2197, 2-2330, 3'2383, thenumberstendingtotheformm-fI thecorrespondingmodes maybegathe: Fig. 52. graphofthefunction /a(z) ;thismaybe asection throughthecentre, normal to t" thesecond oftheabove modes, the radii- given by r/a= 1-2197/2-2330= Fig.53shews inplantheconfigural inthe first three modes ofthetypes MEMBKANES ANDPLATES ingly given bytan=A1/B1.Ifallthe coeffi< lessthan svanish, wehave, forsmall values ofr, =(Ascoss0+B8sins6)Js(&r)... Thenodehasthen sbranchespassing through 0, angles TT/Swith oneanother, their directions b< tans6==AS/BS.This isillustrated intheprecec forinstance thecases s=2,s=8,5=4alloccur i 152 DYNAMICAL THBOBY OF When aforceZperunitarea actsor theequation (1)isreplaced by *+-fo itbeing supposed,forsimplicity,that regardsthedistribution ofZandthecor f. If,further, Zvaryascos(pt+a),we where k? IfZbeindependentofr,sothat t! uniform overthemembrane, thesolutior anddeterminingtheconstant Gsothat Z Theamplitude becomes very greatwhe toarootof(5),i.e.whenever theimpose that ofoneofthesymmetricalfree i other hand, theimposedvibration is small, andwehaveby(4) MEMBEANES ANDPLATES configurationofthenodal lines canbeexhibit ofalittle sandpreviouslystrewn onthesurface particularnormal mode isexcited, thesand is from theplacesofvigorous motion, andaccum neighbourhoodofthenodal lines.Usuallythe into vibration bybowingatright anglestothe desired mode isfavoured bytouchingtheedgewi atoneormore nodalpoints. If,asinthecase of platefixed atthecentre, thepointofsupportis ofseveral normal modes, agreat varietyofbea maybeobtained. Anextensive series ofdiagra obtained inthiswayweregiven byChladni; man; reproducedinthecurrent manuals ofexperiments Inthetheoretical treatment itisassumed th principalaxes ofstrain andstress isnormal toth plate, andthat thecorrespondingstress vanish then, j03=intheformulae (9)of 42,we remaining principal stresses, PI=E'(G!+cre2),p.2=E'(ea where E'= IfjRj,_R2betheprincipalradii ofcurvature atan; plate,when bent,wehave,byaninvestigationsi of45, zdevrKvhp.fi rh'sta.-ncp. from thftinftHml 254 DYNAMICAL THEOEY OF S Ifwesubstitute from(3),andintegrateo find forthepotential energy perunit area Theformulae(4)maybeappliedtotl rectangular section, "uniformly bentby1 MJb,where bdenotes thebreadth. Alo: have 1/2=0,andtherefore Thebending moment isaccordingly by(4).Thisagrees,asitmust, with *2=$h2 .Theformula (7)shews thatwh section isbent inaplane paralleltoonepa or"anticlastic"cur- vature isproducedin theplaneofthecross- section, theratio of thecurvaturesbeing identical with Pois- son's ratio cr.This circumstance has beenmade thebasis ofpracticalmethods ofdetermining cr.by MEMBRANES ANDPLATES Shearingforces willalsobecalled intoplaynorm oftheplate.Thecircumstances aresomewha butthededuction oftheequationofmotion for 1 plateisastraightforward matter, andpresentsnc Amore serious questionariseswhenwecome to tobesatisfied atafreeedge.Itappearstl condition ofstrain which hasbeenpostulateds theformulae (4)of 55cannot beassumed approximately, rightuptotheedge.Intheim bourhood oftheedge,i.e.toadistance inwai with thethickness, apeculiarstate ofstrain in oneremarkable result ofwhich isashearingfo perpendiculartotheedge,ofquiteabnormal am Forthefurther developmentofthesubject bemade toother works*. Wemerely quoteaf importantresults which havebeen obtained, rel plateswhose edgesarefree. Itisfound that foraplateofgivenlateral frequency (w/2-Tr)ofanyparticularnormal mode n*=l.h*.m\3p where, asin46,misaconstant, ofthenature < ofaline,given byacertain transcendental e denotes thevolume-density.Forplateswit? similar boundaries thefrequency accordingly thickness, andinverselyasthesquareofthe lat< Tnthftcase ofanerfectlv free circular disk 156 DYNAMICAL THEOBY OFS( and -842a, and soon,thenumbersvary: with thevalueadoptedfor a.Accordi values ofmfortheabove modes aregiv m2a2=8'8897, 38'36, onthehypothesisthata=. Thecomplete theoryofthefree circul outbyKirchhoff inacelebrated memoii that thegravestofallthenormal m< diameters, andnonodal circle. Itsfreqi accordingasweadoptthevalue o-= ratio. Thefigureshews the configurationofthenodal lines inthesimplestcases ofoneand twonodal diameters. Thetheoryofacircular plateclampedattheedgehas been treated byPoisson and others.Anapproximateestimation ofthe fr symmetrical mode canbemade asfollovi prescribed type thismaking =0,9f/3r=forr=a.The MEMBRANES ANDPLATES Multiplyingthisby2-Trrdr, andintegratingfrorr wefind 9'a2 Forthekineticenergy wehave T=\phI2%7rrdr Jo Thefrequency (n/^ir)isthereforegiven by 2_320^A2 n"" 9pa4 7 or n=5-963f^-'f"-\pja- The correct value ofn,asdeduced from Poisso anumerical coefficient 5*898. The error in(1 slightlyoveronepercent. Theinvestigationisofinterest inrelation signalling,butthenaturalfrequencyofaplatec tureinthesideofashipisconsiderablyreducec ofthewater incontact with theouter face. Fc frequencyofanironplatewhose diameter is7ini nessone-eighthofaninch isreduced from1013 inair,to550.There isalsoconsiderable damp energy conveyed awayinsound-waves inthewat 158 DYNAMICAL THEORY O Fig. 57. Thefrequenciesofawhole series o calculated byRitz(1909)ontheassun thegravestmode hefinds ineffect m4a4=12-43; if2abethesideofthesquare.This j 332A 57.Vibrations ofCurved Shells When weproceedtothevibrat shells, wemeet with furthercomplies noabsolutely sharplinecanbedra extensional modes. This hasbeen al case ofthering (51).Itappears, h ness is(inimagination)reduced the intotwodistinctcategories.Inone tend todefinite limits, thedeformat MEMBRANES ANDPLATES actual bell isofcourse outofthequestion; but remarkable thatnosystematic experimentalstu have beenmade until thesubject was taker Rayleighin1890. Someunexpectedresults A Toquoteatypical case,thenormal modes ofa when arrangedinascendingorder offrequency, have thefollowingnumbers ofnodal meridians andthepitchesindicated: (4,0) (4,1) (6,?) (6,?) <f c" /'+ 6"b Ofthese theonlyonewhich hasanyrelation 1 pitch (d")ofthebell isthe fifth inorder, and anoctave. Amistake ofanoctave injudgin uncommon, forphysiological reasons, but itis s thepresenceofthelower dissonant tones shou disregarded.Itisconceivable thatthemode c beinsomedegree unfavourable totheproducti discordant elements. The vibrations ofanelastic solidwhose din ofthesame order ofmagnitudearefrom ourj view ofsubordinate interest. Theonlycase v worked out isthat ofthesphere.Inthemost i onediameter extends and contracts whilst the diameterssimultaneouslycontract andexpand,res frequencyofthismode is,forsuch values oforas CHAPTER VI PLANE WAVES OFSOUND 58. Elasticity ofGases. Inanyfluid there isadefinite relation between thepressure p,thedensity p,andthetemperature 6,andanytwoofthese quantities accordinglyserve tospecifythephysicalstate of thesubstance. Itisoften convenient touseinplaceofpits reciprocal v,thevolume ofunitmass. Inthermodynamical investigationsthetwoquantities usuallychosen asindependentvariables arepand v.In Watt's "indicator diagram"these aretaken asrectangular coordinates, pbeingtheordinate and vthe abscissa. Any particularstate isthenrepresented byapointonthediagram, andanysuccession ofstates byacontinuous line.Wemay imaginetheunitmass ofthefluid tobeenclosed inadeform- ableenvelope, andthataninfinitesimalchangeofvolume is produced byadisplacementoftheboundaryinthedirection ofthenormal, whose amount is(say)vforanygivensurface- element SS.Thework donebythecontainedgasinthis processisS(p&8.v),orpSv,sinceS(vSS)=Sv.Hence the work done inanysuccession ofchanges, represented byacurve onthediagram,willbegiven byfpdv,i.e.bytheareaincluded between thecurve, theaxisofabscissae, andthe firstand lasfc ordinates. This area isofcourse tobetaken with itsproper sign, accordingasthework ispositiveornegative. There aretwokinds ofsuccessions ofstates which are specially important.Inthe firstofthese thetemperaturedoes notvary,and therepresentativelines aretherefore called "isothermals." Bymeans ofasystemofisothermal lines drawn atsufficientlysmall intervals thepropertiesofthe substance canbecompletely mappedout.The other suc- cessions referred toarethose inwhich there isnogainorloss ofheat tothesubstance, asifitwere enclosed inavessel(of variable volume) whose walls areabsolute non-conductors. The correspondinglines aretherefore called"adiabatics." Inaperfect gaswehave p=Rp0,orpv=R0, (1) where istheabsolutetemperatureonthegasthermometer, andRisaconstantdependingonthenature ofthegas.The isothermal linespv=const, arethereforerectangular hyperbolas asymptotictothecoordinate axes. Asregardstheadiabatics, theheatrequiredtoincrease thepressure byBpwhen the volume isconstant willbegiven byanexpressionoftheform PBp.Ifcdenote thespecificheat(perunitmass)atconstant volume, thismust beequaltocSd,where 80isthe corre- sponding changeoftemperature. Nowwhen Bv=wehave &p/p=S0Jd, whence, comparing, P=cdfp. Again,theheat requiredtoaugmentthevolume bySvwhen thepressureis constant maybedenoted byQSv,which mustbeequaltoc'80, where c'isthespecificheat atconstantpressure. Since, when S/)=wehaveSv/v=S0l0,wefindQ=c'6Jv. Theheat ab- sorbed when bothpressureandvolume arevariedinfinitesimally istherefore PSp+QBv=0(c^-+c'~} (2) andthedifferentialequationoftheadiabatics istherefore ^+^=(3)pGV The ratioc'fcofthetwospecificheats ispracticallyconstant. Denotingitby7,wehave iogp+7logv=const., or ;pv=const., (4) astheequationoftheadiabatic lines. Thevalue of7asfound bydirectexperimentisabout 1'41 forair,oxygen, nitrogen, andhydrogen.Thefigureshews theisothermal andadiabatic lines throughapointPofthediagram,thelatter curvebeing thesteeper. 162 DYNAMICAL THEORY OFSOUND When thepressureandvolume varyinanyconnected manner, the ratio of theincrement Spof thepressuretothe " compression,"i.e.the negativedilatation Sv/v,maybecalled the" elasticityof volume." Its value willdependnotonly ontheparticular state, butonthemanner in which the variations from that state are supposedtotakeplace, i.e.onthe direction ofthecorresponding Fig. 58.curve onthediagram. Ifthetangentatthe pointPmeet theaxisofpinU,andNUbetheprojectionof PUonthis axis,wehave (5) thisprojection thereforerepresentstheelasticity under the particular condition. Ontheisothermalhypothesis,towhich these letters refer inthefigure,theelasticityisequaltothe pressure p,asfollows atonce from(1),orfrom thefactthat thetangenttoarectangular hyperbolaisbisected atthe pointofcontact. Ifthevariations aresubjecttotheadiabatic law,theelasticity,asdeduced from(4),isyp,andsogreater than intheformer case. This isrepresented byNUfinthe figure. Even inthecase ofsolidandliquidbodies weought, instrictness, todiscriminate between isothermal andadiabatic coefficients ofelasticity, butthedifferenceshappennottobe very important. Theworkdonebyunitmass ofagasinexpanding between anytwoadjacent states iseasily read offfrom adiagramas i(P+Po)(vo- v),orp,(v,-v)+$(p-Po)(*,-V),(6) correct tothesecond order ofsmallquantities. When thetwo states areafinite distanceapart we require toknow themanner oftransi- tion.Forchanges alonganisothermal linepv=pvQwehave /vv v V Forvariationsalonganadiabatic .: . (o) 59. Plane Waves. Velocity ofSound. Thetheoryofplane waves ofsound isverysimilar tothat ofthelongitudinalvibrations ofrods(43).Weassume that themotion iseverywhere paralleltotheaxis ofoc,and isthe same atanygiveninstant overanyplane perpendiculartothis axis.Wedenotedisplacementfrom theequilibrium position by.Thesymbols p,p, i;aresupposedtorefer atthetime ttothatplaneofparticleswhose undisturbedpositionis SB;they aretherefore functions oftheindependentvariables xand t.The constantequilibriumvalues ofp,paredis- tinguishedaspQ,p. The dilatation Awasdefined in40astheratio ofthe increment ofvolume totheoriginal volume, viz. (1) Inthepresent branch ofthesubjectitisusual tointroduce asymbolstodenote the"condensation," i.e.theratio ofthe increment ofdensitytotheoriginal density;thus S). (2) Since v=!//>,wehave (3) The stratum ofairwhich wasoriginally bounded bythe planesocand#+&c isattime tboundedbytheplanes#+and 164 DYNAMICAL THEORY OFSOUND a,++8a-+gand itsthickness istherefore changed from &e toSx+8%,or(1+9ff/3aj) &,andthedilatation isaccordingly *=!-;....................<*> Hence, inthecase ofinfinitelysmall disturbances, wehave, by(3), Informingtheequationofmotion weassume that the pressurevaries withthedensity accordingtosome definite law. Wehave then, forsmall values ofs, p=p+Ks,........................(6) where tcisacoefficient ofcubicelasticity. Consideringthe acceleration ofmomentum ofunit area ofastratumoriginally boundedbytheplanesxandx+8%,wehave whereSprepresentstheexcess ofpressureontheanterior face. Hence, by(5)and(6), where c=V(*/ft).........................(8) Thesolution of(7)isasin 23,43 =/(c-0) +^+tf),...............(9) andrepresents twosystemsofwavestravellinginopposite directions with thevelocityc*. Ifweassume, asNewton"f"did,thattheexpansionsand contractions ofagas,asasound-wavepasses,takeplace isothermally,i.e.without variation oftemperature,therelation between pandpisgiven byBoyle's law, viz.p/p=p/p 1+s, whence K=p a,asalready proved.Thismakes c=V(W/=>o)........ ..............(10) Now forairat C.wemay put,ascorresponding values, p=76x13-60x981,/>=-00129, *Theanalytical theoryofplane waves ofsound isduetoEuler(1747) and Lagrange (1759). tTheinvestigationisgiven inProp. 48ofthesecond book ofthePrincipia (1726). inabsolutecentimetre-gramme-second units, whence c=280 metres persecond. This isconsiderablylessthan theobserved velocity. Thediscrepancy was firstfully accounted forbyLaplaceand Poisson*. When agasisrarefied orcondensed the temperature tends tofallorrise, exceptinsofarasthe processismitigated bythesupplyorabstraction ofheat. In ordinary sound-waves thecondensation .9changes signsofre- quently, andthetemperature consequentlyrisesand falls so rapidly,that there isnotime forsensible transfer ofheat betweenadjacent portionsofthegas.The flow ofheat has hardlysetinfromoneelement toanother before itsdirection is reversed, andtheconditions arethereforepracticallyadiabatic. Theformula P/P*=(p/Py (11) becomes, forsmall values ofs, P=p (l.+ys), (12) whence K=jp0)asin 58,and c=V(7po/^o) (13) Putting ry=l-41wefindthat theNewtonianvelocityofsound inairmust beincreased inthe ratio 1*187, whence c=332 metrespersecond at C.This isingood agreementwith direct observation. Astbere isnownoquestionastothesoundness ofthis explanation,andasthedirect determination of7isamatter ofconsiderabledifficulty,theformula(13)isoften used inthe inverse manner, asameans ofdeducingthevalue of7for variousgasesfrom theobserved velocities ofsound-waves in tbem. Forexample,itwas inthiswaythat in1895 thevalue of7forthenewlydiscoveredgasargonwasfound byLord Rayleightoliebetween 1'6and 1'7.Theexperimentalmethod (duetoKundt)isreferred toin62below. Since p/pa=R0U>thevelocityofsound asgiven by(13)is independentoftheactualdensity,but willvaryasthesquare root oftheabsolute temperature. Also, sofaras7hasthe same value, thevelocityofsound indifferentgaseswillvary * Alioufc,orbefore, theyear 1807. 166 DYNAMICAL THEORY OFSOUND inverselyasthesquareroot ofthedensity, providedthecom- parisonbemade atthesamepressure.These conclusions are inagreementwith observation. Theformula(8)willofcourse applytoanyfluidmedium, providedthepropervalue ofKbetaken. Inliquidsthe difference between theisothermal and adiabatic elasticities may beneglected. For water at15C.wemay put K=2'22x1010 ,p=I,inc.G.s. units, whence c=1490 metres persecond. Thenumber found byCollaclon andSturm(1826) bydirect observation, inthewater ofthelake ofGeneva, was 1435, atatemperatureofabout 8C. Another formula forthevelocityofsoundmaybenoticed. IfHdenote theheightofa"homogeneous atmosphere,"i.e.of acolumn ofuniform,density pwhose weight wouldproducethe actualpressure pperunit area,wehavepti=gpQIf,andthe Newtonian formula(10)becomes cf.43(6).Thevelocityisaccordinglythatduetoafallfrom restthroughaheight \H.Itappearsfrom 58(1)that for agiven gas,and atagiven place,Hdepends onlyonthe temperature. Thecorrespondingadiabatic formula is (15) 60.Energy ofSound- Waves. Thekineticenergyofasystemofplane wavesis,perunit area ofthewave-fronts, T^^p^dx,........................(1) where theintegrationextends overthespacewhich wasoccupied bythedisturbed airintheequilibriumstate. Thework donebyunitmass inexpanding throughasmall range wasfound in58tobegiven accurately,tothesecond order, bytheexpression Po(v9-v)+(p-p)(v- v),............(2) where thesuffix refers tothefinal state. Ifweform thesum ofthecorresponding quantitiesforallthemass-elements ofthe system, the firsttermdisappears whenever theconditions are such that thetotalchangeofvolume iszero. Again, inthe second termwemay put,with sufficientaccuracy, pp=KS, vvVc,s,andobtain.^tcs2 .v .Theexpression ^/cs2isthere- fore tobeintegratedover thevolumeoccupiedintheundis- turbed state. Sofarnothingisstipulatedastothehypothesis towhich Krelates; but itisonlyinthecase ofadiabatic expansionthat theresult canbeidentified with thepotential energyinthestrict sense ofthisterm.Wethenhave F=iJV<fcB, .....................(3) where K=yp,perunitarea ofwave-front. If refer tothe isothermal condition, theexpressionontheright hand iswhat isknown inthermodynamicsasthe"freeenergy." Itisunnecessarytorepeat what hasbeen said in23asto theresolution ofanarbitraryinitial disturbance into two wave-systems travellinginoppositedirections. Inasingle progressive wave-system, say -/(<*-*),.....................(4) wehaveby59(5) % cs, ...........................(5) where denotes theparticle- velocityinthedirection ofpropa- gation.Since hasthesamesignass,anair-particle moves forwards(i.e.with thewaves)asaphaseofcondensationpasses it,andbackwardsduringararefaction. Itappears moreover, from(1), (3),and(5),that the total energyishalfkinetic andhalfpotential.This also follows independentlyfrom the general argument givenin 23. Thecase ofasimple-harmonictrain ofprogressivewaves is specially important.Theformula / cc\ %=acosn[t1..................(6) \c/ representsatrain ofamplitude a,frequency n^ir,andwave- length X=%TTc[n.Wefind f (%\T=Api>w2a2sin2n\t ]da> cos2w*-- do;..........(7) Themean value ofthesecond term under theintegral signis zero,andtheaveragekineticenergy perunitvolume istherefore ^po^a?,and theaveragevalue ofthe total energy 168 DYNAMICAL THEORY OFSOUND Since naisthemaximum particle-velocity,weseethat the energyinanyregion includinganexactnumber ofwave-lengths isthesame asthekinetic energyofthewhole masswhen animated with themaximum velocityoftheair-particles.IfST beused todenote themaximum condensation, wehave s:=na/c> andtheaverage energy perunitvolume maytherefore alsobe expressed by />c2 Sj2 . Wecanalsoestimate, incidentally,thenature oftheapproxi- mation involved inthederivation oftheequationofmotion 59 (7).Theapproximationconsisted inneglectingthesquareofs, or_9/9#.Since sa=2-Tra/Xthismeans thattheamplitudeais assumed tobesmall comparedwithX/27T,acondition which is abundantlyfulfilled inallordinarysound-waves. Sofarwehave traced thecourse ofwavesregardedas already existent, without anyreference totheirorigin. Asan example, thoughasomewhat artificial one, ofthemanner in which wavesmaybesupposedtobegenerated, imaginealong straight tube, ofsectional area&>,inwhich apistonismade tomove toand frothroughasmallrange,inanyarbitrary manner. Theoriginofxbeingtaken atthemeanposition ofthepiston,theforced waves inthe tube, totheright, due toaprescribedmotion &=/(*>........................(8) ofthepiston,willevidentlybegiven by Inparticular,if =acosnt, .....................(10) /x\wehave =acos[ )................ (11) \cjv' Therateatwhich work isbeingdonebythepistonontheair totherightis P0)%o=(Po+tCS)w KIlSQ?=-pa>nasinnt-]-- wsin2n.......(12) Themean value ofthe firstterm iszero, whilst that ofthe second is ^Ku-a-M/c=pn,2tt2a>c................(13) which occupyalengthcofthetube,andthat thepiston must asamatter ofcourse supplythecorresponding amount of energy.Itmust beremembered, however, that aninfinitely longtrain ofwaves ofthetype (11)would takeaninfinite time toestablish, andthatinthecaseofafinite train thesuggested line ofargumentwould requireustoexamine intowhat is taking placeatitsfront. Inthepresent instance theresult would, itistrue,beunaffected, butthecasewould bealtered ifthewave-velocitywere different fordifferentwave-lengths, asitisforexampleindispersivemedia inoptics,indeep-water waves inhydrodynamics,andinthecase offlexural waves ona long straightbar(45). There isthen adistinction between thewave-velocity (foraparticular wave-length) andthe"group- velocity" which determines therate ofpropagationofenergy. Intheaboveproblemforcemust beappliedtothepistonin order tomaintain thevibration (8)againstthereaction ofthe air. Ifthepistonbefree,thestore ofenergy which itorigin- allypossessedwillbegraduallyusedupinthegenerationof air-waves.Suppose,forexample,that thepistonisattached toaspring,andthat' intheabsence oftheairtheperiodofits free vibrations would be27T/7?. Under theactual conditions, itsequationofmotion willbeoftheform M^+^)=-(p-pti)co,............(14) where thevariablepartofthepressurealoneappears,since the constantpartmerelyaffects theequilibrium position. From thegeneral theoryofprogressivewaves wehave p-p=KS=K^/c}..................(15) andtheequation (14)becomes This isoftheform discussed in11,andthesolution is ^Ce-^cos(n't+ e),...............(17) 170 DYNAMICAL THEORY OFSOUND providedr=2l/c/*cy==2lT//> a>c,n/2=^2-l/T2(18) When nrislargethe effect ontheperiod maybeneglected. The condition forthis isthat2.M*must belarge compared withpca\/27r,where Xisthewave-length. The inertia ofthe pistonmust therefore begreat comparedwith that oftheair contained inalength A,/2?rofthetube. Thesame law of decaywould begivenalsobytheindirect methodexplained in 12. Wehave seen in(13)thattherateofpropagationofenergy across unitarea ofwave-front inaprogressive systemofwaves ofsimple-harmonic typeis^p?i2a2 c,or^p<?s^ifsldenote the maximum condensation. The result wasobtained forplane waves, butwillhold forallkinds ofwave atasufficient distance from thesource.ConsequentlyifWdenote thetotal emission ofsonorousenergy persecond fromasource near theground, thevalue ofs1}atadistance r,willbegiven bytherelation W=\ poCVX27T?'2=77-poCVV (19) This formula wasapplied byLord Rayleightoestimate the limit ofaudibilityofasound ofgiven pitch. Thevalue ofW, asinferred from thepower spentinactuatingthesource (awhistle),istheproductofthecurrent intothepressure,and if rbethedistance atwhich thesound isjustaudible, theformula willgiveavalue ofSi,which isnecessarily, however, greater than thetrue limit, since thevalue ofWistoohigh,not all theenergy being spentinsound. Inthiswayitwasascer- tained thatsounds could beheard inwhich SLwascertainlyless than 4x10~8 .Thecorresponding amplitudeasdeduced from theformula na=cs1}was 8x10-8cm.Byanindependent method, inwhich theabove source ofuncertainty wasavoided, thelimit ofaudibility wasfixed atabout sl=6x10~9 .Subse- quent experiments byWien(1903) andRayleigh findicate an increase ofsensitiveness with riseofpitch,fortones nearthe middle oftheordinary musical scale. *Thefactor 2woulddisappearifthepiston weresupposedtogenerate waves onboth sides. tPhil.Mag. (G),vol.xiv.(1907). PLANE WAVES OFSOUND 171 61. Reflection. When there isafixed barrier attheoriginthegeneral solution isreplaced,asin 24,by =/(C-*)-/(C* +<B) (1) Considering,forexample,theregiontothe leftoftheorigin, the firsttermmaybeinterpretedasrepresentingaprimary wave-system approachingthebarrier; thesecond term then representsthe reflectedsystem.The latter hasthesame amplitudeatcorresponding points;thevelocityisreversed, butthecondensation s(=9/8#)has itssignunchanged. We have here, initssimplest form, theexplanationofechoes. There isanother case ofreflection which itisimportantto consider. Supposethat atonepoint (sayx0)thecondition ofunvarying pressure (s=0)isimposed. Wemust have then, in59(9), F'(ct)=f(ct), (2) which shews that thefunctions/,Fmust differonlyby aconstant. Since this constant wouldmerely representa displacement common tothewhole mass, which iswithout influence onthequestion,itmaybeignored. Wehavethen =/(c*-*)+/(<* +a), (3) where asbefore the firsttermmaybetaken torepresent an incident, andthesecond areflected wave-system,intheregion lyingtothe leftof0.Thevelocityishere reflected un- changed,butthesignofsisreversed. Theconditions would berealized ifthe airwere incontact attheplanecc=with amediumcapableofexerting pressure,butdestitute ofinertia. This isofcourse anideal case,butthecondition ofinvariable pressureisapproximatedtoinsomedegreeattheopenendof apipe. Thepresent investigationhasalsoanapplicationto thereflection oflongitudinalwaves atthefreeendofarod (43). Thegeneral problemof(direct)reflection atthecommon boundaryoftwodistinct fluidmedia ishardly morecomplicated. Theorigin beingtaken intheboundary,awave-system ap- proachingfrom theleftwillgiverise toareflected wave onthe leftandatransmitted wave ontheright. Wedistinguish 172 DYNAMICAL THEORY OFSOUND quantities relatingtotheincident and reflected wavebythe suffixes 1and 2,respectively,whilst thoserelatingtothe transmitted wave areindicated by(grave)accents. Since the velocity andthepressuremust bethesame forthetwomedia attheorigin, wehave &+&= , *i+Kffa=Y0=0], (4) theequilibrium pressure pubeing necessarilythesame. Now 1=c$i, 2=-cs2,g=cY,whence c(?!- s2)=cY,/c($!+sa)=Y[sc-0] (5) TT K'CKC'v 2/ccr,Hence s2=,Sj,s=- -,s1[x=0] (6)KC+KC KC+KCL J V* These formulae relate inthe first instance tothestate of thingsattheorigin,onthetwosides; but itiseasilyseen that theywillalsorepresenttheratios ofamplitudesatcorrespond- ingpointsintherespectivewaves. Iftheinertia ofthesecond medium were infinite, weshould have S=0,and therefore s=s1,asinthecase ofreflection atarigidbarrier. Onthe other hand, iftheinertia ofthesecond medium were evanescent, weshould have c=ooands2= s1,asabove. Theenergiesofcorresponding portionsofthevarious waves areproportionaltoKS^C,/cs22 c,/cY2c\since thelengths occupied bytheseportionswillvaryastherespective wave-velocities. Theconservation ofenergythereforerequires KSfC KS^C+KS"-C', (7) this iseasilyverified from(6). IfweputK=pc*,K=poC*,wehave, from(6), $f_pnc~Pocxo\ *i~poV +/OoC Asanexample,take thecase ofair-waves incidentnormally onthesurface ofwater. Wehavep,,/pv='00129, c/c'=-222, about; whenceSo/*!='99943. There istherefore almost com- plete reflection, withhardly anytransmission. Thesame result holdswhen theincidence isinthe.opposite direction, from water toair,andinboth cases toastillhigher degreewhen theincidence isoblique. Thecomplete theory was given byGreen(1847). The results arcchieflyofinterest for theoptical analogies,butonecuriouspointmaybementioned. 62. Vibrations ofaCohmm ofAir. When wecome tothefree oscillations oftheaircontained inapipeoffinitelength,thequestion definitelyarises astothe condition tobesatisfied atanopenend. There ishere a transition, more orlessrapid,fromplane waves inthetube todiverging spherical waves intheexternalspace, which itis difficult toallow forexactly. Intheusualrudimentary theory, which dates from D.Bernoulli, Euler, andLagrange,itis assumed thatthevariation ofpressureinthetube, attheopen end,maybeneglected. Asalready stated, thiswould be accuratelythecase iftheexternal airwerereplaced bya substance capableofexerting pressure (p)butdevoid of inertia. There would thenbenolossofenergy onreflectiono/ attheopenend(6l) 3andthevibrations inthetube, once excited, would bepersistent. Thehypothesisisobviously animperfect representationofthefacts; thecondition s=Q canonlybeapproximately fulfilled, andenergy must con- tinuallybespentinthegenerationofwaves divergingoutwards from themouth, sothatthevibrations iflefttothemselves will besensibleonlyforaverylimited time;thismayhowever cover hundreds ofperiods. Weshall return tothesequestions later(Chapter IX) ;atpresentwecontent ourselves with tracingouttheconsequencesoftheapproximate theory. Theperiodiccharacter ofthemotion inafinite pipecanbe inferred from thetheoryofwaves, exactlyasinthecase of strings (24). Supposeforexamplethat awave oflimited extent isstarted ineither direction from apointPofatube AB. After two reflections, atAandB,thewave willpassP againinthesame direction asatfirst. Ifbothendsbeclosed, thesignofsisunaltered ateither reflection, whilst that offis twice reversed. Hence after theinterval2/c,where l=-AB, the initial circumstances areexactly reproduced. Thesame result holds ifboth endsbeopen,since there havenowbeen 2l/c,andafurther interval oflikeduration mustelapsebefore theoriginalstate ofthingsatPisrestored. Theforegoing theory explainsoneortwoimportant points inthetheoryoforgan-pipes.Thus thefrequency,inthe gravest mode, isinversely proportionaltothelength,and is lowerbyanoctave fora''stopped" pipe,i.e.apipeclosed atoneend,than foran"open" pipe,i.e.oneopenatboth ends, ofthesamelength.Itis,again, directly proportionaltothe velocityofsound, andsoincreases with riseoftemperature. Intheanalytical method fordeterminingthenormal modes weassume asusual that varies ascos(nt+e).Theequation 59(7)thenbecomes thesolution ofwhich is ...... (2)G C asin25.Thecorresponding wave-lengthofprogressivewaves infreeairisA,=Zirc/n. Hence inanysystemofstandingwaves there isaseries ofnodes(=0)atintervals of\,andaseries ofloops,orplacesofzero condensation, (d%/dx=0),half-way between these. Foratube closed atboth ends (x 0,so=I)wehave 4=0, sin(^/c)=0, .................. (3) andtherefore n.mirx(m-rrct \ ...=6msinj-cos (-T+e,n1, .........(4) wherem= 1,2,3,...,thenormal modesformingaharmonic series. Forapipeopenatboth ends, thecondition thats=-d%/dx=Q forx=andx=Igives 5=0,sin(nZ/c)=0,..................(5) andthetypicalsolution is n mrrx /nnrct \ =Omcos-y-cos (j-+mj, .........(6) wherem=l,2,3,.... Here, again,thesequenceofnormal modes isharmonic. Thefigureillustrates thecasesm=l,m=2.Thearrows shew thedirection ofmotion attheloops, whosepositionisindicatedbythedotted lines, intwoopposite phases ;thenodes areindicated bythefulltransverse lines. Fig. 60. Inthecase ofapipeclosed atx=andopenatx=l, wehave A=0, cos(?iZ/c)=0, (7) whencenljc=|w7r,theintegersmbeingodd.Wethusobtain -.ntmrx(mirct \ /ONf=C7 B,cos-^-cos(~^-+emj,(8) wherem=l,3,5,....Theabsence oftheharmonics ofeven Fig. 61. order determines thecharacteristic"quality"ofstopped pipes (91). Thefigure shews thecasesm=1,m=3. Theformula(2)canbeappliedalso tothecaseofforced vibrations ofgiven frequency (n/27r). Thus ifaprescribed vibration %=Acoa(nt+e)..................(9) bemaintained atx 0,and ifthetubebeclosed atx=l,the motion ofthegasisgivenby ,.A .n(I (K)<. \ /-,A\fc-,.sm-'cos (nt+)....... (10) * sm(nJ/c)cv ' v Theamplitude becomesabnormally great,evenwhenwetake account ofdissipative forces, ifsin(n/c)=0,or l^m\where misintegral.This istheprincipleofamethod duetoKundt (1868) bywhich thevelocityofsound invariousgasescanbe compared bysmall-scaleexperiments. Thewave-lengthsare found bymeasuringthedistances between thenodes, whose positionisindicatedbytheheaping upoflycopodium powder previouslyscattered inthetube. The vibrations areexcited inthetwotubes(containingthetwogasestobecompared) by disks fitted tothetwoends ofalongitudinally vibratingrod. Iftheendx=Iisopen,theformula(10)isreplaced by -.A nl % andthecondition ofstrongestresonance iscos(nl/c)=0,or l\m\ wheremisanoddinteger. Thepreceding investigations wouldapplyalso tothe vibrations ofacolumn ofwater, orotherliquid,contained inatube, provided thematerial ofthetubewereabsolutely rigid.Inpractice, however, theyieldingofthewalls has anappreciable effect; thepotential energy correspondingto agivenstrain(<?/d#)ofthefluid isdiminished, andthewave- velocityislowered. The facbwasobserved byWertheim (1847), butthetrueexplanationisdue toHelmholtz(1848). The questionhasbeen furtherinvestigated byKorteweg (1878) andthepresentwriter. Owingtothemuchgreatervelocities (44)ofelastic waves insolids such asglassorsteel, as compared with thesound-velocityinwater, the stresses in thewallsadjustthemselves sorapidlythat itislegitimate toassume that thedeformation ofthetube hasthestatical valuecorrespondingtotheinstantaneous distribution of pressureintheliquid.Ifcbethetheoreticalvelocityof sound intheliquid,asgiven by59(8),ctheactualvelocity, itisfound that inthecase ofatube ofsmall thickness h where aistheinternal radius,tcisthevolume-elasticityof fcheliquid,andEisthevalue ofYoung'smodulus forthe material ofthetube. Thus inthecaseofwater(/c=2'22x1010 ) contained inaglasstube(jEr=G'03x 10")-whose thickness is one-tenth oftheradius, wefind c='75900. Even intheother extreme, when thewalls arevery thick, itisfound that */Oo=/*/(+/a),..................(13) where//,istherigidity.Thevalue ofpforglass is,roughly, about 10times thevalue ofKforwater; thiswouldgive adiminution ofabout 5percent, inthewave-velocity. 63.Waves ofFinite Amplitude. Thelaws ofsoundpropagation,astheyareinvestigatedin thisandsucceeding chapters,aresubjecttosomequalifications which maybestbeconsidered inrelation toplane waves, where thetheoryissimplest. Inthe firstplace,ithasbeen assumed that theconden- sation smaybetreated asinfinitelysmall. Thishypothesis isadequateformostpurposes,butthere arecertain"second order"effects which areofsome theoreticalimportance. Itiseasytoshew thataprogressivewave offinite(as distinguishedfrominfinitely small) amplitudecannot bepro- pagatedwithoutchangeoftype, except onthehypothesis ofacertainspecialrelation betweenpressureanddensity. Assuming,foramoment, that awave ofpermanent typeis inprogress, wemayinimagination impressonthewhole mass ofairavelocity equalandoppositetothat ofthe wave. Inthiswayweobtain acondition of"steady motion" asitiscalled, inwhich thevelocity, pressure,anddensityat any pointofspaceareconstant withrespecttothetime. For definiteness wemayfixourattention ontheaircontained inalong straighttube ofunit sectional area. Thevelocity u momentum orthemass which attheinstant considered lies between theplanessoandx+Sac,wehave ud^==_dp/IN ^dx dx' ^' Also, since thesameamount ofmatter crosses each section in unit time,wehave pu=const.=m, (2) say.Hencemdu/dao=dp/dx, and p~Cmu, (3) or p-p=m(MO-u)=ma ( ), \Po PJ.(4) where thesymbols p,pQ,urefer tothepartsofthemedium which intheoriginal form ofthequestionwere undisturbed. Thisgivesthespecialrelation referred to.Interms ofthe volumeperunitmasswehave p-p=m2 (v~v), (5) which istheequationofastraightlineonthe indicator diagram. Arelation ofthistypedoes nothold forany known substance, whether under theadiabatic orthe iso- thermal condition, andcould inanycaseonly applytoa limitedrange,since thevolume would otherwise shrink to nothingunder acertain finitepressure. If,however, therangeofdensity besmall, theequation (5) canbeidentified with 59(6)provided m2=:p.Sincem-ptu0i where uisthewave-velocityintheoriginal form oftheproblem, thisgives ua-=K/p ,inagreement with 59(8).Theprocess isequivalent tochoosing msothat thestraightline(5)shall beatangentatthepoint (v,pa)tothecurve which onthe indicatordiagram givestheeffective relation between_pand u. The condition(5)was obtained indifferentways by Earnshaw(1860) andRankine*(1870). Toascertain thecharacter ofthecontinualchangeoftype *\V. J.M.Eaakine (182072), professor ofengineeringatGlasgow,185572. fl>5--a(6> csV- doc and p^p/(l+A)**pj(l +\(7) Hence, ontheadiabatichypothesisthat P/Po wefindbyelimination ofpandp ?!f=2< where C2=7p /Poasbefore. For illustrativepurposesitissufficient toconsider the isothermal case,which isderived from theabove byputting 7=1, sothat Wehave seen in60thatonthehypothesisofinfinitely small vibrations there isadefinite relation betweenparticle- velocityandcondensation inaprogressivewave. Following Earnshaw, weassume(tentatively)that thesamethingholds inthegeneral case,andwriteaccordingly 3%f(d%\ /-,-,>.~T=/Ur I> (**)dtJ\dx/ where theform ofthefunction istobedetermined. From this wededuce andtherefore Hence(10)issatisfied provided fm\=c/fl^j\ (u) J\dacj /\dv/ or- f=+clog(1+s), (16) by59(3).Another form is P/P=^/C(17) When sisinfinitesimal theformula(16)reduces to=+cs,in agreementwith 60. Tofind therate atwhich anyparticularvalue ofsis propagated,ineither ofthese cases,wenote thatthevalue of df-ftawhich isassociated with theparticle$attheinstant t willhavebeen transmitted totheparticlex+So;attheinstant t4-Si,provided oxdt i.e.by(12)and(14), ................ (18) Thephasesisthereforepropagatedwith thevelocity .................. (19) relative totheundisturbed medium. Tofind the rate of propagationinspacewehave totake account ofthe total variation ofoc+,which is Therequired velocityistherefore +=?c+ ............. (20)dx/dt dt dt' Thelowersignrelates toawavetravellinginthedirection of ^-positive.Itappearsfrom(16)thatpositivevalues of are then associated withpositivevalues ofs,asintheapproximate theoryof60;buttheformula(20)shews that thevelocity ofpropagationisgreater,thegreaterthevalue ofs.Theparts ofthewave where thedensityisgreaterthereforegaincon- tinuallyonthose where itisless. Thus iftherelation between sandxbeexhibitedgraphically,thecurveAintheannexed figuretakes after atimesome such form as5*.Thewave becomes, sotospeak, continually steeperinfront, andslopes moregraduallyintherear, until atime arrives atwhich the gradientatsomepointbecomes infinite. After thisstagethe analysisceases tohaveanyrealmeaning. Fig. 62. The adiabatichypothesisleads toresults ofthesame generalcharacter. Thereader willfindnodifficultyinverifying thefollowingstatement. Theformula (16)isreplaced by andthevelocityofpropagationofaparticularvalue ofsis +c(l+s)*(7+1)..................(22) relative totheundisturbed medium, or .(23) inspace.Inthelatter formula theparticle- velocityisadded tothevelocityofsoundpropertotheactualdensity, which is ontheadiabatichypothesis dependentonthedegreeofcon- densation andconsequent changeoftemperature. Thegeneral conclusions areasbefore. *Itisnotveryimportant herewhether thecoordinate xbesupposed (asin theprevious part ofthisinvestigation)torefer totheundisturbed medium, or tobeanordinary space-coordinate. Ineither casethetendencyisthesame. 1Cmuse oerememoereu imai/ since uueequation01inowon (9)isnotlinear, distinct solutions, such asthoserepresenting wavestravelling rightand left, respectively,which wehave justbeenconsidering,cannot besuperposed bymere addition. Itmayhowever beremarked that, asaresult ofamore complete investigation, Riemann* found(1860)thatalocalized arbitraryinitial disturbance doeseventuallyresolve itself into twowaves oftheabove kinds, travellinginoppositedirections. Tofollowexactlythecareer ofwaves offiniteamplitude generatedinanygivenmanner isaproblemofconsiderable difficulty;butsome indications maybeobtained bymethods ofapproximation.Thisprocedurewasadopted byAiryf(1845) inhisworkonthedynamical theoryofthetides, where similar questionsarise withrespecttotides inshallow seas and estuaries. Suppose,forinstance, wehave along straighttube inwhich apiston (atx0)ismade tomove inanarbitrary manner *=/(*).........................(24) Theequation (9)becomes,ifweneglect terms ofthethird order inthederivatives of, a^.......... (25) Ifweomit the lastterm, wehave asin 60the first approximation (26) Substitutingthis value of inthesmall term of(25)we obtain Thesolution ofthiswhich isconsistent with(26)is asiseasilyverified. Thecorrection tothe firstapproximation *Bernhard Biemann (182666), professor ofmathematics atGottingen 185766. tSirGeorge Biddell Airy (1801 92),Plumianprofessor ofastronomyat Cambridge 182835,astronomer royal 1835 81. isproportionaltox,andtothesquareortheratio otthe velocityofthepistontothevelocityofsound. This latter ratio mayinpracticebeexceedingly small, butaswetravel tothe rightthecorrectioncontinuallyincreases inimportance,until at lengththeneglectofterms ofthethird andhigherorders would nolongerbejustified. This iswhatweshouldexpect from theresults ofEarnshaw'sinvestigation. When themotion ofthepistonissimple-harmonic, say f(i)=acosnt, ..................(29) theformula (28)gives fei^-5?. [l-cos2s (*-f)}.(30) Thedisplacementofanyparticleisnolonger simple-harmonic, but consists ofapartindependentofttogether withtwo simple-harmonic terms, onehavingthefrequencyofthe imposedvibration(29),andtheother afrequency twice as great.This illustrates theimpliedlimitation toinfinitely small motions intheusualtheoryofforced oscillations(17). Again,ifthegivenvibration ofthepiston bemadeupof twosimple-harmonic components, say f(t)=OTLcosU]t+azcosn2t, ............(31)wefind (cc\/ cc\ t 1+a2cosw211 c) \GI +-7-as ]nfoj+M./W- fl-iWcos2wa(t- )OC(\Cj (x\t--1 cJ (&+2njn,,aja2cos(%w2)(t-- (Ct " ..................(32) Wethus learn that inaddition tothevibrations ofdouble frequency,other simple-harmonicvibrations whosefrequencies arerespectivelythe difference andthesum oftheprimary frequencies nowmake theirappearance.Inacousticallanguage, twosimplevibrations ofsufficientamplitude maygiverisenot Onlyu>cnecorrespimumg pure times, uuu uuuueu uuuaves,;is well astocertain "combination-tones," whose occurrence reminds usagain,thattheprincipleofsuperpositionisno longervalid.Weshall have occasion torefer tothis investi- gationatalaterperiod (Chap. X). Theanalogous phenomenonintidaltheoryistheproduction of"over- tides," which areinfactappreciable, andhave tobe providedforintheHarmonicAnalysisreferred toin39. Wehave seen that themain effect offiniteamplitudeis that inaprogressivewave thegradients,both ofpressureand ofdensity, tend tobecome infinite. This hassuggestedthe question whether awave ofdiscontinuity mightnotfinallybe established, analogoustoa"bore" inwater-waves. Toexamine intothepossibilityofsuch awavewetakethequestioninits simplest form,andassume thatthecircumstances areeverywhere uniform, exceptforthesudden transition attheplaneofdis- continuity. Further, bythesuperpositionofacertain uniform velocity, wereduce theproblemtooneofsteadymotion in which theplaneinquestionisfixed. Thesymbols p,p,wwillthenbesupposedtorefer tothe regiontotheleftofthisplane,whilst thevalues ofthecorre- sponding quantities onthe rightaredenotedbyp, p,u. Since inevery unit oftime thesamemass(m)offluid~ .,- ' crosses anyunitareanormal tothedirection offlow,wehave pu=pu=m,oru=mv,ua=mv(33) Again,since inunittime amassmhasitsvelocity changedfrom UQtou,themomentum oftheportionofairincluded between twoplanesinthepositionsindicatedbythedotted lines in Fig.63isincreasingattheratem(u-UQ\whence Pop-m(u-u ), (34) or,invirtue of(33), p-p=*m*(v-va), (35) inagreementwith(5).Jfwenowsuperposeauniformvelocity PLANE WAVES OFSOUND 185 -MO,wegetthe case ofawaveadvancinginto aregion previouslyatrest. Thewave-velocityisgiven by uf=mV=^1^2V=P-^ , (36)V-V p-po poV} asfirst found byStokes (1848), andafterwardsindependently byEarnsbaw, Riemann, andEankine. Adifficulty,firstpointed outbyLordRayleigh, arises, however, astotheconservation of energy. The rate atwhich work isbeingdoneontheportion ofairabove considered ispuapu,whilst that atwhich the kineticenergyisincreasingis%m(uz u<f).Thedifference is pu-pu-m(w2-w2 )=im(Pt+p)(vt-v}....(37) Ifthetwopoints (v,p),(v,jt?)ontheindicatordiagram be denotedbyP,P,respectively,theexpression (37)ismtimes thearea ofthetrapeziumboundedbythestraightlinePP, theaxis ofv,andtheordinates p0)p.Ifthetransition be effected withoutgainorlossofheat, thepointsP,Pwill lie onthesame adiabatic, andthegainofintrinsicenergywillbe represented bytheareaincluded between thiscurve, theaxis ofv,andthesame twoordinates. Since theacliabatics arecon- caveupwards,thelatter area is(inabsolute value) lessthanthe former. Hence ifv>vtheworkdone ismore than isaccounted forbytheincrease ofthekinetic andintrinsicenergies,whilst ifVQ<vtheworkgivenoutwould bemore than isequivalent totheapparentlossofenergy. Itisevident that acomplete theoryofwaves ofeven approximate discontinuity must takeaccount ofbothviscosity andthermal conduction, since atthetransition thegradients ofvelocityandtemperaturearevery great.Thequestionhas beenvery fullydiscussed byKayleigh. Itdoes notappear probablethatunderordinaryconditions themodifications due tofinite amplitudeareofserious im- portance.Inequation (30),forinstance, the ratio ofthe amplitudeofthevibration ofthesecond order tothat ofthe primaryvibration iscomparablewithn'axfo'2 ,orwithn?afg.x/H, whereHistheheightofthehomogeneous atmosphere. With ordinary amplitudes a,andordinarydistancesas,thisratio will beverysmall. Inthree dimensions the effect must bevery Theessence ofviscosityisthat inamovingfluid thestresses differ fromastate ofpressureuniform inalldirections about a point, byquantities dependingontherates ofdeformation. It isusually assumed thatthesequantitiesarelinear functions ot the rates ofstrain; from ourpresent standpointthis is sufficiently justified bythefactthat thestrain-velocities are regardedasinfinitelysmall. Asin40there will atany instant, andatanygiven point,bethreeprincipalaxes ofthe deformation which istaking place,andthese willnaturallybe theprincipalaxes ofthecorrespondingstress. Wetherefore write, byanalogywith 42(1), ...............(1) J where1;e2,e3aretheprincipal strain-velocities, and ......................(2) Bythesame kind ofproofasin 41, /*'isrecognizedasthe coefficient ofviscous resistance toashearing motion inparallel planes;viz. if^denote therate ofshear, and OTthecorre- sponding stress, wehave r=/A'i7.........................(3) Thevalue ofphasbeen determined with considerableaccuracy foranumber offluids, gaseousaswell asliquid. Itwillbenoticed that themeaningofthesymbol p,and consequentlythevalue ofV,issofarindeterminate, since nothingisaltered intheshapeoftheformulae(L)ifwe incorporateinpanyconstantmultipleofA.Inthecase of liquidsitisinfactusual sotoincorporate thesecond terms in (1).Intheapplicationtogasesitisconvenient toregard pas defined bythegaseouslaws(p=Rp6). There isatpresentno experimentalevidence astohow farthemean stress about a point,viz. (Pi+p+p s}=-p+(V PLANE WAVES OFSOUND 187 differs, inamoving gas,fromp,asthus fixed ;butfrom considerations based onthekinetictheoryofgases Maxwell* inferred (1866; that thetwothingsareidentical, andthat accordingly X'=-$/*' (4) Asweareinterestedchieflyintheorder ofmagnitudeof the effects, theprecisedetermination ofVisnotofmuch consequencetous;accordinglyMaxwell's view isadoptedfor simplicityinwhat follows. The dimensions ofp!arethose ofastressmultiplied by atime, or[ML^T*1 ].Itisfound that/u'isindependentof thedensity,but(ingases)increases with riseoftemperature. Itsvalue forairat C.isabout '000170 inabsolute c.G.S. units. Itwillappearhowever immediatelythattheeffect of viscosityinmodifyingmotion dependsnot somuch onthe value of/i'asonitsratio totheinertia ofthe fluid. This ratio v=*!*' Ipe (5) istherefore called byMaxwell the"kinematic"coefficient of viscosity;itsdimensions are[LZT~1 ].For airat C.itsvalue isabout 132 C.G.S. The rate atwhich thestresses onthefaces ofaunitcube aredoingwork inchangingitssizeandshapeisgiven by &+^'A2+V (<?r+ea34-es2 ) {(e,-e3)24-(e3-e3)2+(i~<Q2 }--(6) The term->Arepresentstherate atwhich theintrinsic energyisincreasing.Theremaining terms, which areessenti- ally positive,indicate adissipationofenergyattherate |X{(e2-es)2+(e,- e,)2+(e,-e2)2 } (7) perunitvolume. Themechanical energythus lost isconverted into heat. Itwillbenoticed that(7)vanishes inthecase of uniformexpansion (ea=e,=3);this isanecessary consequence ofourprevious assumptionastothevalue oftheconstant V. *James Clerk Maxwell (183179), professorofexperimental physics at Cambridge (187179);author oftheelectromagnetic theoryoflight. takes theshape tsf)=ytif (8) Inplane waves ofsound wehave e2=0,e3=0,andtherefore from(1)and(4) pl=-p+^f l=-p,-KS +^fj,'e l (9) Moreover, inthenotation of 59, Theequationofmotion, viz. cQjpi /i-\ PQ=~(.**/ therefore becomes Toobtain asolutionappropriatetothecase offreewaves weput fPcosfcr, ..................... (13) wherePisafunction oft,tobedetermined. Wefind that (12)willbesatisfied, provided |**P+tW>_0............. (M) This hastheform of11(3),andthesolution istherefore P^e-^cos^-he),...............(15) providedr=3/2^2 ,rc2=&2c2-1/r2............. (16) Inallcases ofinterest CTisaconsiderablemultipleofthe wave-length (\=2-7T/&),sothatn=A;c,practically,the i'riction havingasusual noappreciableeffect ontheperiod. Thus =Ce~"Tcos(kct+e).coskac..........(17) Thisrepresentsasystemofstandingwaves with fixed nodes andloops.There isasimilar solution inwhich coskx is replaced bysinlex,andbysuperpositionofthetwowecan construct aprogressive wave-system =ae-"Tcos&(ci-#)................ (18) Puttingv'132 forthecase ofair,wefindr='288A.2 ,the unitsbeingthesecond andthecentimetre. PLANE WAVES OFSOUND 189 The solution of(14)mayalsobeeffectedconcisely bymeans ofimaginary quantities. Thus ininvestigatingforcedsimple- harmonic vibrations ofprescribed frequency weassume that g=ae'inmx , (19) whence, onsubstitution, m'=-^-~^(20) Theratiovn/c"isusually verysmall; thus forn=1500 itsvalue is,withprevious data, about 1'8x10~7 .Hence Takingthelowersign,whichcorrespondstowavestravellingin thedirection of^-positive,andrejectingtheimaginary partof (19),wefind ............... (22) provided Z=3c3 /2i>w2...................... (23) Thisrepresentsasystemofwavesgeneratedtotherightof theorigin byaprescribedmotion f=acosntatthispoint (as byapistoninatube ifweneglectthefriction atthesides). Thewaves advance, with(sensibly)theusualvelocity c,but dimmishexponentiallyinamplitudeasthey proceed*.The linear magnitudeImeasures thedistance overwhich thewaves travel before theamplitudeisdiminished intheratio l/e. In terms ofthewave-length wehave I=(3c/8vrV).X2 ,.................. (24) or,withprevious data,I=9'56X2x103 .The effect ofviscosity instiflingthevibrations istherefore very slight exceptinthe caseofsounds ofveryhigh frequencyandconsequentlyshort wave-length. Even forX=10cm.thevalue of Iisnearly10 kilometres. When wecome tothediscussion ofthree- dimensional waves itwillbeclear that theeffect ofviscosity mayformostpurposesbeignoredincomparisonwith the diminution ofintensity due tospherical divergence.Itis, however, ofsome interest toobserve that there isa *Thin calculation was firstmade byStokes(1815). limit tothefrequencyofvibrations which arecapableof propagationformore than averymoderate distance. Theviscosity being small, therate atwhich work isdone perunit areabythepistoninmaintainingthewave-system (22)musthavesensiblythevalue pn*a?cfound in60.Since theenergyinthemedium totherightisnow finiteandonthe average constant, thismust beequaltotherateofdissipation ofenergy byviscosity. Theequalityiseasilyverified. The dissipation is,by(7), rcc fVS^V tfdx^tfjf (/-!- ^Jo6 Jo\doidt !,...(25) approximately,ifwekeep onlythemostimportantterm. Writing /x\ fx\cos2nit =1+Acos2n(t. VG) \cj' andtakingthemean value withrespecttothetime,weobtain f/*'~. il=i^^,............... (26) by(23). 65. Effect ofHeat Conduction. Afurther cause ofdissipationofenergyistobefound in thethermalprocesses consequent onthealternateexpansions andrarefactions ofthe air. Ifindeed these succeed each other with sufficientrapidity,thevariations arealmost accurately adiabatic, asexplainedin59 ;but,aswas firstpointedoutby Kirchhoff(1868), theresidual conduction ofheat isinanycase ofequal importance withviscosity. Onthekinetic theory of gasesthecoefficients of"thermometric" conductivity (z/)and ofkinematicviscosity areinfactofthesame order ofmagnitude; accordingtoMaxwell therelation isv'=v.Forthisreason thepreceding calculations oftheeffect ofviscosity onair-waves must notbelookeduponasmorethan illustrative. Acomplete investigation, inwhich both influences aretaken intoaccount,shews that the effect isequivalent toanincrease inthe kinematicviscosity, buttheorder ofmagnitudeisunaffected. j.ionlineotuer liana thealternations ofdensity were to Lakeplace withextreme slowness, asinthecase ofverylongwaves ofsimple-harmonic type,there would betime for practical equalization oftemperature, and thedissipative influence ofconduction aswell asviscosity wouldagainbe insignificant. Since theexpansionsareherenearly isothermal, thewave-velocity will approximatetotheNewtonian value (S59(10)). Inintermediate cases thetheory shews that thewave- velocity would nolongerheconstant, butperceptibly dependent onthefrequency. Since nosuch effect isobserved, weinfer that inallordinary cases theconditions arepracticallyadiabatic. Itappearsalsothat insuch intermediate cases thedissipation would beverygreatlyincreased. TheinvestigationofStokes (1851), which ishere referred to,relates tothe effect of radiation;theextension toconduction wasmadeindependently byKirchhoff and Lord. Rayleigh.Itisprobablethat the effects ofradiation alone areofsubordinateimportance. The detailed calculation must bepassed over, butthe general explanation ofthemanner inwhich thermalprocesses may operatetoproduce dissipationofenergyhasbeen stated with suchadmirable clearness byStokes that itisworth while toreproducethepassageinquestion.Theexplicitreference is toradiation, butth.esameprinciplesareinvolved inthecase of conduction also. "Conceive amass ofaircontained inacylinderinwhich an air-tight piston fits, -which iscapableofmovingwithout friction, andwhich has itsouter faceexposedtoaconstantatmospheric pressure;andsu.pposethe airalternately compressedand rarefied bythemotion ofthepiston.Ifthemotion takeplace with extreme slowness, there willbenosensible changeof temperature,and therefore thework doneonthe airduring compressionwillbegivenoutagainbytheairduring expansion, inasmuch asthepressureonthepistonwillbethesamewhen thepistonisatthesamepointofthecylinder,whether itbe movingforwards orbackwards. Similarly,thework done in rarefyingtheair-willbegiven,oubagain bytheatmosphereas thepistonreturns towards itspositionofequilibrium,sothat 192DYNAMICAL THEOBY OFSOUND themotion wouldgoonwithout anypermanent consumption oflabouringforce. Next, supposethemotion ofthe piston somewhat quicker,sothatthere isasensible changeoftempera- tureproduced bycondensation andrarefaction. Asthepiston moves forward incondensingtheair,thetemperaturerises, and therefore thepistonhastowork againstapressure greaterthan ifthere hadbeennovariation oftemperature. Bythetime thepiston returns, agood portionoftheheatdeveloped by compressionhaspassed off,andtherefore thepistonisnot helpedasmuch initsbackward motion bythepressureofthe airinthecylinderasithadbeenopposedinitsforward motion. Similarly,asthepistoncontinues itsbackward motion, rarefying the air,thetemperature falls, thepressureofthe airinthe cylinderisdiminished more than corresponds merelytothe changeofdensity, andtherefore thepistonisless helpedin opposingtheatmospheric pressurethan itwouldhave beenhad thetemperatureremained constant. Butbythetime the pistonisreturningtowards itspositionofequilibrium,thecold hasdiminished inconsequenceofthesupplyofheat from the sides ofthecylinder,andtherefore theforceurging1thepiston forward, arising,asitdoes,fromtheexcess oftheexternal over theinternalpressure,islessthanthatwhich opposedthepiston inmovingfrom itspositionofequilibrium. Hence inthiscase themotion ofthepistoncould notbekeptupwithout a continualsupplyoflabouringforce. Lastly, supposethepiston tooscillate withgreat rapidity,sothat there isnottime forany sensiblequantityofheat topassandrepass between theairand thesides ofthecylinder. Inthiscase thepressures would bu equalwhen thepistonwasatagiven pointofthecylinder, whether itweregoingorreturning, andconsequentlythere would benopermanent consumption oflabouring force. Ido notspeak ofthedisturbance oftheexternal air,because Iam notnowtaking intoaccount theinertia oftheaireither within orwithout thecylinder. Thethird case, then, issimilar totho first, sofarasregards thepermanence ofthemotion; butthere isthisdifference; that, inconsequence oftheheatproduced by compression andthecoldproduced byrarefaction, the force urgingthepiston towards itsposition ofequilibrium, on PLANE WAVES OPSOUND 193 whichever sideofthatposition thepiston mayhappentobe,is greaterthan itwould havebeenhadthetemperature remained unaltered. "Now the first case isanalogoustothat ofthesonorous vibrations ofairwhen theheatandcoldproduced bysudden condensation andrarefaction aresupposedtopassaway with great rapidity.Forweareevidently concernedonlywiththe relative rates atwhich thephaseofvibrationchanges, andthe heatcausingtheexcess oftemperatureQpasses away,sothat ifcisperfectlyimmaterial whether wesupposethechangeof motion tobevery slow, orthecoolingofheated airtobevery rapid. Thesecond case isanalogoustothat ofsound, whenwe supposetheconstants q*andncomparablewitheach other; and wethus seehow itis,that,onsuchasupposition, labouringforce would besorapidly consumed, andthesound sorapidlystifled. Thethird case isanalogoustothat ofsound whenwemake the usualsupposition,that thealternations ofcondensation and rarefaction takeplacewith toogreat rapiditytoallow agiven portionofairtoacquireorloseanysensibleportionofheatby radiation. Theincrease intheforce ofrestitution ofthepiston, arisingfromthealternate elevation anddepressionoftempera- ture, isanalogoustotheincrease intheforces ofrestitution oftheparticlesofairarisingfrom thesame cause, towhich correspondsanincrease inthevelocityofpropagationof sound." 66.Damping ofWaves inNarrow Tubes andCrevices. Asomewhatgreatereffect ofviscosity maybelooked for when theairisincontact with asolidbody,asatthewalls of apipeorresonator, owingtothepracticallyinfinite resistance which thesurfaceopposestotheslidingofthefluidimmedi- atelyincontact with it.Itseems infact tobewell-established that therelativevelocityvanishes atthesurface, whereas in ourtheoreticalinvestigations weassume forthemostpartthat slidingtakesplace quite freely. Acloser examination shews however that inthecase ofrapid vibrations, such asweare concerned with inacoustics, the effect ismainly local, being * [gisaconstant ofradiation.] 194 DYNAMICAL THEOKY OFSOUND confined, practically,toaverythinlayerofairnear the surface, and isexceptinverynarrowspaces unimportant. Thematter maybesufficientlyillustrated byaverysimple case. Supposethatthefluidabove theplan-e yisacbed on byaperiodicforceX=fcosnt, (1) perunit mass, paralleltoOx,theplane formingarigid boundary. Theconsequentmotionbeing everywhere parallel toOxandindependentofthecoordinatex,there isnovariation ofdensity,andthedeformations which aretaking placeareof thenature ofshearingmotionsparalleltoy=0.Denotingthe velocity byu,therate ofshear willbe andtheshearingstress onaplane paralleltoy= isaccordingly pdu/dy. Thestratum boundedbytheplanes yandy+By thereforeexperiencesaresultant force 3/,dux~ (A oy \ perunit area, parallelto#,andtheequation ofmotion isofthe form du t Wehave tosolve thisunder thecondition thatw= for 3/=0. For conciseness weputX=feint ,andreject (inthe end)theimaginary partofourexpressions. Theequationis then satisfiedby (4) provided m2=in/v, or m=(l+f)/3, .....................(5) =V(w/2i/) .........................(6) Since wearelookingforasolution which shall befinite for y=oowetake thelowersign. Also, thecondition that u,= fory=requires thatA=-f/in. Hence ifW=-- PLANE WAVES OFSOUND 195 or,keeping onlytherealpart, f f-$mnt-e~Py$mn(8) aresult which iseasilyverified. Whenftyismoderately large thevalue ofureducespracticallytothe first term, which is thesame asifthere hadbeennofriction. Therigidboundary accordinglyactsasadrag onlyonathinstratum;forexample when y=27T//3thevelocityfalls short ofitsvalue atagreat distance from thesurfacebyabout onepartin535. In .actualproblemsofacoustics(relatingforexampleto vibrations inpipes)theforcepXperunitvolume isreplaced bythenegative pressure-gradient-dpI'dso,andwehave ofcourse changesofdensitytotake into account, buttheresults have asimilarinterpretation. Thelinearmagnitude /i=27r/=V(47"/.29r/n)............... (9) maybetaken tomeasure theextent towhich thedragging effectpenetratesintothefluid. With thepreviousdata itsvalue incentimetres isabout r29/JVT ,whereNisthefrequency;thus for#"=256wefindh=-80mm. Wemayapplytheaboveinvestigationtoobtain anestimate oftheeffect ofviscosityonthewave-velocityinatube, onthe suppositionthat thediameter issmall comparedwith the wave-lengthbutlarge comparedwith thequantityA.The tangentialstress onthefluid attheboundary y=is,inthe caseof(Y), ......00) by(9),thetime-factor eintbeingunderstood. The total tan- gentialforce exertedbythewalls ofacylindricaltube ofradius aonthecontained airmaytherefore beequatedto (1 i)ha .dp/doc perunitlength,where pdenotes themeanpressureover the section (?ra2 ).Hence ifubethemeanvelocity, wehave, calculatingtheforces ontheaircontained inanelement Sac ofthelength, du 1dpf, ,,.vh},.,. or =^Ji-(i_t)_L(11)dtpo3ao(^ ' 27rajx' Tothiswemust addtherelations p=po +<?(>& (12) ds_du dt fa Theelimination ofpand 5between theseequationsleads to Jl_(1_*},f*(14) at-( ITTGL)doc* Itisalready assumed that thetime enters throughafactor eint ;andthesolution of(14)istherefore ofthetype u=Ceint+m *, (15) h"^-'IH1-^}" "-iT^-K1-")^' approximately,onaccount oftheassumed smallness ofhfa. Forwavespropagatedinthedirection of^-positivewetake thelowersign,nndwrite m=info' a, (18) e' and a=nh/4<7rac (20) Wehave, then u=Ce~ax,ein(t~xlc'\ (21) _ /x\ or,inrealform, uCQ-^ cosn (t -,) (22)V cj Thewave-velocityistherefore diminished intheratiogiven by(19). Theexponentialfactor in(22) expressesthelawof decayofthewaves asthey advance. IfIbedefined asin 64(23)itwillbefound that alisoftheorder\2 /ah.The rate ofdecayistherefore muchgreater under thepresent conditions than inthecase ofsound waves intheopen.Aformulaequivalentto(19)waspublished without demon- strationbyHelmholtz in1863. Theaboveproofisavariation ofthatgiven byLordRayleighinhisTheory ofSound. PLANE WAVES OFSOUND 197 Amorecomplete investigation was instituted byKirchhoff (1868)inwhich thermalprocessesareconsidered, aswell as viscosity. Theeffects arethereby increased, asalready explained, butremain ofthesame order ofmagnitude. Asalready stated, itisimpliedintheabove calculation that thediameter ofthetubegreatlyexceeds thequantityh.When ontheother hand thediameter iscomparable with, orless thanh,thewalls haverelativelyamuchgreaterholdonthe vibrating mass, andthecharacter ofthemotion isentirely altered bythe friction. Inparticular, when hislargecom- pared with thewidth themere inertia ofthe fluid ceases to haveany appreciable influence, themeanvelocityover a cross-sectionbeingdetermined byanapproximatelystatical equilibrium between thepressure-gradient (inthedirection of thelength)andthefriction ofthewalls.Wehave, then, whereRisacoefficient ofresistance, dependingonthenature ofthe fluid, andontheshapeand size ofthecross-section. Again, byBoyle's law, j?=j>o(l +5),..................... (24) theisothermalhypothesis being adoptedasnow themost appropriate, since, owingtotheassumed narrowness ofthe tube, transfer ofheatcantakeplace freely. Eliminating pand sbetween(13), (23),and(24),wefind dt~Rdx* Thishasthesame form astheequationoflinear conduction of heat.Assumingthat u=Ceint+mx ,..................(26) wehavem?=inJR/p,andtherefore ro=(l+i)w,..................(27) if Ta=nJX/p...................(28) Takingthelowersignweobtain ussCe*1****<"**#, ...............(29) or,inrealform, u-Ge~^xcos(nt-TSX)............. (30) 198 DYNAMICAL THEOKY OFSOUND Thevalue ofRwillbesensiblythesame asifthefluidwere incompressible.Itsdetermination istherefore thesame asin thecase ofthesteadyflow ofaliquidunderpressure through acapillarytube. Inthis case, ifthesection becircular, the shearingstressperunitlengthonacoaxialcylindricalsurface ofradius risSTT?-.pdu/dr,andtheresultant ofthelongitudinal forces onthetwocurved faces ofacylindricalshell ofthick- ness Bristherefore _ 9/du\ ZTT/I. x-[r-z-] or or\ drj perunitlength. The sectional area oftheshell being 27rr3r, therequisite pressure-gradientis dp a'd _ y* dx rdr\dr which isindependentofx.Therebeingnoradial motion, we havedp/dr=0,sothatp,andthereforedp/d.v,isalsoindependent ofr.Theequation (31)isthen satisfied byu=A+Br^ provided Sbeproperlydetermined. Theconstant Aisfixed bytheconsideration that there isnoslippingatthewall (r=a)ofthetube. Inthiswaywefind w=_|.<^=;2 (32) Themeanvelocityovertheareaofthesection istherefore dx' 8fjf Hence, foracircular section, R=Sfjffa? (34) Theformula(33)contains Poiseuille's*lawofefflux ofliquid throughacapillary tube, viz.that thedischarge persecond varies asthepressure-gradient andasthefourthpowerofthe diameter. Itmaybemade thebasis ofanexperimentalmethod ofdetermining p!. *J.L.M.Poiseuille(17991869),apractising physicianinParis,whowas interested inthecapillary circulation oftheblood. Thedate ofthememoir referred tois1844. PLANE WAVES OFSOUND 199 The case ofanellipticsection canbesolved inasimilar manner. The result, firstgivenbyBoussinesq (1868),is J2=V(a9+&a)/a&,.................. (35) where a,barethesemi-axes. Ifweputa.=oowegetthe case ofanarrow crevice, bounded byparallel planes,the breadth being 26, viz. J2=V/&......................... (36) Thiscanofcourse beobtained moreeasily byanindependent process. The formula(30),when combined with (34)or(36), agrees with theresult ofthemorecomplete investigation given by LordEayleigh (1883).Itappearsthatugoes throughits cycleofphasesinadistance27r/vr,butthatwithin thisspace theamplitudeisdiminished intheratio e~~*"=1/535. Inthe caseofcircular section wehave by(28)and(34). Hence when thecircumstances aresuch that theratiov\na?islarge,thedistance inquestionissmall com- paredwith thewave-length (A,=2,7rc/n)intheopen;forwe have (XCT/27r)2= r2c2 /tt2=4,v/na*............. (38) Hence inasufficientlynarrow tube thewaves arerapidly stifled, themechanicalenergylostbeingofcourse converted into heat. Theinvestigationhasbeenemployed byLordEayleighto illustrate theabsorptionofsound byporousbodies. When asound-waveimpingesonaslabwhich ispermeated byalarge number ofveryminute channels, partoftheenergyislost, so farassound isconcerned, bydissipationwithin these channels, inthewayjust explained.The interstices inhangings and carpetsactinasimilar manner, and itistothiscause thatthe effect ofsuchappliancesindeadeningechoes inaroom istobe ascribed, acertainproportionoftheenergy beinglostateach reflection. Itistobeobserved that itisonlythroughthe action oftrue dissipative forces, such asviscosity andthermal CHAPTER VII GENEEAL THEORY OFSOUND WAVES 67. Definitions. Flux. Divergence. Inrespectofnotation itisconvenient now totake apoint ofviewsomewhat different from thatadoptedinthepreceding chapter. Wedenote by u,v,wthecomponentvelocities, paralleltorectangular axes, considered asfunctions ofposition (x,y,z)and oftime t.With eachpointofspacethere is accordingly associated, atanygiven instant, avector (u,v,w\ andthewholeassemblageofsuch vectorsgivesaninstantaneous pictureofthedistribution ofvelocity*.On.theother hand thevariations ofu,v,wwith thetime, forgivenvalues of x>y>z>giyethehistoryofwhatgoesonataparticular placef, butsupplyinthe first instance noinformation astothe careers ofthevariousparticles which(sotospeak) successively cross thescene. When weproceedtocalculate thecomponentaccelerations oftheparticle which attheinstant tisintheposition (as,yyz) wehave totake account ofthefactthat after thelapseofa time however short itsvelocitiesu,v,wwillbegiven bythe respectivefunctions ofthealteredpositionaswell asthealtered epoch. Supposethat attwosuccessive instants t1}t.,aparticle occupiesthepositions PandP',respectively,and that the correspondingvalues ofthe-component ojfthevelocityare *M.Marey andothers have takenphotographs,ofshort exposure,ofatwo- dimensional current ofwatercarrying suspended motes. Theimago ofeach mote isdrawn outintoashort line,which indicates thedirection andmagnitude ofthecorresponding velocity. tAsifwewere toview thesurface ofastream through anarrow tube, GENERAL THEORY OPSOUND WAVES 201 Wi,WaatPand w/,u2'atP'.The^-componentofthe acceleration ofthisparticlewillbethelimit of oj-i __ gy-j tg~""~ Cj Z*2""""Cj vj*""" wj The limit ofthe firsbterm ontherightisdu/dt,therate of changeofuatP.Again u^ u^isthedifference ofsimul- taneous velocities atthepoints P,P',sothat, ultimately, , du,du ..>f where9w/9sisaspace-differentiationinthedirection PP',and qistheresultantvelocity \f(u"+v2+wz ).The finalexpression fortheaccelerationparalleltoxistherefore du du Similar values areobtained inlikemanner fortheother components.If(I,m,n)bethe direction-cosines ofPP', wehave du__dudxdudydudz ds "docds"byds9^ds jdudu du=l^-+mr~+Wr-, ..................(4)da<Jydz^' whilst u=lq,v=mq)wng................ (5) Hence wemaywrite(3)intheform du du du du ,^+u+v ^.u, ...............(0)ot oxdyois^' which isfamiliar tostudents ofHydrodynamics. Ithasbeenthoughtworth while, asamatter ofprinciple, toaccentuate thechanged pointofview, butintheapplication tomotions which aretreated asinfinitelyslow thedistinction loses itsimportance.Thesecond term in(3)isthen ofthe second order inthe velocities, and thecomponent particle- accelerations maybeidentified withdu/dt, dv/dt, dw/dt. The extent oftheerror here involved, inacousticalquestions, may beestimated asin60byareference toplanewaves ofsound If #),.....................(7) 202 DYNAMICAL THEORY OFSOUND theratio ofthemaximum value ofudu/dastodu/dtisko.The restriction to"infinitelyslow" motions therefore means that theamplitudemust besmallcomparedwithX/2?r. Ifwe fix:ourattention onanygeometricalsurface, openor closed, drawn intheregion occupied bythe fluid,theexpression (lu+mv+nw)$ .Bt, where(I,m,n)isthedirection ofthenormal drawn from an elementaryarea &Sfofthesurface, towards oneside, measures thevolume which intheinfinitelyshort time Stcrosses 8 The coefficient of$tinthis'expressioniscalled the"flux" across 88,and itsintegral (lu+mv+nw)dS, (8) taken overthesurface,iscalled thetotal fluxacross thelatter towards thesideonwhich thenormals aresupposeddrawn. It measures therate atwhich fluid isbeingcarried across the surface, expressedinterms ofvolume perunit time. Tocalculate thefluxoutwards across theboundaryofan elementary rectangular region SxSySz havingitscentrePat thepoint (#,y,z),wenote thattheaveragevelocitiesparallel tox,overthefacesSySz, being equaltothevalues ofutitthe centres ofthese faces, willbe respectively. The difference ofthefluxes, from left toright, across these faces isaccordingly du/dx.SxBySz. Addingthe correspondingterms fortheotherpairsoffaces,weobtain the result (dudv.dw\ &s5U-+T'+^~ OXM/02 ................ (9)\9# dy dzjJ ^' Theexpressioninbracketsgivesasortofmeasure oftherate atwhich thesubstance intheneighbourhood ofPisonthe wholeflowing away from P. It istherefore called the "divergence"ofthevector(u, v,w),and isdenotedby div(u,v,w) ;thus Bydividing anyfiniteregionintorectangularelements we seethat thetotal fluxoutwards across theboundary must be equaltothevolume-integralofthedivergence,or du dvdw\7j, ff/7., \ja /ii\ ;r- -f-;r-H-TT-decaydz=ll(iu+nw+nw)dS.(11)pady dzj JJ' Thiscanofcourse beproved mathematicallywithoutattributing anykinematical meaningtothesymbols. 68.Equations ofMotion. Toform thedynamical equations, wefixourattention on thatportionofmatter which attheinstant toccupiesthe rectangular space Bx&ySz. Onthehypothesisofinfinitely slow motion itsacceleration ofmomentumparallelto asis p8#$y$z .du/dt, wherepisthedensity. Themeanpressures ontherespectivefacesmaybetaken tobethepressui'esatthe centres ofthose faces, andthetotalpressuresonthetwofaces perpendiculartoooaretherefore The difference givesaforce dpJdx.Sx&ySzinthedirection of ^-positive. Equatingthistotheacceleration ofmomentum, we obtain thefirstofthefollowing systemofequations: du dp~~~ ___ _Pdt~d'Pdt~ Since thevariations ofpwhen multiplied bydujdt, ..., ...may beneglected, wemayreplace pbyitsequilibriumvaluep,but itwillnotalwaysbenecessarytopreservethesuffix. Asin59wewrite p=p+KS,........................ (2) where sdenotes thecondensation(p po)/pf),and isthecubic elasticityofthe fluid. Ifwefurther write <*=*//>*........................(3) asbefore, weobtain du ds dv ds3w__Js dt dz.......V' uneinstant tmistnespace bxbyoz, ascomparedwith its equilibrium condition, weevidently have ~ dj=div(u,v,w\ (5) orsince, inthecaseofsmall motions, s=A, dsfindvdw\51=[5 r~rx (u) 9* \9o> s?/a^yv' Theequations (4),(6)arefundamental inthepresentbranch of oursubject. ThepurelyIdnernatical relation(6)issometimes called the" equationofcontinuity." 69.Velocity-Potential. Ifweintegratetheequations (4)of68withrespecttot weobtain 9[t3 ~c~^-l dyj o * where ua>v,warethevalues ofu,v,watthepoint (as,y,z)at theinstant t=0.Inalargeclass ofcases, these initial values ou,v,w canbeexpressedasthepartialdifferential coefficients ofasingle-valuedfunction of(x,y,z\thus %=-?& %=-?& w--d^(9\ UQa^5v dy'~ Tz() Throughout anyregiontowhich thisstatementapplies, the values ofu,v,watanysubsequent instant tcanbesimilarly expressed; thus, from(1), dx' ty' /t where 6= . This functionq5>iscalled a"velocity-potential," owingtoits analogy withthepotential -function which occurs inthetheories ofAttractions, Electrostatics, &c. Itwasintroduced into hydrodynamics byLagrange. asregardsbothmagnitude and direction. Supposetwo consecutive surfaces tobe drawn, forwhich thevalues of </>differby Bcf>.LetPP'bedrawn normal tothese, and PP, parallelto#;and letPP'=&/. Ac- cordingto(3)thevelocityatP,resolved inthedirection PP15isFig. 61. .(5) ultimately,ifIdenote thecosine oftheanglewhich thenormal PP'makes with Ox.From this,andfrom theanalogousforms ofv,ui,itisseen thatthevelocityatPisnormal totheequi- potentialsurfacepassing throughthatpoint,and isequalin magnitudetothelimitingvalue ofB<ft/&v. Hence ifasystem ofsurfaces bedrawncorrespondingtovalues of <which differ byequalinfinitesimal amounts, thevelocityiseverywhere orthogonaltothese, andinversely proportionaltoSv,thedistance between consecutive surfaces. Moreprecisely,thevelocityis everywhereinthedirection inwhich <decreases* mostrapidly, and isequalinabsolute value tothegradientof<. Ifwedraw alinear element PQ(=Bs)inanyother direction, thevelocityresolved inthedirection ofPQisequaltothelimit of (6) ord$/ds. The cases inwhich avelocity-potentialexists include all those where, intheregion considered, thefluidwasinitiallyat rest, forwemaythenputfa 0,simply,andthesubsequent value is * sdt.........................(7) This willholdwhenever themotion hasbeenoriginated bythe vibration ofsolid orother bodies. *Itshould bementioned that inmany books<pistaken withtheopposite sign;thusu=d(j>jax, &o. The realmeaningofthepropertywhich differentiates the present typeofmotion from allothers ismostclearly expressed interras ofthe"circulation" round aclosed curve. Ifwedivide thecurve into infinitesimal linear elements, andmultiplythe lengthofeach element bythetangential componentofthe velocity;estimated alwaysinthesame direction round the curve, theresult isthe"circulation" referred to. Itmayhe denoted by dx dy dzU-*~-pV-?-pW~T~as as asr ds,or I(udoc+vdy+wdz)....(8) J Onthepresent hypothesisthetangential velocityis3$/9s,and theintegralofthis, taken, round thecircuit, iszero,thefirstand lastvalues of$beingthesame. Thecirculation istherefore zero ineverycircuit which canbedrawn intheregionin question. Forareason which maybeunderstood byreference tothecase ofaninfinitesimal circuit, thetypeofmotion now under consideration iscalled "irrotational." Thename hasthe advantageofcallingattention toageometrical propertyrather than toananalytical form ofexpression. Adynamical interpretationcan alsobegiventothe velocity-potential.Theequations (3),when written inthe forms /JOM=-p^jdx, pnv=-pfifldy, p,w=-p^/dz, (9) shew that <isthepotential perunitmass ofasystemof extraneousimpulsiveforces which wouldgeneratetheactual motion ofthefluidinstantaneouslyfrom rest. Thetheorem astothepersistenceoftheirrotational character ismostimportant;but itisnecessarytoobserve therestrictions under which ithasbeenproved.Itwasimplied,inthe first place,that the fluid was frictionless, and this isessential. Againthemedium hasbeensupposedfreefrom extraneous forces, buttherestriction iseasily removed inthecaseofforces which, likegravity,have apotential (perunitmass). Finally, theassumptionhasbeenmade that themotion isinfinitely small. Thissimplifiestheproof,andcovers most caseswhich are ofinterest inacoustics. Amorerigorous investigation would shew that thecirculation is(under theabovecondition)still constant roundany circuit, provided weimagine thecircuit to move with the fluid. Ifinitiallyzero foreverycircuit which canbedrawn inafiniteportionofthefluid, itwillremain zero foreverysuch circuit. 70.General Equation ofSound Waves. Wepostulatehenceforth theexistence ofavelocity potential, atallevents inthecase ofauniform medium, towhich we confine ourselves forthepresent. Wehave then, from 68(6) =V2<> (1} dt9>................... .......W 2232 32where V= -+~+~...................(2)da?dy*oz*x' Thissymbol V2iscalled the"Laplacian operator," from its constant occurrence intheanalytical theoryofattractions as firstdeveloped byLaplace. Again, bydifferentiation of69(4) withrespecttotweget Finally, byelimination ofs, 9^="V^.........................(4) Thismayberegardedasthegeneraldifferentialequationof sound waves inauniform medium. Ifasolution canbe obtained whichgives prescribedinitial values to</>and s (or3$/9i), and satisfies theother conditions oftheproblem,the subsequentvalue ofsisgiven by(3),andthevalues ofu,v,w by69(3). Wemaystopforamoment tonotice theformassumed by theequations when thefluid isincompressible.Thismaybe regardedasanextreme case, inwhich cismade infinite, whilst siscorrespondingly diminished, insuch awaythat c2 s,which =(ppo)fp,remains finite. Theequationofcontinuity, 68 (6),takes theform du dvdw 5-+5-+o-=0,..................... (5)dxoyoz^J Inthecaseofirrotational motion, thisbecomes V^(f>=0, (6) which isidentical with"Laplace's equation"inthetheoryof attractions. Thesameequationoccurs inthetheoryofsteady electric(orthermal) conduction inmetals. If,forexample, <f> denote theelectricpotential,theformulae(3)of69givethe componentsofcurrent, providedthespecificresistance ofthe substance betaken tobeunity.This analogywillbefound useful inthesequel. Thetheoryofthemotion ofincompressiblefluids iscapable ofthrowing morelight, occasionally,onacoustical phenomena thanmightatfirstsightbeanticipated. Weareapttoforget thatthevelocitywithwhichchangesofpressurearepropagated inwater isafter allonlyfour orfivetimes asgreatasinair, andthatthevisible(oratalleventseasily imaginable)motions ofwater, under circumstances where thecompressibilityhas obviouslylittle influence, maysupplyavaluable hint astothe behaviour ofagaseoussubstance under similar conditions. This remark willhavefrequentillustration inthefollowing chapters. Thekinetic energyofasystemofsound waves is T=<rP111(u*+tf+w'}dxdydz Thepotential energy,asgiven bytheargumentof60,is ^)2 d*dyd*. ...(8) Theintegrationsextend overtheregionaffected. 71. Spherical Waves. Inthecase ofplanewaves with frontsperpendiculartoOx theequation. (4)of70reduces to m ^' whence ^=f(ct-x) +F(ct+a;)................(2) Thisneed notbefurther discussed. GENERAL THEORY OFSOUND WAVES 209 The case which conies next inimportanceisthat of symmetrical sphericalwaves. If(j>beafunction ofthe distance rfrom theoriginandoft,only, thevelocityisd(j>/dr outwards, inthedirection oftheradius, and isuniform over anysphericalsurface havingtheoriginascentre. Instead ofapplyingthegeneral equationtothepresent circumstances itissimplertoform thekinematical relation correspondingto70(1)denow. The fluxoutwards across asphereofradius ris3</>/3r.47rra ,andthedifference offlux across theouterandinner surfaces ofasphericalshell ofthick- ness &risaccordingly Aa(29<^*4?r5-r2 -g-}8r.dr\ dr/ Thevolume oftheshell being4>mA &r,thismust beequalto A .4iirr-8r or s.4>Trr28r,whence Since c95= ........................(4) i v &$c2 asusual, wehaveg^=^ Thismayalsobewritten The solution ofthisequation,viz. r<j>^f(Gt-r) +F(ct+r')i...............(7) represents thesuperpositionoftwowave-systems travelling outwards andinwards, respectively,with thevelocityc.In thecase ofadiverging wave-system r<=/(c-r)...... ...............(8) wehave, by(4), crs=*f(ct-r)...................... (9) Any value ofrsispropagated unchanged;thecondensation s therefore diminishes intheratio1/rasitproceeds,andthe potential energy perunitvolume diminishes as1/r2 .Forthe particle-velocity wehave r)....... (10) Thelawofdependence ondistance isheremorecomplicated, butasthewavespreads outwards the firsttermultimately predominates ;thevelocityatcorresponding pointsofthewave then varies asl/r,andthekineticenergy perunitvolume asI/?-2 . Inadiverging wave-system wehave, from(9), crs=-~- (r<f>), (11) andsimilarly,inaconverging wave-system. These relationscorrespondto(5)of 60,which isindeed a particular case, since asrincreases oursphericalwaves tend to becomeultimately plane. Thegeneral argumentof23canbeadduced toprovethat inadiverging (oraconverging) wave-system byitself the energyishalfkinetic andhalfpotential. The solution(7)canbeappliedtoaregionincluded between concentricspheres, ortoaregion having onlyone finitespherical boundary, internal orexternal. Inany case, theconditions tobesatisfied attheboundaries, whether finite orinfinite, must begiveninorder thattheproblem maybe determinate. Inparticular, evenwhen theregionisotherwise unlimited, thepointr= istobereckoned asaninternal boundary; thispoint mightforinstance beoccupied bya "source" ofsound(73).When there isnosource there, the flux across asmallspherical surfacesurrounding must vanish, i.e.wemusthave Whenappliedto(7)thisconditiongives /(cO+JXeO-O, ..................(14) forallvalues oft,andthegeneral solution therefore takes the shape r<}>=F(ct +r)-F(ct-r) .............(15) This formula maybeused todetermine themotion con- sequent onarbitraryinitial conditions which aresymmetrical 111 O/U. U.i.J.11 UJGUJLU.UU. IUIU, UWI1C11 wehave .(16) Theformer ofthese functions determines theinitial distribution ofvelocity,andthelatter that ofcondensation. Thefunction Fmustnowsatisfytheconditions F(r)-F(~r)=r<j )o(r), (17) .(18) Itistobenoted thatthevariable risessentially positive ;this explains whytwoequationsarenecessarytodetermine Ffor positiveandnegativevalues oftheargument. Suppose,forexample,that there isnoinitialvelocity anywhere,butonlyaninitial condensation, sothat <(r)=0. From(17)and(18)wededuce Thecondensation attime tisgiven by r)-F'(ct-r)S~c*dt cr(20) This takes different formsaccordingasctislessorgreater than r.Intheformer case {(r+cOxo(r+ct)+(r-ct)%0(7-- ct)},...(21)2cV andinthelatter ,(fifr lp\<y(Q^_yAl f22^ Asaparticular case,suppose wehaveaninitial condensation which isuniform(=s)throughouttheinterior ofasphereof radius a,and vanishes forr>a;and letusexamine the subsequentvariations ofsatpointsoutside theoriginally disturbedregion.Since%(^)vanishes byhypothesisforr>a, the firstpartofthesolution(21)or(22) disappears inthe 212 DYNAMICAL THEOBY OFSOUND presentcase. Solongasct<ra,thesecondpart of(21)will alsovanish, butwhen ctliesbetween raandrweshallhave s=~(r~ct) (23) When ct>r,thesecond formula(22) applies, andwefind that, solongasct<r+a,theresult(23)will still hold. Finally, when ct>r+awehaveagains=0.The results areshewngraphicallyinthefollowing figurewhich exhibits thevariation ofswith tataparticular point,andthespace- ct-ct/r:<*a Fig. 65. distribution ofsataparticular instant, respectively.It appearsthat after thelapseofacertain time(2a/c) wehave a divergingwave intheform ofasphericalshell ofthickness 2a, andthat 5ispositive throughtheouter half,andnegative throughtheinner halfofthethickness. Thechangesinthe velocity maybeinferred bymeans oftheformulaq=d(f>/dr. Forvalues oftbetween(r a)/cand(r+a)/c,i.e.duringthe time oftransit ofthewave across thepoint considered, wefind whilst forother values oftwehave <=0.Hence within the aforesaid limits oftimewehave .(25) When rislarge comparedwithathischanges signfort=r/c, approximately,thevelocity beingdirected outwards inthe outer half,andinwards intheinner half ofthe shell. Atthe boundaries ofthedisturbedregion, where rcta,wehave q=cos/2r.Asthediverging wave reachesanypointthe velocity suddenlyrises from zero totheformer ofthese values, and asitleaves itthevelocityfallssuddenly from thelatter (negative) value to0.The oriein ofthediscontinuities inthis GENERAL THEOUY OFSOUND WAVES213 solution istobesoughtofcourse inthediscontinuityofthe initial distribution ofdensity. Any difficulty whichmaybe feltonsuchgrounds mayingeneralberemovedbysubstituting inimaginationaninitial distribution inwhich thediscontinuity isreplaced byaveryrapidbutcontinuous transition. The solution of(6)interms ofthegeneral initial con- ditions(16)maybeinvestigatedinasimilar manner, but it must suffice toquotetheresults. Itmay easily beverified thatthey satisfyalltheconditions ofthequestion. Theyare r<jf>=\(r+ct) (j>(r+ct}+%(r-ct)<- ct) irr+ci +W-J*&()& -.(26)&(jjr-ct forct<r,and r(j>=|(ct+r)<(ct+r)-$(ct- r)<(ct-r) ct+r ct-r forct>r. Since theorigin evidently occupiesanexceptional position inthetheoryofsphericalwaves itisdesirable tocalculate the value of<pthere, moreespeciallyastheresult willbeofservice presentlywhenwecome tothesolution ofthegeneral equation 70(4)ofsound waves. The result maybededuced from (27),ormore directlyfrom(15).We find M. ...(28) r=0r and therefore from (17)and(18) <=tx(ct)+<(ct)+cttjto (ct) (29) Forexample,inthespecial problemabove considered, where <(r)=0,whilst %(r)-C2sor accordingasr$a.,wefind =c\toraccordingast$a/c.The consequentvalue of sat issfort<a/candzero for t>a/c,whilst attheinstant t=a/citisnegativeinfinite. Toescapethis resultwemust slightly modifythedata, replacingtheoriginaldistribution ofdensity byacontinuous one. Thefigureisanattemptto 214 DYNAMICAL THEORY OFSOUND shew aninitial distribution ofswhich variesrapidlybut continuouslyfrom sato intheneighbourhoodofr~a, togetherwith theconsequenttime-variation ofsat0, Fig. 66. Theproblemwhich wehave discussed exhibits amarked contrast with thetheoryofplane waves, inthat thewave resultingfrom anarbitrarydisturbance contains both con- densed and rarefiedportions,evenwhen there isnoinitial velocity andtheinitial disturbance ofdensityhaseverywhere thesamesign. Thestatement iseasily generalized bymeans ofequations (1)of 69. Ifwetake theintegralofthevalue ofsatanypointPoveratimewhich covers thewhole transit ofthewave, sothatthevalues ofu,v,wvanish atbothlimits, wefindthat itsspace-derivativesare allzero. Theintegral hastherefore thesame value forallpositionsofP.Andby takingPataninfinite distance, sothat sbecomesinfinitely smallbyspherical divergence, weseethat thevalue isinfact zero,i.e. fsdt=Q(30) GENERAL THEORY OFSOUND WAVES 215 72.Waves resulting from agiven Initial Disturbance. Wehavenext totrace theeffect ofinitial conditions inan unlimitedregion,inthegeneralcase.Wesupposethat atthe instant twehave where thefunctions arearbitrary. Todeduce theeffect atany subsequent instant, atanyassigned point P,weconsider inthe first instance theaveragevalue of <overasphereofradius rdescribed withPascentre. This willbedenoted by if8(0representtheelementarysolidangle (8$/r2 )subtended at PbyanyelementaryareaBSofthesphere.Inthesameway wewrite This, like(2),willbeafunction ofthevariables rand tonly. Ifin70(3)wemultiply both sidesby8<w/47r, andintegrate overtheaforesaidsphereofradius r,wefind dt Itisalsoevident that theaveragenormalvelocityover the spherewillbe3</9r. Theargument bywhich therate ofchangeofswas in71inferred from theconsideration ofthetotal fluxoutoftheregionbounded bythespheres randr+Srcanthen beappliedtoprovethat inthepresent case dtrzdr\ dr Eliminating s,wehave df~r* which isidentical informwith(5)of 71.Werecognize then 216 DYNAMICAL THEORY OFSOUND which would result from initial distributions ofvelocity and condensationexpressed by these functions ofrbeingtheaveragevalues of <(,y,z)and %(x,y,z)taken over theaforesaid sphere.Itfollows from 71(29)thatthevalue of <atPisgiven by (8) Thisgivesarule forcalculatingthevalue of$forapointPat anygiveninstant t.Itmaybestated inwords asfollows : Tofindthepartof<f>due tothegiveninitial distribution ofcondensation, wedescribe aboutPasphereofradius ct,and calculate theaverageofthegiveninitial values ofd<j>/dt,i.e.of thefunction ^(as,y,z\atthepointsofspace throughwhich thissurfacepasses,andmultiply byt.Tofindthepartdueto theinitial velocities wereplacetheaverageofthegivenvalues of9(/>/3<bytheaverageofthegiveninitial values of<,i.e.of thefunction <(or,y,z\and differentiate the result, asthus modified, withrespectto t. Thetheorem contained in(8)wasgiven byPoisson (1819); theactual form(8)andtheinterpretationaredue toStokes (1850).Itwillbeseen that theresult, asthus stated, isin reality very simple,ifregardbehad tothegreat generalityof thecircumstances which aretaken intoaccount. Totrace thesequenceofevents atPweemployaseries of sphereswhose radii(ct)increasecontinuallyfrom zero. IfP beexternal totheregionwhich isthelocus ofthe initial disturbance, noeffect isproducedsolongasthespheres do notencroach onthisregion.Ifr1}i\betheleastandgreatest distances ofPfrom theboundary, thedisturbance atPwill beginafter atimer^c,will last foratime(ra?-I)/c,andwill then cease. Ifwith thevariouspointsoftheboundaryoftheoriginally disturbedregionascentres wedescribe aseries ofspheres of radius ct,theouter sheet oftheenvelopeofthesesphereswill mark outtheboundaryofthespace which hasbeeninvaded by GENERAL THEORY OFSOUND WAVES 217 thedisturbance uptotheinstant t.Theenvelopescorre- spondingtosuccessive values oftwillform aseries ofwhat are known ingeometryas"parallel surfaces"; inother words, the boundaryofthedisturbedregion spreads everywherenormal toitself with theconstantvelocityc. Asasimple applicationoftheformula(8)wemaytake the problem alreadydiscussed in 71,where aninitial uniform condensation swassupposedtoextendthroughouttheinterior ofasphereofradius a havingtheoriginas centre. When aspherical surface ofradius ct,de- scribed withPascentre, intersects theboundaryof theoriginallydisturbed Fig. 67.region,asinthefigure, the area oftheportion included within thelatter is2-rr .PQ2(1 cosOPQ), andthe averageofthegiveninitial values ofsoverthewhole surface (47r.PQ1 )istherefore _a?-(ct-r)2 so> where r=OP. Hence, bytherule, .(9) inagreementwith 71(24). 73.Sources ofSound. Reflection. veryusefulconceptionofa" point-source"The was introduced intothesubject byHelmholtz. Wemayimagine (withMaxwell andLordRayleigh)that atsuchapointfluid isintroduced orabstracted atacertain rate, aridthat the " strength"ofthesource ismeasured bythevolume thus introduced perunit time. Thewave-train due toasource of strength/ (4)attheoriginisaccordingly represented by (1) 218 DYNAMICAL THEORY OFSOUND (oi \ -^.47rr2)=/(0................(2)OT J Ifwedifferentiate thegeneral equationofsound waves((4) of70)withrespecttoxoryorz,werecognizethat if <isa solution soalso isd(f>/dx,or9</>/3y,ordcf)/dz.Thus from(1)we derive thesolution47r<j!>=5-.- ,..................(3)05? ?* which satisfies thegeneraldifferentialequation exceptatthe singular pointr=0.Thevalue of$thusobtained maybeinter- pretedasthevelocity-potentialofa"double source" duetothe juxtapositionoftwosimplesources which arealwaysinopposite phases.This willbeexplained morefullyin 76,intheparti- cular casewhere thevariation withtime issimple-harmonic. Theproblemofreflection ofsound byarigidinfiniteplaneis readilysolved bythemethod of"images."Ifwithevery source Pofsound onthenear sideoftheboundary weassociate asimilar source atthegeometrical imageP'ofPwithrespecttothe plane,itisobvious thatthecondition ofzeronormalvelocity over theplane would stillbefulfilled iftheboundary were abolished. Hence, intheactual case, themotion onthenear side willbe madeupofthatduetothegivensourcesPandofthatdueto theimagesP'. Itmaybementioned thatthepresent caseofa rigid plane boundaryistheonlyonewhere thephysical "image" ofapoint-sourceisitselfaccuratelyapoint-source. Insubmarinesignalling,ontheother hand, reflection atthe freesurface ofthewater canberepresented bythefiction ofa negative image.Thus inthecase ofthesource(1)thecombined effect willbe 4s7rd>=(-ft--}--,f(t- ~] ,............ (4) r\rJCJr'J\ CJ'\J where r'denotes distance from P'.Forthismakes atthefree surface, where r=r'.Atagreat distance thecom- bination isequivalenttoadouble source. Inparticular, at depths which aresmallcomparedwith thedistance there is almostcompleteneutralization oftheoriginalsourcebyits image. GENERAL THEORY OFSOUND WAVES 219 74.Refraction duetoVariation ofTemperature. Questions relatingtowave-propagationinheterogeneous media canonlybediscussed inageneral way,andwith the helpofconceptionsborrowed fromgeometrical optics.Ifat anysurface there isanabrupt changeofpropertiesthelawof propagationisofcourse altered. Ifthedimensions ofthe surface, and itsradii ofcurvature, arelarge compared with thewave-length,wehavephenomenaofregularreflection and refraction, asinoptics.Cases ofabsolutediscontinuityareof course notmetwith intheatmosphere,butthetheory would bepracticallyunaffected ifthechangeofproperties were effected within aspacewhich issmall compared with the wave-length. When ontheother handwehave acontinuous variation such that thechangeofpropertieswithin awave-lengthis negligible,the case isanalogoustothat ofatmospheric refraction oflight,which isdiscussed inbooks onoptics and astronomy.Inanatmosphereofthesamegas,atrest, a variation inthevelocityofsound canonlyarisethrougha variation oftemperature (59).The refraction duetovaria- tionoftemperaturewith altitude was firstdiscussedbyOsborne Reynolds (1876). Suppose that, asusually happens,the temperaturediminishesupwards.Since thevelocityofsound varies asthesquarerootoftheabsolutetemperature,thelower portionsofawave- front willbepropagatedfaster thantheupper ones, sothatafrontwhich wasoriginallyverticalgetstilted upwards moreandmore asitproceeds. Thesound willthere- fore, forthemostpart, passover thehead ofanobserver at asufficient distance, such residual effects ashoperceives being referable todiffraction. Ontheother hand, whenever- the temperatureincreasesupwardsthewaves willhetilted down- wards, andtheeffect atadistance willbegreaterthan ifthe temperaturehadbeen uniform. This latter condition ofthe atmospheresometimesprevailsonaclearnight following a warmday,when, owingtothecoolingoftheground by radiation, thelower strata oftheatmospherearereduced in temperature relativelytotheupperones. 220 DYNAMICAL THEORY OFSOUND equal wave-velocity being supposedtobehorizontal, eachray will travel inaverticalplane. The cur- vature ofaraymaybecalculateddirectly byamethod duetoProf.James Thomson*. IfRbetheradius ofcurvature, thetwo wave-frontspassing throughtheextremities ofanelement Bsofthepathwillbe inclined atanangle BsjR,and ifBsbethe length interceptedonanadjacent rayin thesame verticalplane, wehave Bs'=(l-^j}Ss,(1) where Bndenotes thedistance between thetworays,thestandard casebeing thatshewn inthefigure.Since theelementsBs,Ssarc described inthesame timewehave __= c+Bc G whence, bycomparison with(1),(2) .(3) R cdn When thetemperaturediminishesupwards, dc/dnisnegative andthecurvature l/Rispositive,asinthefigure, andtherays arecurvedupwards. But ifthetemperature increaseupwards, thecurvature isdownwards, sothatanobserver atthelevel of thesource mayhearsounds which would otherwise havebeen intercepted byobstacles. Theformula(3)leads totheordinarylawofrefraction. If i|rbetheinclination oftheraytothehorizontal wemaywrite 3c dc 5=-7-cosondydcdc ifybethevertical coordinate. Hence, along thecourse of aray, d-^r_1_1do dsM cds'' (' *James Thomson(182292), professor ofengineering atBelfast 185772andatGlasgow 187289.' GENERAL THEORY OFSOUND WAVES 221 or csecvjr=const., .....................(6) which isthelaw inquestion. Conversely,from(6)wecan derive theformula(3).When cisknown asafunction ofy theequation (6)determines thepaths. Thesimplest hypothesisisthatthetemperature decreases (orincreases) upwardswithauniform gradient. This includes theparticularcase ofanatmospherein"convectiveequili- brium"undergravity, where thegradientis Hbeingtheheightofthehomogeneous atmosphere (59) correspondingtothetemperature6*.This isattherate of about 1C.per100metres. Ifalawofuniform decrease were toholdwithout limitation, weshould atacertain altitude meet withazerotemperature (absolute).Ifforamoment wetake theoriginatthis level, anddraw theaxis ofy downwards, thetemperaturewillbeproportionaltoy,andthe wave-velocityctoy*.Hence by(6)wehave, along anyray, ......................(8) Thepathsarethereforecycloids,thegeneratingcircles ofwhich rollontheunder sideoftheliney=0.Ifontheotherhand thetemperatureincreases upwardswithauniformgradient,the pathsarethecycloids whosegeneratingcircles rollontheupper side ofthelinewhichcorrespondstothezero oftemperature. Inanypracticalcaseweareconcernedonlywith theportions ofthecurves near the vertices. The arcsmaytherefore be taken tobecircular, with aradius double thedistance below (orabove) thelevel ofzerotemperature.Intheextreme case ofupwarddiminution towhich theformula(7)refers, this radius willtherefore be(roughly)2x273x100=54600 metres, foratemperatureof C. *Itwaspointed outbyLord Kelvin (1862)that this isthecondition into which theatmosphere would bebrought bythefreeplayofconvection currents alone, without conduction orradiation. Itistherefore oneofneutral equilibrium. Ifthetemperature diminish upwardsatagreaterrate theequilibrium becomes unstable. 222 DYNAMICAL THEORY OFSOUND 75. Refraction byWind. Anotherinteresting questionisthat ofrefractionbywind. Auniformmotion ofthemedium introduces ofcourse no complication,therelative motion ofthesound wavesbeing exactlythesame asifthemedium were atrest.Usually, however, thewind-velocitynear thegroundislessthan above, themotion ofthelowerlayersofairbeingobstructed. Hence when awave-front travels with thewind, theupper portions arepropagated (inspace)somewhat faster than thelower, the velocityofthewind being superposedonthat ofsound. The front isthereforecontinually beingtilted downwards. Fora similar reason awave-fronttravelling againstthewindgets tilted upwards,sothat thesound tends topassover the head ofanobserver atadistance. Thisexplanationofthe familiar factthatsound canbeheard better, andfurther from thesource, when this lies towindward thanwhen itisto leeward oftheobserver, was firstgiven byStokes(185 "7). Theonly previous suggestionhadbeen that asound which has travelled acertain distance with thewind hasreally traversed ashorter lengthofair,andhasconsequently become lessattenuated byspherical divergence,than ifthewindhad been absent. Owingtothesmallness ofwind-velocities in comparisonwith that ofsound, thiscause isquite inadequate toexplaintheverymarked effects which areobserved. The truetheory wasdiscoveredindependently byReynolds (18*74), andconfirmed byanumber ofinteresting experiments. Ifweproceedtoapply opticalmethods tothequestion, itisnecessarytodis- tinguish,asinthetheory ofaberration, between the direction ofarayandthat ofawave-normal. Let$1 representthepositionofa wave-front attimet,S' thepositionattime t+St ofthoseparticles which were on$1}and S.2the newpositionofthewave- GENERAL THEOEY OFSOUND WAVES 223 front. LetPxbeanypointonSi,andP1thecorresponding pointonS',sothatPJP' isthepathofaparticle ofthe medium inthetime St.Ontheprinciplesofoptics,thenew position $2ofthewave-front isobtained astheenvelopeof asystemofspheresofradius c&t,described with thevarious pointsP'ofS'ascentres. IfP2bethatpoint onthe envelope whichcorrespondstoP',PiP awillbeanelement of aray,andP'P2anelement ofthewave-normal. Also since P1P/=Ufa,whereUisthevelocityofthemedium, the" ray- velocity" (PiPz/Bt)istheresultant ofthewave-velocity and thevelocityofthemedium. Inthepresent questionthevelocity Uishorizontal, and afunction ofthe altitude(y)only.If-v/r, </>denote the inclinations tothehorizontal oftherayandthewave-normal, respectively,wehave sin(<-^)=-sin^, (1) or =ty-\ sinvjr, (2) ifU/cbesmall, aswillusually bethecase. Toascertain thelawgoverning thechangeofdirection oftheray, consider firstthecase ofrefraction at thecommon horizontal boundaryof twouniform currents U,U'. If<,$' betheinclinations ofthewave-normal onthetwo sides oftheplaneof discontinuity, wehave csec</>+ U=c'secfi+U', (3) each sideexpressingthehorizontalvelocityofthetrace ofthe wave-front ontheplaneinquestion.LordRayleigh pointsout that sincesec<^>'<jc 1,<fiwillbecomeimaginaryif 7'-?7>csec<jb-c' (4) There istherefore total reflection ofallwave-fronts whose inclina- tion(<)tothevertical fallsshort ofacertain limit. Since acontinuous variation ofUcanbeapproximatedto 224 DYNAMICAL THEORY OFSOUND byaseries ofsmall discontinuities, weinfer that(3)will still hold if<,c,Uand <',cf ,Vrefer toanytwopositions onthe sameraj.Theequationis,infact,ageneralizationofArt.74(6). Ifweconfine ourselves totheeffects ofwind alone, sothat cisconstant, wehavealong anyoneray sec -f=const., (5)c or,by(2), secA/Msec2ty=const., (6)c provided ^benottoogreat.Ifwedifferentiate thiswithrespect tothearcs,andput d\fr(ds l/R,dy/ds=sini/r,wefind IdU ..____(7) Therayistherefore curved downwards orupwards, according asdUldyispositiveornegative. If,asusually happens,the wind increases inforceupwards,these two cases will arise accordingasUispositiveornegative,i.e.accordingastheray istravellingwith oragainstthewind. Ifthegradient dUjdy beuniform, therayshave allthesame uniformcurvature, approximately, owingtofchesmallness oftheratioU/c,unless indeed theinclinationtybecomes considerable. Forinstance, araystartingnear thegroundatanelevation a,down the wind, attains analtitude R(1cosa)andhasahorizontal range 2Rsin,whereRc/m}mdenotingtheupward gradientofthewind.Again, raysstartingfrom aheight h,and travelling againstthewind, haveamaximum horizontalrange. This isfound, with thesameapproximation, bydrawingacircle throughthesource totouch thelinerepresentingtheground; thus,ifxbetherange, x2=2Rh, ...(8) since hissmallcompared with R.Itistobenoticed that inalltheseproblemstheFlg'71t pathofarayisnotreversible(seeFig. 71). The effects onthepropagationofsound ofawant ofuniformity intheatmosphere, whether oftemperatureorvelocity,areoften veryremarkable. Thesound ofanexplosion,forinstance, has GENERAL THEORY OFSOUND WAVES 225 sometimes been heard atagreat distance, whilst oversome interveningareasnothinghasbeen observed. Afamiliar instance istheintermittent orfluctuatingcharacter ofthesound from anaeroplane. This isduetorefraction ofthesoundraysasthey traverse masses ofairatunequal temperatures. Theraysareat oneinstant deviated more orlessfrom theearoftheobserver, andthenbrought back ingreater intensity,theaverageeffect being unchanged.Thephenomenonisanalogous,onadifferent scale, tothescintillation ofstars*. 75a.Acoustic Properties ofBuildings. Itisnotaltogetheraneasymatter topredictinadvance the acousticqualitiesofaroom ofgiven designanddimensions, but itmay safelybesaid thatdisappointmentshavebeen often caused bytheneglectofsomeverysimpleconsiderations. Itisa matter ofcommon observation thatspeechintheopenairdoes notcarry veryfarwithoutgreat effort, owingtotheattenuation ofthesound waves asthey diverge.Inanenclosedspacea speaker'svoice isreinforced byechoes from thewalls andthe ceiling,and this is(sofar)tothegood,but itisobvious that theechoes should notfollow theoriginalsounds attoogreat an interval, andshould bequickly extinguished.Echoes areliable tobeprolongedifthewalls areofhighly reflecting quality, marble(forinstance) being speciallyunsuitable. Ontheother hand thepresenceofareflectingsurface close behind the speaker,orover hishead ifthere isalofty roof, isadvantageous. When echoesprovetobeexcessive induration, aremedy may befound intheuseofmore orlessporous materials, e.g.in theform ofhangings,ontheprinciple explainedin66.But itispossibletocarrythisdevice too far,inwhich casethevoice fallsdead, asinopen-air speech. Ithassometimes beenattemptedtosuppressechoes from a roofbywires hanging vertically,with theidea of"breakingthe sound waves." Testimonyastotheefficacyofsucharrange- ments isoftenconflicting,butinthelightofthetheoryof scattering byacylinder (81)itisdifficult toseehowany considerable effect canbeobtained insuch away.Itistrue *This explanationisduetoEayleigh. 226 DYNAMICAL THEORY OFSOUND that, intheinvestigationreferred to,viscosityisneglected,but thiscanhardlyaffect thegeneralconclusions. Amore difficultquestion mayarise throughtheunequal focussing,asitwere, ofsound indifferentpartsofaroom. This isnowbeginningtobestudiedsystematically, bymeans ofmodels, both inthis country,attheNationalPhysical Laboratory,andinAmerica. 75b.Doppler's Principle. This isaconvenientplaceforareference towhat isknown as"Doppler's principle"*.Suppose,forinstance, thataperiodic source ofsound isapproachingastationaryobserver. The number ofmaxima of(say)thecondensation swhich strike the earofthelatter inasecond isincreased, andthepitchisthere- fore raised. Thediminution intheperiodistotheperiodwhen thesource isatrest intheratio ofthevelocityofapproachto thevelocityofsound. When thesource recedes from theob- server, this ratio isnegative,andthepitchislowered. When themotion ofthesource isobliquetotheraysbywhich the sound isheard, thecomponentofitsvelocityinthedirection of therayisalone effective.Analogouseffects areproducedwhen thesource isatrestandtheobserver inmotion. Theprinciple isexemplifiedintheapparent changeofpitchofthewhistle of alocomotive asatrain dashesthroughastation; but itsmost striking and fruitfulapplicationsaremetwith inthetheoryof radiation. *Christian Doppler (1803 5-i),anAustrian mathematician, professor of physicsatVienna 1851. CHAPTER VIII SIMPLE-HARMONIC WAVES. DIFFRACTION 76. Spherical Waves. Point-Sources ofSound. From thispointitisconvenient toconsiderspeciallythe case ofsimple-harmonicvibrations. Inproblems; relatingto theimpactofsound waves onobstacles, ortheir transmission byaperturesinascreen, and soon,theresults willvaryin character with thepitch,thedeterminingelementbeingthe relation between thewave-lengthandthelinear dimensions of theobstacles, &c. Itwillbedesirable, forthesake ofconciseness, touse imaginary quantities somewhat morefreelythan inthepre- ceding chapters. Thusweassume thatthevelocity-potential <varies aseint ,oreikct ,where &=?i/c=27T/X, (1) ifXbethewave-lengthofplanewaves ofthesameperiod 27r/?i. Thegeneral equationofsound waves(70(4))therefore be- comes V2 <+/c2=(2) Inthecase ofplanewaves whose fronts areperpendicular totheaxisofa-,wehave ^+7^=0, (3) dsc-T thesolution ofwhichmaybewritten cf>=Ae-ikx+Beikx , (4) or =Gcoslex+Dsin lex, (5) <f)=Aei(nt-ka......................(6) When weproceedtocalculations ofenergyitisofcourse necessarytorevert toreal forms. Thus, takingtherealpart of(6),wehave =Acosk(ct x)..................(7) Themeanenergy perunit volume, asgiven by70(7),(8), is^pkzAz ,andthemean energytransmittedperunit time, perunit area ofthewave-front, is kp&cA*,or%pn*/c.Az................ (8) Wemaycallthisthe" energy-flux"inthewave-system (7). Theequationofsymmetrical spherical waves, 71(6),now takes theform 2^+W>-o...................(9) andthesolution is eikr , .................. (10) orr<p=Ccoskr+Dsinkr, ............(11) thetime-factor beingunderstood asbefore. Thetwoterms in(10)correspondtowavesdiverging from, orconverging to, theorigin, respectively.Inparticular,thedivergingwaves due toasource Aeikctattheoriginarerepresented by ..................(12)\/ or.inrealform. <=icosnU )................ (13) r4nrr \cj' This isofcourse aparticularcase of73(1). Themaintenance ofsuch asource inanunlimited medium requiresacertainexpenditureofenergy. Thework doneper unittime atthesurface ofasphereofradiusr,onthefluid outside, istheproductofthepressure,the area, andthe outwardvelocity,or4.(14) SIMPLE -HARMONIC WAVES. DIFFRACTION 229 Itisevident thatpcontributes nothingtotheaverage effect, since themean value of9</3ratanypointiszero. Ifwe substitute from(13)wefindthattheaverageoftheremaining partis (> ThisquantityWisindependentofr>aswastobeanticipated, since themean energyinthespaceincluded between two concentricspheresisconstant. Itmeasures theemission of energy (perunittime) bythesource. Theformula mayalso beinferred fromtheconsideration thatatagreatdistance the waves mayberegardedasplane.Ifin(8)wereplace Aby theAf^irrof(13),andmultiply by4nrr*,weobtain theresult (15). Itmustberemembered that thiscalculation oftheenergy emittedapplies onlytoanisolated source infreespace.A sourceplacedinanenclosure withrigidwalls doesnowork on thewhole, since theenergyofthegasisconstant. Even inan open spacetheemission ofenergy maybegreatlymodified by theneighbourhoodofanobstacle. Thus inthecaseofasource Pclose toarigid plane boundarytheamplitudeofvibration at anypointisdoubled bythereflection asfrom, theimageP' (73);theintensityisquadrupled,andtheemission (ononeside) istherefore twice that ofanequalsource infreespace. p Theequation<= fi<**+), ..................(16) or,inrealform,</>=- >cos%U+-],...............(17)TJ7T?" \ C/ maylikewise beinterpretedasrepresentinga"sink"ofsound, i.e.apointwhere energyisabsorbed, under similar conditions, attherateri^B^/Sirc.This conceptionishowever ofnogreat assistance inacoustics. Thenotion ofasimple source, valuable asitisfortheoretical purposes,isseldom realized evenapproximatelyinpractice. Avibrating bodysuch asamembrane, oreitherprongofa timing fork,istendingatanyinstant toproduceacondensation oftheairincontact with itontheonesideandararefaction 230 DYNAMICAL THEOEY OPSOUND ontheother, and istherefore moreadequately represented,in thesimplest cases, byacombination oftwosimplesources near togetherbutinopposite phases. Idealizingthisalittle further weareledtothemathematical conceptionofa"double source." Webeginwithasimplesource ofstrength matapoint 0, andasimplesource ofstrength +111atanadjacent point 0', thesigns indicatingtheoppo- sition ofphase.Ifwenext imagine mtobecome infinitely great,whilst thedistance 00' becomesinfinitely small, insuch awaythattheproductin.00' remains finite,wehave theideal "double source"oftheory.The direction 00' iscalled the "axis," andthelimit ofm.00' iscalled the"strength."Theresultingmotion isevidently symmetricalabout the axis. Ifthedirection 00'bethat oftheaxis ofx,and be taken asorigin,thevelocity-potentialatPdue tosimple sources mat'and 0,respectively,willbegiven by 47TG&=mf-,rf .(18) where rOP,r'=O'P. IfwedrawPP'equal andparallelto O'O,wehave/=OP',andtheexpressioninbrackets isequal tothechangeofvalue ofthefunction e~ikr /rcaused bya displacementof_ptoP'.Hence, ultimately,ifP'P ^x, 4,7rd>=m&# .;r-( J ^dx\rJ.(19) PuttingmSiT=l,wededuce theformula foraunit double source at0,havingitsaxisalong Ox, viz. .(20) this isaparticularcase of73(3).When xalone isvaried, whilstyandzareconstant, itappearsfrom thefigurethat SIMPLE-HABMONIC WAVES. DIFFEACTION 231 Sr=cos68x,where 6denotes theinclination ofOP to One. Henced/dx=cosdfdr,and 9/gikr\ (21) Performingthedifferentiation, wefind Forsmall values ofkr, i.e.within distances from which are smallcomparedwithX/2?r,thisbecomes Ontheother hand, forlargevalues ofkr, Q-ikr=ik cos6,..................(24) sothatalong anyoneradius vector thecondensation(s=</c2 ) variesultimatelyas1/r.Theradial andtransversecomponents ofthevelocityare tobefound bytheformula(6)of69;- viz.theyared(f>/dr and~d(j)/rd0, respectively.Itappears thatneartheoriginthese areofthesame order ofmagnitude, whilst atagreatdistance thelateralvelocityislessthan the radial intheratio1/kr. Introducingthefactor Ceintin(24), andtakingthereal part,wefindthatthevelocity-potentialduetoadouble source, ofstrengthcosnt,atagreat distance, is <f>= -r-^-smnit -}cos#......... ,...(25) ^4nrr \ C/^' Thewaves sentoutinanydirection 6arethereforeultimately plane,ofthetype (7),provided A=kOcos6/4i'irr>themere difference ofphase being disregarded;andtheflux ofenergy (acrossunitarea)willtherefore bep/c4c(72cos2 0/327r2r2 .Multi- plying by277Tsin 6.rS0,which isthearea ofazone ofa sphericalsurface ofradius rbounded bythe circles whose angularradii are6and+89,andintegratingfrom 6=to Q=TT,wefindthatthetotal emission ofenergy bythedouble source Ccosnt is 232 DYNAMICAL THEOBY OFSOUND Itwillbenoticed that asthewave-length Xisincreased, andkaccordingly diminished, thefundamentalequation (2) tends toassume theform Vs*=........................(27) which ismetwith inthedynamicsofincompressible fluids, and inthetheories ofattractions andofelectric andthermal con- duction. This assimilation maycome about intwoways, either throughadiminution inthefrequency (?i/27r),orbyanincrease intheelasticityofthemedium andconsequentlyinthewave- velocity. Under thesame condition the formula(12) approximatestotheform which istheexpressionforthepotentialofamagnetic pole,or forasource ofelectricity,and soon ;whilst inthecaseofthe double source (21)thelimitingform is(23),which isrecognized asthepotentialofaninfinitelysmallmagnet. Afurther remark ofgreat importanceisthatwithinany region,freefrom sources, whose dimensions aresmallcompared with X,theconfigurationoftheequipotentialsurfaces <=const. isatanyinstantsensiblythesame asifthefluidwere incom- pressible.Forthevalue of<f>duetoanexternal source differs from itsvalue inthecase ofincompressibility chiefly bya factor e~ikr ,where rdenotes distance from the source. If bdenote thegreatestbreadth oftheregion,this factor can at mostvaryiritheratio e~'ikb ,which differsverylittle fromunity when kbissmall. 76a.Reflection ataPlane Surface. The reflection ofwaves from asourcebyarigidwallcanbe represented bymeans ofanimage,asexplainedin 73.Thus inthecaseofaprimarysource ofstrength Gwehave ... (1) wherer*i,rzdenote distances from thesource andtheimage, respectively.Ifweputforamoment r.,+r,=2r,v,-r 1=2,y, ...............(2) SIMPLE-HARMONIC WAVES. DIFFRACTION 233 thismaybewritten 47rd>=Ce~ikr\(-+-}cosks+i(---}sinks].(3) r(Virj Vnrj ]v' Atdistantpointsraandr2willbenearly equal,andthe first term inthelargebracket willbethemoreimportant. Themaximum value ofsisthedistance(h)oftheprimary source from thewall,and ifthewave-length (X=2?r/^)islarge comparedwith thiswehavepracticallyatdistantpoints (4) Thecombination isthusequivalenttoasource ofstrength 2(7aswas tobeexpected,butsince waves aregenerated only ononesideofthewall,thesupplyofenergy requiredtomain- taintheoriginalsource istwicewhat itwould beIDanunlimited space. Inthegeneralcasetheintensitywillvaryasthesquareof theamplitudeoftheexpression ontheright-handsideof(3), i.e.as (5) ra'2''2) This liesalways between (- )and C If,incontrast totheformersupposition,A.issmall compared with h,theupperlimit oftheintensityisattained atpoints such that r2rjisamultipleof\,andthelower limitwhen rai\isanoddmultipleof|X.Thecircumstances areanalogous tothose ofFresnel's interferenceexperimentinOptics. Theproblemofreflection fromanysystemofsources, andin particular from adouble source, canbedealt with inasimilar manner, buttheresults willvaryincharacter. For instance, suppose wehave adouble source atadistance from thewall which issmallcomparedwith thewave-length.Iftheaxis of thissource isparalleltothewall, theeffect atdistant pointsis practically doubled, whereas ifitisatright angles pointing (say)from the wall, theimagewill alsopointfrom thewall 234 DYNAMICAL THEORY OFSOUND inthefictitious medium ontheother side,andtheeffect at distantpointswillbepracticallyneutralized. Theproblemofreflection atthefreesurface from asource ofsound inwater hasalreadybeen referred to.Ifweneglect thewavesgeneratedinthe air,forthereason indicated in61, wemaymake useoftheconceptionofanegative image. Thus in placeof(1)wehave /v* ft*^'ri rz since thismakes atthefreesurface where rl=rz.This isequivalentto ---)cosJcs +i(-+-)smks,(8) (9)x''2 andtheintensityisconsequently proportionalto Near thesurface this isalways relativelysmall. If\besmall comparedwith h,wehave, atapoint whosedepthiszand horizontal distanceas, (z-Kf (z+A)'2 providedzandAareboth small comparedwith as.Hence 7.2/Jl 47T/1 2 ^(r 2-r])=--^= .^............. (H) Forpointsonthesame vertical theminima ofintensity occurwhen k(r2r,)=0,2ir, 4-Tr,... ,or -e -'r 2/1 /r2;? 77.Vibrating Sphere. Bymenns ofthe fiction ofadouble source, ofsuitable strength,atthecentre itispossibletocalculate thewaves generatedinthesurroundingairbyavibratingsolidsphereof anyradius. Asthis isalmost theonlyproblemofthekind SIMPLE-HARMONIC WAVES. DIFFRACTION 335 which canbecompletelysolved wedevote somespacetoit.The work issimple,andtheresults throw agooddeal oflighton other cases. Forreasonsjustreferred to,itisinstructive tolook firstat thecasewhere thefluid isincompressible. Wetaketheorigin atthemeanpositionofthecentre ofthesphere}andtheaxisof (Kalongthe lineofitsvibration; andwedenote itsvelocity by7.Thevelocityofthe fluid incontact with thesphereat anypoint P,resolved inthedirection ofthenormal, must be equaltothenormal componentofthevelocityofthepointP ofthesphere itself,i.e.to?7cos 0,where 6istheangle POx. Thisgives [r=a],..................(1) ifabetheradius. Thevelocity duetoadouble source at in anunlimited mass ofincompressiblefluid isoftheform (h=-5- ocose;.....................(2) r4-Trr2 v' andinorder that thismaybeconsistent with(1)wemust have G=2-rra5U. ........................(3) With thisdetermination ofGthe effect ofthesphereonthe fluid isexactlythat ofthedouble source, andthesolution of ourproblemis C7a3 ,,,.,.6=-,-cos9......................... (4) T2r2 Thisdepends onlyontheinstantaneous value ofU,asweshould expect,since under thepresent hypothesisdisturbances are propagatedwith infinitevelocity.Itshould alsobenoted that there issofarnoassumptionthatUissmall. Thedirections ofmotion atvariouspointsofthefieldmay beshewn bytracingthe"lines ofmotion," which arelines drawn frompointtopoint, alwaysinthedirection ofthe instantaneousvelocity.Inthecase ofsmallvibratory motion, which wehaveespeciallyinview, eachparticleoscillates backwards andforwardsthroughashort distancealongtheline onwhich itissituate. If8r,r&dbetheradial andtransverse DYNAMICAL THEORY OFSOUND projectionsofanelement ofsuchaline, thesequantities must beproportionaltotheradial and transverse componentsof velocity,viz.-3</3rand -d<j>[rdd, respectively.Hence theintegralofwhich is ^^ where 6isaparameterwhich varies from onelineofmotion toanother. The curves, which areidentical inform with the lines offorceduetoasmall magnet,areshewn inFig.73. Fig.73. Tocalculate thereaction onthespherewodivide thesurface intozones byplanes perpendiculartoOx. The, area ofazone being27ra2sin0S0, theresultant force onthesphereinthe direction of^'-positiveis X'=-I"pcos6 .27T(t" sinOd&(V) Jo SIMPLE-HARMONIC WAVES. DIFFRACTION 237 The constantpartofthepressurecontributesnothingtothe resultant. Thevariablepart is,ifterras ofthesecond order in thevelocities beneglected, since tentersonlythroughU.Substituting wefind (9) Theremarkablepointhere isthat the force isindependent ofthevelocity, anddepends onlyontheacceleration ofthe sphere.Ifthemass ofthespherebeM,and ifitbesubjectto other extraneous forceX,itsequationofmotion willbe ' .....................(10) or (M+^pa^=X................(11) This isthesame asifthefluidwere abolished, andtheinertia ofthespherewere increased byfTrpa3 ,i.e.byhalfthat ofthe fluidwhich itdisplaces.Itwasshewn byStokes(1843)that thisconclusion isaccurate evenwhen therestriction tosmall motions isabandoned. Thereis,asweshall see(79),nothing peculiartothe sphereinthegeneralcharacter oftheabove result, butthe apparentaddition totheinertia willvaryofcourse with the shapeaswell asthe sizeofthe solid, and willusuallybe different fordifferent directions ofmotion, ase.g.inthecase of anellipsoid. Thetheoryheretouched uponhashadagreat influence onrecentphysical speculations,andinparticularwas responsibleforthesuggestionthat theapparentinertia of ordinarymatter mightbepartlyoreven wholly duetothat of asurroundingaetherial medium. Turning now totheacoustical problem,letthevelocityof thespherebeexpressed symbolically by U=Ae*nt.........................(12) Thesurface-condition willhave thesame form(1)asbefore. Thevelocity -potentialofadouble source Geintat is nA iaikr\_JL(L_).coB0,............... (13) r4-Tror\r/ 238 DYNAMICAL THEORY OFSOUND by76(21),thetime-factor eintbeingomitted. The ratio ofG toAisthendetermined by(1). Themostinterestingcase iswhere theradius aofthe sphereissmall comparedwith\/27r,whereXisthewave-length. Intheimmediateneighbourhoodofthespherekrwillthen besmall, andtheformula (13) is,forthisregion, practically identical with(2). Itfollows that a=27ra4, ........................(14) nearly, andfurther thatthelines ofmotion nearthespherewill havesensiblytheconfigurationshewn inFig.73.Theapparent addition totheinertia ofthespherehasvery approximately thesame valuefvrpa8asbefore. Ontheother hand, atdistances Twhich arecomparable with, orgreater than, \,themotion of the fluid isaltogethermodified bythecompressibility.At sufficiently greatdistances wehave,by(13)and(14), <b=xika?A coB9, ............... (15) v2 r or,inrealform, ,r\a. . '............(^ correspondingtoavelocity J7=Acosnt .....................(17) ofthesphere. Theamplitude now variesultimatelyas1/r, instead of1/r2 ,asinthecaseof(4). Theinvestigationsofardisclosesnothing analogousto africtional resistance, whereas weknow thatowingtothe generationofwavestravelling outwards acontinual abstraction ofenergy must takeplace. Tocalculate either thedissipative resistance, orthework done, atthesurface ofthesphere, we should have tousethecompleteformula(13); buttheemission ofenergy maybeascertainedindependently from theformula (26)of 76.Thestrengthoftheequivalent double source being given approximately by(14),wefind W=%7rpk*aecA2...................(18) Ifp'denote themeandensityofthesphere,itsenergy when vibrating under theinfluence of(say)aspringwillbe SIMPLE-HARMONIC WAVES. DIFFRACTION 239 If,followingaprocedure explainedin 11,weequatetherate ofdecayofthisenergytoW,wefind andtherefore A=A e~t/T ,.....................(20) Q/ providedr= ,. .,.......................(21) rn(ka)3p^ The ratio(wr/27r)ofthemodulus ofdecaytotheperiodis thereforeusually very great. 78.Effect ofaLocal Periodic Force. Correspondingresults can,with thehelpofmore orless intuitive considerations, beobtained forother forms ofvibrating solid,butthework ismuchsimplified byapreliminary theorem, which hasalsoanindependentinterest. This relates tothe effect ofaperiodicextraneous force concentrated about apoint inagaseousmedium. Anelementary proofcanbederived atoncefrom thepre- ceding investigation.The result willobviouslybethesame if theforce beimaginedtoactonaninfinitelysmallsphere havingthesamedensityasthesurroundingfluid. The effect istherefore that ofadouble source; and ifwenowdenote the concentrated force, supposed acting paralleltosc,byPeini ,we find,puttingM tsirpa?in77(11), P=^iirpkca?A,.....................(1) andtherefore, by77(15),forlargevalues ofkr, pp-ikr cos0 ...................(2)^'r Comparingwith 76(24)weseethat thestrengthofthe double source is i,Peint jpkc, or,takingtherealpart, j-sinnt. plcc Itfollows from 76(26)thatthework donebyaperiodicforce Psin.nt ingeneratingthedivergingwaves is /r2P2W= .. . (3) K 247TPC........................ w perunit time. Foragivenmedium thisvariesinverselyasthe squareofthefrequency. 7.waves generaceu ayviuratiug ooixu. Wereturn totheproblemofinvestigatingthewaves generated byavibrating body, Inorder nottocomplicatethe questiontoomuchwewillassume thatthebodyhassome sort ofsymmetrywithrespecttoanaxis;thus itmaybeaform ofrevolution about this axis, oritmayhave twomutually perpendicular planesofsymmetry meetinginthis axis, or (again)asingle planeofsymmetry perpendiculartothe axis. Inanycase this axis istaken tobethedirection(Ox)of vibration. Thedimensions ofthesolidbeing supposedsmall incom- parison withX/27T,themotion ofthefluid intheimmediate neighbourhoodwillbesensiblythesame asinthecase of incompressibility,andtheprincipaleffect onthebodywillthere- forebeequivalenttoanincrease ofinertia. Toestablish this latterpointinageneral manner, wenote thatthe(irrotational) motion ofafrictionlessliquid duetothemotion ofasolidinit willhave thevelocityatevery pointinadeterminate ratio tothevelocity Uofthesolid, andthatthetotal kineticenergy ofthefluidmaytherefore beexpressed by^pQ'U*, where Q'is aconstant, ofthenature ofavolume, depending onlyonthe sizeandshapeofthesolidandthedirection ofitsvibration. Hence ifMbethemass ofthebody,theequationofenergy takes theform ...............(1) where theright-hand memberrepresents therate atwhich work isbeingdonebytheextraneous forceX.Thus X, .................. (2) which shews thattheinertia ofthebodyisapparentlyincreased bytheamountpQ'.Anequivalent statement isthat the reaction oftheliquidisequivalenttoaforcepQ'dU/dt. Intheactual case ofthegaseous medium, itisplainthat if the solid were removed, and itsplace supplied byfluid, the motion atadistance would beveryapproximatelythesame as would beproduced byasuitableperiodicforce from without, thevolumedisplaced bythesolid, aswell astobalance the reactionjustreferred to,itsamount would be P^p(Q +Q')d ^^^p(Q+Q')Aei^, ......(3) if U=Aeint.............. ...........(4) By78(2),thevelocity-potentialatagreatdistance rwill therefore be e cose= Comparingwith 76(24)weseethattheeffect ofthevibrating solid isequivalenttoadouble source ofstrength C=(Q+Q')A, andthattheemission ofenergyisaccordingly by76(26).Inthecase ofthesphere wehaveQ=|7ras , Q'=%Q,andtheresultaccordingly agreeswith 77(18).It canbeshewn that foracircular disk ofradius a,moving broadside on,Q'^^vra3 ,whilst Qofcourse =0. 80.Communication ofVibrations toaGas. Thecircumstances whichgovern theefficiencyofavibrating bodyingeneratingsound waves, andthecomparativeeffects in differentgases,were elucidated byStokes inaclassical memoir "OntheCommunication ofVibrations from aVibrating Body toasurroundingGas*." Thestarting pointoftheinvestigation wasanobservation byProf. J.Leslie (1837), whofound thatthe sound emitted byabellvibratinginanatmosphereofhydrogen wasextremelyfeeble ascomparedwith the effect inair.No satisfactory explanationofthisphenomenonwasforthcoming uptothetime ofStokes'paper.Theessence ofthematter is conveyedinthefollowing quotation: "When abodyisslowly moved toand froinanygas,the gasbehaves almostexactlylikeanincompressible fluid, and *Phil. Trans. 1868. Thepassage which follows below isfrom "abstract"intheProc. Roy. Soc.the anterior becomesposterior,ittrierate 01alternation orwie body's motion betakengreaterandgreater, or,inother words, theperiodic time lessandless,thecondensation andrarefaction ofthegas,which inthe first instance wasutterly insensible, presently becomes sensible, andsound waves(orwaves ofthe same nature incasetheperiodic timebebeyondthelimits of audibility)areproduced,andexistalongwith thereciprocating flow.Astheperiodictime isdiminished, moreandmore ofthe encroachment ofthevibrating bodyonthegasgoestoproduce atruesound wave, lessandlessamere localreciprocatingflow. Foragiven periodic time, andgiven size, form, andmode of vibration ofthevibrating body,thegasbehaves somuch the morenearlylikeanincompressiblefluid asthe velocityof propagationofsound initisgreater ;andonthisaccount the intensityofthesonorous vibrations excited inairascompared withhydrogen maybevastly greater than corresponds merely withthedifference ofdensityofthetwogases." These remarks areexemplifiedintheresults of77(13), (14). Ifwefixourattention onapointatadistance from the sphere, supposed vibrating with thevelocity U=Acosnt, .(1) themotion there isgiven, when theperiodissufficiently long, bytheformula Aas $=7-7cos0.cos?z, .................. (2) asifthefluidwereincompressible. Butwhen thefrequencyis increased until thewave-lengthissmallcomparedwith the distance rfrom thecentre, theappropriateformula is kci?A cos' .sinn(t c andtheamplitudeisaccordingly greater than intheformer caseintheratio kr,or27rr/X. Forthesamefrequency,the amplitude, whichdepends onkjcorn/c2 ,willindifferentgases SIMPLE-HARMONIC WAVES. DIFFRACTION 243 nowvaryinverselyasthesquareofthewave-velocity. Again, theemission ofenergy is,by77(17), W=^7rpIciaecA2=^7rpniaB /cs .A\ (4) and sovaries(forthesamegas)asthefourthpowerofthe frequency.Theemission indifferentgaseswill(forthesame frequency) vary inverselyasthe fifthpowerofthewave- velocity,ifweassume (59) thatthelatter variesinverselyas thesquareroot ofthedensity. Forinstance itwillheabout 1000 times lessinhydrogen than inoxygen. Inorder further toillustrate theeffect ofthelateral motion ofthegas,nearthesurface ofthesphere,fromthehemisphere which isatthemomentmovingoutwards tothatwhich is moving inwards, inweakeningtheintensityofthewaves propagatedtoadistance, wemaycalculate what theemission would beifthis lateral motion wereprevented. For this purposewemay (after Stokes) imaginealargenumber offixed partitionstoextendradially outwards from near thesurface. Inanyoneofthenarrow conical tubes thus formed, themotion willbeofthesame character asinthecase ofsymmetrical sphericalwaves. Now auniform, radialvelocity Ccosntover thesurface ofasphere would beequivalenttoasimplesource 47ra2cosnt,andthecorrespondingemissionperunit area would beP2a2pcC\by76(15). Ifwenowput0=Acos0, andintegrateoverthesurface, wegetthetotal emission inour systemofconical tubes. Theresult isW=fTrkWpcA*, (5) since theaverageofcos2foralldirections inspaceis .If wecomparethiswith(4),weseethat the effect ofthelateral motion istodiminish theemission intheratio|7e2a.2 . When, asforexampleinthecaseofaplateorabell,the surface isdivided bynodal lines intoanumber ofcompart- ments vibratinginopposite phases,theopportunityoflateral motion isincreased, andtheemission ofenergy correspondingly weakened. Forfacilityofcalculation Stokes took thecase of aspherical surface, with varioussymmetrical arrangementsof nodal lines. Intheproblemoftheoscillating sphere wehave onesuch line,viz.thegreatcircle &=|TT,andtheemission, as 244 DYNAMICAL THEORY OPSOUND wehavejust seen, isdiminished bythelateral motion inthe ratio%k'W. Forasphericalsurface withtwonodalgreatcircles meetingatright anglesthe effect ismuchgreater,theratio being ^-tea*. And asweincrease thenumber ofcompartments intowhich thesphereisdivided, the ratio, already very small, decreases withenormousrapidity. Forthesake ofsimplicityithasbeen assumed inthe precedingstatements that theperimeter2-rra ofthesphereis small comparedwith X.The influence oflateral motion is however notconfined tothis case, but willmake itself felt whenever thedimensions ofthecompartmentsreferred toare small comparedwith X,.Inthecaseoftheoscillating sphere there isnodifficultyinworkingouttheresult without any restriction tothevalue ofka,startingfrom theformula(13)of 77. Stokes hasalsoinvestigated mathemabicallythecase ofa cylinder vibratingatright anglestoitslength,where thesame cause isofcourseoperative.Inthiswayanestimate is obtained ofthedirect effect ofavibrating stringingenerating air-waves. This involves the ratio oftheperimeterofthe cross-section ofthestringtothelengthoftheair-waves, and isinanypracticalcaseextraordinarilyminute. Asexplained in24,almost thewhole ofthesoundgivenoutwhen apiano stringisstruck comes from thesoundingboard. 81. Scattering ofSound Waves byanObstacle. Wehave next toconsider thedisturbance producedina train ofsound waves byarigidobstacle whose dimensions are small compared with thewave-length. The scattered waves which aresensible atadistance areduemainlytotwo causes. Iftheobstacle were absent thespacewhich itoccupieswould betheseat ofalternate condensations and rarefactions. The effect oftheobstacle inrefusingtoexecute thecorresponding contractions andexpansionsofvolume is,atadistance, approximatelythesame asifinamedium otherwise atrest its volume were toundergoaperiodic changeofjusttheopposite character. The result isequivalenttoasimplesource. On SIMPLE-HARMONIC WAYES. DIFFRACTION 245 ave-system, which isduetotheimmobilityoftheobstacle. fthelatter werefreely movable, and ifithadmoreover the imedensityasthesurrounding air, itwouldswingtoand fro dththeair-particles,andthesecondwave-system would be bsent. Thissystemisaccordinglythesame aswould be roduced iftheobstacle were constrained tooscillate with motionexactly equal andoppositetothatofthe airinthe rimarywaves when undisturbed. The effectis,aswehave 3enin79,that ofadouble source. Itmight appear,atfirst ight,thattheformer ofthesedisturbinginfluences would be inch lessimportantthan thesecond, but initseffect ata istance itbecomescomparable, owingtothegreater attenuation ylateral motion ofthewavesproceedingfrom adouble source. IfQbethevolume oftheobstacle, thestrengthofthe iinplesource due tothe firstcause is dsQd rhere s,<j&refer totheprimarywaves. Inthecaseofasystem fplanewaves <=(7e-to........................(2) icident onasmall obstacle at0,thisgivesavelocity-potential (3)\' Asregardsthesecond cause,wewillassume forsimplicity tiattheobstacle hasthedegreeofsymmetry postulatedin 79withrespecbtothedirection (Ox)ofthevibration inthe ir-waves. Ifthewave-system (2)were undisturbed, the elocityoftheair-particlesat would berepresented ymbolically byikC,andthestrengthofthedouble source due otheobstacle movingwith thisvelocityreversed would be -ik(Q+Q')0,inthenotation of79.Thescattered waves at distance, duetotheimmobility,arethereforerepresented by r. ' >y76(24). Thecompleteresult isgiven by<=fa+</>2. Itfollows thattheamplitudeofthescattered waves atany 246 DYNAMICAL THEOEY OFSOUND distantpoint is,forsimilar forms, directly proportionaltothe volume ofthe obstacle andinversely proportionaltothe squareofthewave-length.This latterparticular raighthave been foreseen without calculation. The ratio totheoriginal amplitude mustnecessarily vary directlyasthevolume Q, andinverselyasthedistance r,and inorder that theresult maycome outapurenumber wemust divide byX2 ,sinceX istheonly other linearmagnitudeinvolved. Theemission ofenergy, being proportionaltothesquareoftheamplitude, willthereforevaryasX~4 .This lawoftheinverse fourth power holds alsoinoptics,and forasimilar reason, withrespect tothescatteringoflight byparticles whose dimensions are smallcompared with thedimensions ofthelight-waves.The blue ofthesky,forinstance, isattributed totherelative preponderanceoftheshorter waves inthelightscattered by themolecules ofair,andpossibly byotherparticles;inthe transmittedlight,ontheother hand, thelongerwavespre- dominate. Thetheoryisdue toLordRayleigh,whohasalso pointedtoanacoustic illustration in\vhat arecalled"harmonic echoes." Ifacomposite musical note, consistingofafunda- mental tone with itsoctave, &c.,besounded near agrove oftrees, forexample,theratio oftheintensityoftheoctave to that ofthefundamental willinthescattered sound be16times what itwasintheoriginalnote. The scattered sound may therefore appeartoberaised inpitch byanoctave. The actualscatteringofenergyisfound byaddingthe results duetothesimpleandthedouble source. Thismaybe proved bycalculatingthework done atthesurface ofasphere oflargeradius r.Theterms duetothecombined action ofthe twosources contain afactor cos9,and sodisappear when integratedover thesurface. Hence, by76(15), (26), Theenergy-fluxintheprimary wavesbeing ^pk2cC2 ,by 76(8),theratiowhich theenergyscatteredpersecond bears tothis is SIMPLE -HABMONIO WAVES. DIFFRACTION 247 Inthecase ofthesphere wefound Q'=$Q=|7ra3 ,andthe expression (6)therefore reduces to Kka^.Tra? .........................(-7) Inother words, thesphere scattersonlythefraction(/ca)4of theenergywhich fallsuponit.Forexample,ifthewave- lengthbeametre (which correspondstoafrequency ofabout 332),andthediameter ofthesphere1mm., thefraction is roughly7'6x10~".Inthecase ofthecirculardisk,where Q'=f-7ra8 ,Q=0,the ratio ofthescattered totheincident energyis$(ka)*. Themathematical theoryofthescattering bycylindrical obstacles ismore difficult. Wewillmerely quote theresult, based onLord Rayleigh's calculations, thatwhenplane waves areincident onacircularcylinderofradius athefraction ofthe incident energy which isscattered isf7r'2 (/i;a)3 ,approximately, itbeingassumed asusual that lea issmall. Forawire of diameter 1mm.,andawave-lengthofametre, this Itistobeobserved however thatinthecaseofveryminute obstacles theorder ofmagnitudeoftheresults maybecon- siderablymodified byviscosity. Thedeterminingelement here isthe ratio ofthediameter oftheobstacle tothequantity hwhich wasintroduced in66asameasure ofthethickness of theair-stratum, atthesurface ofthe obsbacle, whose motion isappreciablyaffected bythe friction. When the ratio in questionismoderately largetheinfluence ofviscosityonthe results willbevery slight. Thedistribution ofvelocityintheimmediateneighbourhood oftheobstacle willbesensiblythesame asinthecase ofa uniform current ofincompressible fluidflowing pastthebody. Inthecaseofthesphereitcanbedeterminedcompletely,but thefollowing approximationwillbesufficient. Weassume , (8) where the firsttermrepresentstheincident waves, andthe 248 DYNAMICAL THEORY OFSOUND second istheform which thevelocity-potentialofadouble source assumes(76)when krissmall. Thismakes -d ^-=iJcCcos0e-ikr+~cosd, ......... (9)g.. rs> \j andthecondition ofzeronormalvelocityforr=aistherefore approximatelysatisfiedprovidedB=$ikasO.Hence inthe neighbourhoodofthesphere wehave / ns\ 1l-ik(r+^)cos0k.........(10) - nearly. The velocities arethereforenearlythesame asifthe fluidwereincompressible. Thepressureisgiven by p=p Q+p^p +inp(j).............(11) This differs from thepressure (p+inpC) which would obtain attheoriginiftheobstacle were absent byaterm which issmall, oftheorder kr,incomparison. Atpoints whose distance ris a,moderatemultipleofa,whilst still small comparedwithA,,thepressure approximatesevenmoreclosely tothatduetotheincident waves alone. 82.Transmission ofSound byanAperture. Indiscussingthetransmission ofsound waves byanaperture inathin screen wewillsuppose,inthe first instance, thatthe dimensions oftheaperturearesmallcomparedwiththewave- length.This isofcourse themostinterestingcasefrom an acousticalpointofview. Thescreenbeing supposedtooccupytheplane&=0,and theorigin beingtaken intheaperture ($),letawave-train represented by <=Ce-<''* ........................(1) beincident from the left. Ifwedistinguishthefunctions relatingtothetwosides ofthescreenbythe suffixes 1and 2, weshould have, ifthescreen werecomplete, fa=Ce~ikx+Ceikx ,<a=0, ............(2) thesecond term in<j ,whichrepresentsreflected waves, being chosen soastomakedfa/dx=forx=0. Intheactualproblemthedisturbance duetotheaperture willbeconfinedmainlytotheimmediate neighbourhoodofS, andmaybetaken tobeverysmall atdistances from which, SIMPLE-HAEMONIO WAVES. DIFFRACTION 249 though largeascompared with thelinear dimensions of8, aresmallcomparedwith X.Lettwo surfaces bedrawn, on thetwo sides, atsome such distance from 0,eachabutting onthescreen inthemanner indicatedbythedotted lines inthefigure.Within theregion thus bounded, the fluid oscillates backwards andforwards almost asifitwere in- compressible, andthe total flux(G7)throughtheaperture will therefore bear aconstant ratio tothedifference ofthe velocity-potentialsatthetwo surfaces. This willperhaps be understood moreclearlyifwehave recourse totheanalogyofelectric conduction. Suppose wehave a largemetallic mass, severed almost intwobyanon-conducting parti- tionoccupyingtheplaceofthe screen. Ifthismass formpartof anelectric circuit, there willbe little variation ofpotentialinit exceptintheneighbourhoodofthe narrow neckwhich connects thetwo portions. The electricpotentials atadistance onthetwosidesbeing faandfa,thecurrent throughtheneck willbePig. 74. K(fa-fa), .(3) whereKmaybecalled the"conductivity"oftheneck, the specific conductivityofthesubstance beingtaken tobeunity. Inthehydrodynamical question, also, thequantityKmay appropriatelybecalled theconductivityoftheaperture.Itis easilyseen that itisofthenature ofalength. Atthetwosurfaces shewn inthefigurewehavefa=20, fa=0,approximately,andthetotal fluxthroughtheaperture istherefore 2KG. Ifanequalfluxwere directedsymmetrically from theaperture ontheleft-hand side, thecombination would beequivalent,inanunlimited medium, toasimplesource of strength4<KC. Hence, by76(12), KGa-ikr .(4) 250 DYNAMICAL THEOEY OFSOUND Thecorresponding velocity-potentialonthenear side is evidentlyKG61=Ce~iJtx+Geikxe"ikf . (5) ^irr Theenergy (W)transmitted bytheaperture persecond is bytheabovereasoningone-half thatdue toasimplesource 4,KC at0,whence, by 76(15), W=pk<-cK*C~/7r (G) Theenergy-fluxintheprimarywaves(1)being ^pkz cC'*,the ratio ofWtothis is2J^2 /7r.Itistobenoted that this is independentofthewave-length X,solong,ofcourse, as\is large compared with thelinear dimensions ofS. Theexact calculation ofKforvarious forms ofapertureis naturallyamatter ofsomedifficulty. Foracircularaperture ofradius aitisfound thatK2a; forother formsdiffering little from acircle thevalue issensiblythesame asfora circularapertureofthesame area, the circle being evidently a" stationary" form, inthesense inwhich thisterm isused inthetheoryofmaxima andminima. Itappearsthen that acircular(ornearly circular) aperturetransmits thefraction S/7r2 ,or'816, oftheenergy propagatedacross anequalarea (ira?}intheprimarywaves. Thisis,under thepresent limitation astosize, very great comparedwith theenergy intercepted byadisk ofthesame dimensions(81).The figure opposite givestheshapesofthe surfaces ofequal pressure (<2=const.), drawn forequidistantvalues of<j62,inthe immediateneighbourhoodofacircularaperture,andshews howrapidlythese tend toassume thesphericalform. The directions ofvibration oftheair-particlesareofcourse normal tothese surfaces. Withregardtofurtherproblemsofthekindwemust content ourselves with afewstatements ofresults. Inthe case ofanapertureintheshapeofalongnarrowslit,whose breadth issmallcomparedwith X,theenergytransmitted is again comparable with, andmayevenconsiderably exceed, thatcorrespondingtoanequalarea ofwave- front inthe primary waves. Inthecase ofagrating composedofequal, SIMPLE-HARMONIC WAVES. DIFFRACTION 251 parallel,andequidistantslits inathin screen, thefraction of thetotal incident energywhich istransmitted isfound tobe 1/(1+k"P\"where k=2?r/Xasusual, and Ia+b, irb -logSecYT~-r., TTa '2(a,+b)(7) where adenotes thebreadth ofanopening,and 6that ofeach intervening portionofthescreen. Asanumericalexample, Fig. 75. supposethewave-lengthtobetentimes theinterval a+b between thecentres ofsuccessiveapertures; theneven ifthe apertures formonly one-tenthpartofthewhole area ofthe screen, 88percent, ofthesound willgetthrough.Inthe 252 DYNAMICAL THEOEY OFSOUND caseofagratingformed byequidistantbarsofcircularsection, thecorrespondingvalue ofIis Z=7r62 /a, (8) where bistheradius ofthesection, andathedistance between theaxes ofconsecutive bars. Itisimplied, however, thatthe ratiob/amust notexceed(say). 83. Contrast between Diffraction Effects inSound and Light. Influence ofWave-Length. Intheinvestigationof82anaperture wasfound toactas asimplesource fromwhich sounddivergesonthefarther side uniformlyinalldirections. This isinstrikingcontrast with what isusuallyobserved inthecase oflight.Wehave sofar noindication ofanythingofthenature ofbeams orraysof sound, justaswhen sound waves were incident onanobstacle wefoundnothingofthenature ofasound-shadow. The difference intheresults isduetothefactthatthedimensions oftheaperture (orobstacle) have beensupposedsmall in comparison with thewave-length,whereas withlightthe relation isusuallythereverse. Wehave avoidedtrespassingonthedomain ofOptics, but asthedynamicalconditions areinthepresent subject perfectly definite, itmaybepermissibletoexamine thisquestionofthe influence ofwave-lengthalittlemorefully. Consider theregionofspace lyingtotherightoftheplane a;=0.Ifthisplanewere afixedboundary,and ifthere were nosources ofsound intheregion, anydisturbance would ultimately pass away. Any steady periodicmotion inthe regionmust therefore intheabsence ofinternal sources bedue tomotion oftheboundary,andwillbedeterminate when the value ofthenormal componentofthevelocityatevery pointof thelatter isgiven.Itcan,moreover, beexpressedinterms of thisdistribution ofnormalvelocity,asfollows. The fluxout- wards fromanelement 88oftheplaneisd<j>/dn.8S,ifBn denote anelement ofthenormal drawn inwards from BS,and ifinimagination weassociate with thisanequalfluxinthe oppositedirection ontheother side, theresult isequivalent SIMPLE-HAEMONIO WAVES. DIFFEACTION 253 toasource23</3w.8Sininfinitespace. Thecorresponding velocity-potentialatapointPis where rdenotes thedistance ofSSfrom P.Integratingover alltheelements SSoftheplane,wehave 1rrAth e-ikr ^P^~~\\fe~~dS, ...............(1) r2irJJonrx' which istherequired formula. Themotion totherightoftheplaneso=isalsodeterminate when thevalue of <atevery pointoftheplane, andthence thepressure,isgiven, these twoquantities being connected by therelation p=p+p(j>= <pQ+ikcp<j>. Supposeforamoment that inanotherwise unlimited medium wehave athinmassless membraneoccupyingtheplanex=0,andthatoneachelement ofthisanormal forceXperunit area isexerted, which is adjustedsoastoproducetheactualperiodic pressure,and therefore theactual value of<j>,onthepositiveface ofthe membrane. Bythetheorem of 78(15),the effect foran element SS,willbeequivalenttoadouble source, andthe corresponding velocity-potentialatapointPwillbe -................(2) 4nrkcp ox\rJ The variablepartsofthepressuresonthetwo faces ofthe membrane, viz.+pcj}=+ilccpfy, must balance theforceX,so thatX=Zikcptp. Substitutingin(2),andintegratingoverthe planex0,weobtain The structure ofthointegralsin(1)and(3)recalls the process bywhich"Huygens' principle" isappliedinopticsto findthedisturbance atanypointPinterms of"secondary waves" supposedtoissue from thevarious elements ofawave-front. There wasatonetimemuch discussion astotheexact character tobeassignedtothesesecondary waves, moreespeciallyasto thelawofintensityindifferent directions. Wenowrecognize that theproblemhasmathematically more than onesolution ; either oftheabove formulae willlead toanexact result, and wemighteven useacombination ofthetwo, inanyarbitrary proportions.This resolution ofahistoriccontroversyisdueto Lord Rayleigh. Asaverification of(3),supposethatthevalue of$>atas= isthatduetoatrain ofplanewaves <e~ikx .Let CTdenote thedistance of$Sfrom theorthogonal projectionofthepoint Pontheplanex0,sothat r*=x*+tar2 .Fortheaggregate ofelements &Sformingacertain annulus oftheplane we maywrite 277-73- ckr=27rr8r. Wehave alsodr/dx=%/r.The formula(3)thereforegives 1f8=- uc- 5~ "Z-rrJ odrI =-*jdv*'"*"""-W Inthecaseofwaves transmitted byanapertureinaplane screen(x=0),wehave, in(1),d<p/dn exceptoverthearea of theaperture. If,further, thedimensions oftheaperture Sare small comparedwith X,then atapointPwhose distance ris large comparedwith X,thefunction e~ikr /rwillhavesensibly thesame value for alltheelements ofS,andwemaywrite CFrirh pikr P=_^dSe-~, (5) ^JJdn %7rr^' where the first factorrepresentsthe total fluxthroughS. Under these circumstances theapertureacts likeasimple source, asin 82. Itisunderstood ofcourse that theexpression dcj>/dnin (1)or(5)representsthenormalcomponentofthevelocity,as 'modified bytheactionofthescreen. When asinthecasejust considered theapertureisrelatively small, thedistribution of normalvelocityover itwill differconsiderably from thatdueto theprimarywaves alone. This distribution canbeascertained approximately,inthecase ofplane waves incidentdirectly on acircularopening,from theelectricalanalogyof 82.The lines offlowhave thesameconfigurationasthe lines offorce duetoanelectrifieddisk*, andthenormalvelocity hasthe distribution dnV<>2-O'.................. where -ordenotes thedistance ofanypointoftheaperturefrom itscentre. Thevelocity becomesverygreatneartheedge,and ismathematicallyinfinite fattheedgeitself(=a),but it appears onintegrationthatthepartsoftheareaneartheedge contribute little tothetotalflux,which is (7) Iftheincident waves berepresented by <=<>-*** ........................(8) thesame flux will asin82beexpressed by2/fG,or4a<7. Hence, comparing,5=2/7T.a........................(9) Intheother extreme, where thewave-lengthisonlya minute fraction ofthedimensions oftheaperture,theeffect of thescreen inmodifying thedistribution ofnormalvelocityover thelatter ispracticallyconfined toadistance ofafewwave- lengthsfromtheedge,andthecorresponding partoftheintegral in(1)isquite unimportant.Inthis case, theincident waves beingstillexpressed by(8),wecanput -~B<f>/dn=i/cCwith sufficientaccuracyoverthewhole area oftheaperture, whence Forthemethods ofapproximatingtothevalue ofthisintegral, bytheuseofHuygens'orFresneFs"zones," orotherwise, we must refer tobooks onOptics,Itisfound thattheamplitude isnearlyuniform within thespacebounded byacylindrical surface whosegeneratorsarenormal tothescreenthroughthe edgeoftheaperture,and isnearlyzero inthesurrounding region.Near thecylindrical boundciry, oneither side,wohave *SeeFig. 75,p.247,whichrepresentstheconfiguration oftheequipotential surfaces. tTheawkwardness ofthisconclusion maybeavoided bygiving thescreen acertain thickness, androunding theedges. 256 DYNAMICAL THEOEY OFSOUND thediffraction effects which areespeciallystudied inthetheory ofLight. Thequestionoftheimpactofwaves onaplane lamina can betreated inasimilar manner. For thispurposetheformula (3)ismost convenient. Thelaminabeingintheplane %=0, andtheprimarywavesbeing represented by(8),wemay write $=Ce-iJex+x>.....................(11) where%isthevelocity-potentialdue toavibration ofthe lamina normal toitsplanewith thevelocity ikG,equal and oppositetothat intheprimarywaves. Itisevident that the values ofthisfunction atanytwopointswhich aresymmetric- allysituated withrespecttotheplanex= willbeequalin magnitudebutoppositeinsign.Wehave then, totheright ofthelamina Thisonlyrequiresaknowledgeofthevalue of^atthepositive face ofthelamina, thevalue atallotherpointsoftheplane x=Qbeing obviouslyzero. Thecasewhere thedimensions of thelamina aresmallcomparedwith A,hasbeen noticed in 81 ;thescattered waves have then amuch smallerintensity than those transmitted byanapertureofthesame sizeand shape. Intheopposite extreme, thevalue of%near the positivefaceis.,exceptnear theedge,thesame asinthecase ofaninfinitevibrating plate,viz.%=Geikx ,sothatwehave with sufficientaccuracy nrra/>-av\ *-//!( )**............... (13 > Adetailed studyofthisintegralwouldindicate, inthecomplete solution expressed by(11), theexistence ofasound-shadow to therightofthelamina. Forlargevalues ofkrtheformula (13)maybereplaced by and forsmallobliquities6wemayfurtherputcos0=1. The formula then becomes, exceptastosign, identical with(10), shewingthatthedisturbanceproduced bythelaminais,under SIMPLE -HAKMONIC WAVES. DIFFEACTION 257 theconditionspostulated, exactly oppositetothattransmitted byanapertureofthesame dimensions. This isafamiliar factinOptics ;buttheprecedingconsiderations shew that it maybeutterlywide ofthemarkwhen thewave-lengthisno longersmall comparedwith thelinear dimensions concerned. Itneedhardlybesaidthatthere areacoustical phenomena where, asinthecase oflarge reflectingorobstructing surfaces, opticalrelations areapproximatedto.The results arethen analogous,theresemblancebeing morecompletethehigher thepitchofthenotesounded. Bytheuseofasource ofvery high pitch,andofasensitive flame asadetector, LordRayleigh hassucceeded inimitating some ofthemost delicate phenomena ofphysical optics. Intheabove theoreticalinvestigation wehavebeenobliged torelytosome extent onintuitive considerations, ase.g.inthe assumed distribution ofvelocityoverthearea ofanaperture when thewave-lengthisrelativelysmall. Itistherefore desirable thatsuchassumptionsshould betested ifpossible by exact calculation. Theonly instance, atpresent,where thishas beensuccessfullycarried out isthat ofwaves incident ona planescreen with astraight edge.The reflection bythescreen, thetransmissionpasbtheedge,theformation ofashadow behind the screen, andthe diffraction phenomenanear the boundaries OAtherespective regions,allcome outinpractical accordance with theusualtheory. Theinvestigationwas published bySommerfeld in1895*. *Asimplified versiou isgiven intheProc, Land. Math. Soc.(2),vol. iv. (1906). CHAPTEE IX PIPESANDRESONATORS 84.Normal Modes ofRectangular and Spherical Vessels. Themainobjectinthischapteristodevelopthelaws of vibration ofaircontained incavities, such asthose ofresonators andorgan pipes,which areincommunication with theexternal atmosphere. Alittlespacemayhowever bedevoted inthe firstinstance tosomeproblems relatingtothevibrations ofair inspaceswhich arecompletelyenclosed byrigidwalls. These will atalleventssupplysomeinteresting examplesofthe general theoryofnormal modes(16). Theanalytical processconsists infindingsolutions ofthe equationV2+A;2 (= ........................(1) consistent with thecondition whichexpressesthat thecomponentofthefluidvelocityin thedirection ofthenormal(n)vanishes attheboundary.It appears that, asinformeranalogous problems,this isonly possibleforacertainsequenceofvalues ofk,which determine thenature andthefrequencyoftherespective normal modes. Inthecase ofarectangular cavity wetake theoriginata corner, andthecoordinate axesalongtheedgeswhich meet there. Ifthelengthsoftheseedgesbea,b,c,thecondition (2)isfulfilled by PTTX q-rry TTTZ-,n._. d>-Ccos-cos 2-r^.cos- ............. (3) ra b ov' PIPES ANDEESONATOBS 259 wherep,q,rareanyintegers;andtheequation (1)isalso satisfied provided Ifwepufcq=0,r0,thecasedegeneratesinto that ofthe doublyclosedpipe (62).Amoreinterestingcase isthat ofaspherical cavity. The symmetricalradial vibrations come under themethods of 71, 76.Theformula(15)of 71,whichimpliesthat there isno source attheorigin, gives,inthecase ofsimple-harmonic vibrations, or,say,r sin&r.(5) .(6) 4-TT 'XiiiliUUX UJtf forr=a,theradius of -(7)Thecondition(2)requiresthatdtf>/dr= thecavity. Hence tanka ka, This isatranscendentalequationtofindk,andthence n(=kc). Theroots areobtainedgraphically (seeFig.76)astheabscissae oftheintersections ofthelinesy=tanas,y=x,thezero root beingofcourseexceptedasirrelevant. Wehave, approximately, ka (m-{-^}Tr, wherem= 1,2,3,....More accurate values of the firstthree roots are ka/TT=1-4303, 2-4590, 3'4709(8) Thenumbersgivetheratio ofthediameter 2aofthecavityto thewave-length.Inthemodes after thefirstthere areinternal sphericalnodes(i.e.surfaces ofzerovelocity)whose relative positionsareindicated bytheroots ofinferior rank. Inthe highermodes thenodal surfaces tend, asweshouldexpect,to becomeequidistant,since theconditions, exceptnear thecentre, approximatetothose ofplanewaves. Equationsofsomewhat similar structure to(7)occur(as wehave seen)invariouspartsofoursubject,aswell asin other branches ofmathematicalphysics, andprocessesof numerical solution havebeen de- vised byEuler, LordRayleigh andothers. There isonemethod, ofvery general application, which issoelegant, and atthesame time solittle known, that itmay beworth Avhile toexplainit.Ib isgiven byFourier inhisTheorie delaChaleur(1822). Starting with arough approximation, say xx1,toaparticularroot of(7), wecalculate insuccession the quantities#2>z>&*, determined bytherelations #2=tan"1#j ,#3=tan~zx. Thefigureillustrates themanner inwhich theseconverge towards thedesired root asalimiting value, nomatter fromFig. 77. (9) PIPES ANDRESONATORS 261 which sidewe start. Somefairly obviousprecautions are necessaryinusingthemethod, and itiseasilyseen that the convergencewillbeslow ifthetwocurves havenearly the same inclination(inthesame orinopposite senses) totheaxis of 0.ExpressedasmultiplesofTT,thesuccessiveapproximations obtained inthiswaytothe first rootof(7)are* 1-5,1-433435, 1-430444, 1-430304, 1-430297, .... Thesameanalysiscanobviously beappliedtothetheoryof vibrations inaconicalpipewhosegeneratinglinesmeet in0. Ifthetubeextend from theorigintor=a,theusualapproxi- mate condition(s=0)tobesatisfied attheopenendgives sinfca=0, (10) thesame asforadoubly openpipeoflength a(62). Forthe case ofatubeextendingfrom r=atorb,andopenatboth ends,werequirethecompletesolution r<f>=Acoskr+B&inkr(11) Theconditions give Acoska+Bsinka 0,Acoskb+Bsinkb=0,(12) whence sink(b-a)=0, (13) asinthecaseofadoubly open pipeoflengthb a. If^beanysolution ofthegeneral equation (1),itappears ondifferentiation throughoutwithrespecttoxthattheequation isalso satisfied by (/>=9%/cte.Wehavealready hadanexample ofthis inthegeneraldouble source of 73.From(6)we derive inthiswaythesolution or,ifx=rcos9, 9/sin IGT\/iC'/i 1 7\ //-iir\ cb=(7-- cos=-(krcoskr sinkr)cosd.(15)rdr\rJ r-^ This leads toanother series ofnormal modes oftheaircon- *Incalculations ofthiskind, and forthepurposesofmathematical physios generally, trigonometricaltables based ontheceutoHimul division ofthe quadrantaremost convenient. Afour-ligurotable ofthistypeiincluded in J.Hoiiel'a llecueil deFornnilns titdeTablets A'uwcn'gites,!kdcd.,Paviw, 1885. 262 DYNAMICAL THEORY OFSOUND tained inaspherical cavity. Thecondition9<jE>/9r=issatisfied forr=a,provided tanTea= g-_^(16) The solution canbecarried outasinthecase of(7).The annexeddiagramofthecurves ycotos,y=(2 z?)/2x, shews thattheroots tend afteratime totheform mir. Approximate values ofthe firstfewroots are kaJTr=-6625, 1-891, 2'930, 3'948, 4'959, ...(17) Fig. 78. the firstofwhich alonegivesanytrouble. This rootcorresponds tothegravestofallthenormal modes ofthecavity. The air swingsfrom side toside,much asinthecase ofadoublyclosed pipe, and thewave-lengthisX=27r/&=T509 x2a.The forms oftheequipotentialsurfaces, towhich thedirections of vibration oftheair-particlesareorthogonal,areshewn inFig.79. Inthenextmode theradialvelocity vanishes over thesphere r/a= -6625/1-891='350. Thestudyofthemorecomplicatednormal modes ofvibra- tioninasphericalvessel would leadustoo far.Theproblem isfullydiscussed inLordRayleigh'streatise. PIPES ANDEESONATOES 263 Fig. 79. 85.Vibrations inaCylindrical Vessel. Thetheoryofthepurelytransversal vibrations oftheair enclosed byacircularcylinderisverysimilar. Asin54,the equation whereso,yareCartesian coordinates intheplaneofacross- section, becomes inpolarcoordinates rdr(2) andthetypical solution, when there isnosource atthe origin,is (f>=CJm(Jcr}cosmd (3) Theadmissible values ofkaredeterminedbythecondition that d(f>/dr=forr=a,or Jm'(ka)=0 (4) Fortheradial vibrations (m 0)theearlier roots aregiven by /ca/7r=1-2179, 2-2330, 3'2383, (5) 264 DYNAMICAL THEORY OFSOUND thelimiting form"being integer +.Inthecasem=1,which includes thegravest mode, &a/7r=-586, 1-697, 2-717,..., (6) thelimiting formbeing integer J. Thepurely longitudinalmodes ofaclosed circularcylinder comeunder 62.There remain thevibrations ofmixedtype. Theequation (2)hasnow tobemodified bytheinclusion of aterm 92 <jb/9^2 ,where zisthelongitudinalcoordinate. Itis found that theequationissatisfied by 7T2 $=Gcos TJm(J3r)cosm6, (7)L provided^2=yS2+m/27r2 /^, (8) theorigin being taken atthecentre ofoneend.Thecondition ofzeronormalvelocity (d^fdz)attheother end (z=Z)is satisfied ifm'beintegral. Thecorrespondingcondition atthe cylindricalsurfacerequiresthat/3should bearootof Jm'(/9a)=(9) 86.Free Vibrations ofaResonator. Dissipation. Theforegoing examplesareoftheoretical rather than practical interest, since thevibrations ofamass ofairenclosed byrigidwalls would becompletelyisolated. Foracoustical purposesthevibrating massmust havesome communication with theexternalatmosphere ;ontheother hand itisessential that thecommunication should besorestricted thatthe frac- tion oftheenergy which isusedupinasingle periodin thegenerationofdiverging waves shall stillbevery small. Otherwise the free vibrations couldhardlyberegardedas approximately simple-harmonic, andmight even resemble the"dead-beat"type(11). Thetheoryissimplestinthecase of"resonators"such as wereemployed byHelmholtz inhisresearches onthequality ofmusical notes. These arenearlyclosed vessels, with an aperture, andareused tointensify, bysympathetic vibration of theenclosedair,theeffect ofasimpletoneproducedinthe neighbourhood. Theprecise form isnotimportant;itmay besphericalorcylindrical, oralmostany shape,solongas PIPES ANDKESONATOBS 265 the least diameterconsiderablyexceeds thedimensions of theaperture.Inhissynthetic work onthevowel sounds Helmholtz usedcylindricalresonatorshavingacircularopening atthecentre ofoneend.When theobject wastodetect and toisolate aparticularovertone inacomplex sound, heused themore convenient formshewn inFig.80.Thesmallopen nipple oppositethemouth isinserted into theearcavity,so thatthetympanicmembrane becomespartoftheinternal wall oftheresonator. Fig.80. Fig. 81. Thetheoryofresonators wastreated mathematicallyforthe firsttimebyHelmholtz in1860, andwasafterwardsgreatly simplified byLord Rayleigh (1871). Supposeinthe firstplace thatwehave avessel with anarrowcylindricalneck which is occupied byaplugorpiston freelymovable toand fro(Fig. 81). LetQdenote thecapacityofthe vessel,Ithelengthof theneck, itssectional area, p'thedensityofthepiston. We willassume that theperiodofvibration issolongthat the corresponding wave-length (X)inair islarge comparedwith thediameter ofthe vessel. Under thiscondition thecon- densation swillatanyinstant bealmost uniform throughout theinterior, andwemayputs=mx/Q,where xdenotes the small displacementofthepistonoutwards from itsmean position.The resultingexcess ofpressureonthebase ofthe pistonis/oc*sa>,or-pcWaj/Q,andtheequationofmotion of thesystem is,approximately, p'tolas=pc"co2xJQ (1) 266 DYNAMICAL THEORY OFSOUND Themotion isaccordingly simple-harmonic,withaperiod 2-rr/n, provided c^o>W?'rt, nz= Thenature ofthepistonisoflittleimportance, provided itsmassbesufficientlysmall.Wemayevenreplaceitbyair, ifthelengthIbesmall comparedwith X,forunder this condition thecolumn ofairintheneck willbehave almost asif itwereincompressible. Wehavethenp'=p,and (3) Even inthecase ofaresonator whose mouth consists of amereopeninginthewall,without aneck, thetheoryisnot verydifferent. Itisonlyaquestionofobtainingaproper measure oftheinertia ofthemass ofairintheimmediate neighbourhoodofthemouth, inside and outside, which takes theplaceofthepistonintheabove problem. The flow throughtheapertureatanyinstant isstillregulated, ap- proximately, bythesame laws asthat ofanincompressible fluid, orofelectricityinauniform conducbor. Therebeing little motion inthe interior, the value of <there willbesensibly uniform;wedenote itby X.Out- side, atashort distance beyondthe mouth, weshall have<p=0,nearly. Ifqdenote thevolume ofairwhich haspassed throughtheaperture outwards uptotime t,thecurrent, orflux,outwards atthisinstant willn ' beq,andwehave,bytheelectricanalogy, whereKisthe" conductivity" (82),whichdepends,ofcourse, ontheshapeand sizeoftheapertureandtheconfigurationof thewall initsneighbourhood.Itistobeobserved that this relation(4)ispurely kinematical; from thepointofview ofthe generalized dynamicsofasystemofonedegreeoffreedom PIPES ANDEESONATOES 267 (7),itexpresses themomentum (which may"besymbolized by p^i)interms ofthevelocity q.Thedynamical equation c2s=&........................... (5) of70(3)mayinlikemanner beinterpretedasexpressing therelation between changeofmomentum and force. Ifthe zeroofqcorrespondtotheequilibrium state,wehave s=-q/Q.........................(G) Eliminatingsandfabetween(4), (5),and(6),weobtain Themotion istherefore ofthetype <7=(7cos(n +e),.....................(8) providedn*=Kc*/Q.........................(9) Ifwewriten=kc,thisgives k*=K{Q, X=27rV(Q/A')................(10) Thewave-length depends,asweshouldexpect, solelyonthelinear dimensions oftheresonator and itsaperture.Forresonators which aregeometricallysimilar inallrespects,itvariesdirectly asthelinear dimension. This isinaccordance with ageneral principlewhich maybeinferred from thedifferentialequation (2)of|76,orotherwise. Theformula(9)indicates further that thepitchoftheresonator islowered bycontractingorpartially obstructingtheaperture,whilst itisraised bydiminishingthe internal capacity. Thekinetic energy, being mainlyresident intheneighbour- hood ofthemouth, maybecalculated from theprinciples applicabletoanincompressiblefluid. Iftheactual motion were generated instantaneouslyfrom rest, theworkrequired would bothesum ofhalf theproductsoftheimpulsesinto thecoiTospondingvelocities. Theequations (9)of69shew thattherequisite impulsive pressureispfa;hence Thepotential energy is,by70(8), The coefficients intheseexpressions being known, thespeed nofthe oscillations canbeinferred atoncebythegeneral 268 DYNAMICAL THEORY OFSOUND formula(7)of 7.Itwasunder thisform thatthetheory was presented byLordRayleigh.Itistobenoticed that the inertia-coefficient isproportionaltothe"resistance"ofthe aperture (inthe electrical sense), whilst the coefficient of stability,orelasticity,varies inverselyasthecapacity Q. Thepreceding theory applies onlytothegravest mode ofthe resonator. Inthehighermodes theinternalspaceisdivided intocompartments byoneormore" loopsurfaces" (i.e.surfaces ofconstantpressure,where <=0),andthefrequenciesare muchgreater. Thewave-lengthisthen atmostcomparable with thelinear dimensions, asintheproblemsof 84. Asalreadystated (82) thecalculation ofJKisusuallydifficult. Foracircular apertureinathin wallKisequaltothe diameter, and foranyformdifferingnottoomuch from acircle wemayputK=^^(W/TT), approximately,where toisthearea. Thefrequency,asdetermined by(9),willthenvaryasT /()-. Itisremarkable that thislawwasarrived atempirically by Sondhauss atadate(]850)anterior tothetheory. When the apertureisfitted with acylindrical neck, theconductivityis limited mainly bytheneck itself, andwemayputK= ro[l, approximately, where Iisthelength. Theformula(9)then agreeswith(3).Itisimpliedthat Iissmallcompared with X, andatthesame timelarge compared with thediameter ofthe channel. Wehave intheabovetheoryallowed fortheinertia ofthe externalatmosphere,butnot foritscompressibility, andthe vibrations asgiven by(8)areaccordingly persistent. Inother words, wehaveneglectedtheapparent* dissipationofthe energyoftheresonator due toair-wavesdivergingoutwards from theneighbourhoodofthemouth. This will have, in general,noappreciableinfluence ontheperiod,but wil' manifest itself byagradual decayoftheamplitude. The effect canbeestimated with sufficientaccuracyin- directly. The fluxoutwards atthemouthis,by(8), e)...................(13) *Truedissipativeinfluences such asviscosity andthermal conduction are ignoredinthepresent investigation. They probably playasaruleawholly subordinate part. PIPES ANDRESONATORS 269 Iftheresonator werepracticallyisolated inspace, thenon account oftheassumed smallness ofitsdimensions ascom- paredwith A,,theeffect ofthefluxatadistance would bethat ofasimplesource ofstrength nO,andtherate ofemission of energywouldaccordinglybe (14) bytheformula (15)of 76.TheenergyEofthemotion, being equaltothepotential energyatitsmaximum, is, approximately, (15) by(12). Equating,ontheprinciplesof 11,therateofdecay ofthisenergytotheemission W,wefind andtherefore q=0^-^ cos(nt+e),...............(17) providedT=87rc/n*Q=87rQjK* C> ............(18) invirtue of(9).The ratio ofthemodulus ofdecaytothe period (27r//i'c)isgiven by SinceKisatmost comparablewith themean breadth ofthe aperture,this ratio isusually very great,andthepreliminary assumptions impliedintheaboveprocessareamply justified. Ifthemouth oftheresonator were furnished with an infinite flange,i.e.onewhose breadth islargecomparedwith X, theequivalentsource would, asexplainedin82,have double thestrengthabove assumed, andtheemission ofenergy, now operativeinonehalfofthesurrounding region,would betwice asgreat.Themodulus(18)wouldaccordinglybehalved. Asanumerical illustration ofthetheoretical results, take thecase ofasphericalvessel 10cm.indiameter, withacircular aperture1cm.inradius, sothatQ-523'6,K-2.Thewave- length,calculated from(10),is101'6;andthefrequencythere- foreabout 327. Themodulus ofdecay,asgiven by(18),is about one-tenth ofasecond. 270 DYNAMICAL THEOEY OPSOUND 87. Corrected Theory oftheOrgan Pipe. Thesameprinciplescanbeappliedtoobtain acorrection totheimperfect -theoryoftheopen pipewhich wasgivenin 62.Wemaybegin byabrief examination oftheslightly simpler problemofreflection atanopenendofaninfinitely longpipe (61). Kg. 83. Near theopenendthere isacertainregion, whose dimensions aresmall comparedwith thewave-length,within which the transition takesplacefromplane waves within thetube to diverging sphericalwaves outside*. Wetaketheorigininside thetube, nearthemouth, butintheregionofplane waves, and thepositivedirection oftheaxisofxalongthetube. Forthe regionofplanewaves wemaywrite </>-Aeikx+Be~ikx , (1) where the firsttermmaybetaken torepresentatrain ofwaves approachingtheend,from theright,whilst thesecond term *The figure, which isbased onformulae given byHelmholtz inanother connection, relates tothetwo-dimensional form oftheproblem. Inthree dimensions thetransition toastate ofuniform radial flowoutwards from the mouth would bestillmore rapid. PIPES ANDEESONATOES 271 representsthereflected waves. Theoutwardvelocityat is thereforerepresented byik(A B),andtheflux is q=ika(A-B), .....................(2) where coisthesectional area. Thevelocity-potentialat is A+B.The"resistance"between thesection so andthe external regiontothe leftmaybespecifiedasequivalentto that ofacertainlengthaofthepipe, and isaccordingly denoted bya/to.Hence, bythe electricalanalogy, -xik(o(A-B),...............(3) , B 1ikawhence -7=-- ~A Ifweputka==tan7c/3, .....................(5) thismaybewrittenJ5/J.=-e~2^......................(6) Hence <}>=A{eijcx-e- *+*)} ................(7) The reflected train isthereforeequalinamplitudetothe incident one,aswastobeexpected,since theinertiaonlyofthe external airissofartaken intoaccount;butthere isadifference ofphase.Inthetheoryof61thecondition tobesatisfied at anopenendwass0,or<p=0.Hence ifwewrite(7)inthe form (j>=Ae-W[eilc<*+#-e~ik{x+ft) }............(8) werecognizethatthecircumstances arethesame asifthepipe wereprolongedtotheleftforalength /3,andthereflection atthe mouth were totakeplace accordingtotherudimentary theory. Thewave-length being assumed tobelarge comparedwith the diameter ofthepipe,kotwillusuallybesmall, sothatft=a, nearly.But ifthepipebeverymuch contracted orobstructed atthemouth, kamaybeconsiderable, andk/3willinthat case approach -^TT.Wethen haveB=A,nearly,andthecircum- stances approximatetothose ofreflection ataclosed end. The actual determination ofaisaprobleminelectric conduction which has atpresent only been solved, even approximately,inaveryfew cases. LordRayleighestimates that foranaccurately cylindricaltube iitted with aninfinite flangethevalue ofaisabout '82ofthesectional radius. For 272 DYNAMICAL THEORY OFSOUND anunflanged cylindricaltube experimentseems toindicate a value ofabout '6oftheradius. Wewillnextsupposethepipetobeoffinitelength,and to beclosed atas=I,theorigin beingchosen asbefore, near the mouth, intheregionofplanewaves. Forthislatterregion we mayassume <f)=Acos&( #),..................(9) since 3$/3#must vanish forx=l.The fluxoutwards at themouth istherefore q(ed^fix=kwA sinkl,............ .....(10) andthepotentialat isAcoskl.Hence with thesame meaningofaasbefore wehave Acoskl=xko)Asinkl, or c,oikl ka......................(11) This equationdetermines thewave-lengths (27T/&)ofthe various normal modes.Usually,kaissmall, andthesolution of(11)isthen kl=(m+|)TT ken, or k(l+a)=(m+&7r,............... (12) wheremisintegral.Thecharacter ofthenormal modes isthere- forethesame asontherudimentary theory (62),provided we imaginethelengthofthepipetobeincreased bythequantity a.Inparticular,thefrequenciesare astheoddintegers 1,3,5,...,solongasthewave-lengthremainslarge compared with thediameter. Iftheaperturebecontracted thevalue ofaisincreased, andtheresult tends tobecome lesssimple.Inparticular,the harmonic relation ofthesuccessivefrequenciesisviolated, as may easilybeseenfromagraphicaldiscussion oftheequation (11).When thepipeisalmost closed, aisrelatively great,and thesolution of(11)iskl=l//ca,or /c2=I/la.Thisagreeswith theformula(10)of86,ifweputcol=Q,ai/a=K. Inthecase ofapipeopenatbothends theperiod equation isfound tobe tan JfcZ=-&(+'),............... (13) wherea,a'arethecorrections forthetwoends,butthecalcula- PIPES ANDEESONATOES 273 tionimpliesthat leaandku?aresmall. Itis,however, onlyon thiscondition thattheconductivities atthetwoends can, asa rule,beestimatedindependentlyofoneanother. Theequation isthenequivalentto sink(I+a.+a')=0, (14) andthefrequenciesaretherefore those which areassignedto apipeoflengthI+a.+a!bytherudimentary theory. The harmonic relation between thevarious normal modes ispre- served, but itmust beremembered that theapproximationis themoreprecarious,thehigher theorder oftheharmonic. Thewave-lengthsofthepropertones areinallcases fixed bythelinear dimensions, butthefrequencies, whichvaryas thevelocityofsound, will rise orfallwith thetemperature. An"open" organ pipeistunedbymeans ofacontrivance which increases ordiminishes theeffectiveapertureattheopen end,i.e.theendremote from the"mouth" proper. Thepitch ofa"closed" pipeisregulated byadjustingthepositionofa plugwhich forms thebarrier. Tocalculate therateofdecayofthefreevibrations itwill besufficient totakethecaseofthestopped pipe. Thekinetic energy correspondingto <=AcosTc(l x^cosnt (15) isgivenwith sufficient accuracy by l.A*co&*nt, ...(16) ifktx.besmall, since cosId=0,nearly. Amore careful calcula- tion,taking account ofthetransitionregion between theplane andthespherical waves, replacesIbyI+a,approximately,in this formula, butthecorrection isnotimportant. The total energy, being equaltothekineticenergyatitsmaximum,is accordingly E^lpfralA* (17) Ifthemouth beunflangeditacts, inrelation totheexternal space,asasimplesource ofstrengthkcoAsinM, orJewA,nearly, andtheconsequentemission ofenergy persecond isaccordingly W=pkt <o*cA*/&irl (18) 274 DYNAMICAL THEOEY OPSOUND by76(15). EquatingtherateofdecayoftheenergytoIf, weareledtotheequation dA Jfac andthemodulus ofdecayistherefore T=47rl/k*ooc...................... (20) The ratio ofthis totheperiod (27r/&c)is2//ca>,or(inthe gravest mode)4Z2 /7r&>, nearly.Since themoduli ofthevarious normal modes areproportionaltothesquaresoftherespective wave-lengths,thedecayisthemorerapidthehighertheorder. Foraflanged pipetheresult (20)would behalved. 88.Resonator under Influence ofExternal Source. Reaction ontheSource. Thetheoryofforced vibrations duetoanexternal source ofsound, towhich wenowproceed,involves some rather delicate considerations, and isoften misunderstood. That themass ofaircontained inaresonator oranorgan pipe should besetintovigorousvibration byasource inapproxi- mate unison with itisintelligible enough; but itisfurther desirable tohavesome estimate oftheamplitudeoftheforced vibration, andinparticulartounderstand whythesound which isapparently emitted bytheresonator should under certain conditions enormously exceed thatwhich would beproduced bytheoriginalsource alone. Forsimplicity wewillsupposethat thissource ismain- tained atconstantamplitude byasuitablesupplyofenergy, sothat thevibration ofthe airiseverywhere steady.Itis evident atonce thatunder thiscondition nowork isdone, ontheaverageofawholeperiod,atthemouth ofaresonator onthecontained air,theenergyofthelatterbeing constant, andconsequentlythatnowork can inturn bedonebythe reaction ofthismass onthe externalatmosphere. Any increasedpropagationofsound toadistance must bedueto thechanged conditions which theaction oftheresonator has introduced intheneighbourhood oftheoriginalsource. If thissource benotmaintained constant, butmerelystarted withaninitial fund ofenergy (asinthecase ofatuning PIPES ANDEESONATOBS 275 fork), thisfund willunder theinfluence oftheresonator be morerapidly consumed. Inorder totreat thequestioninaform freefrom unessential details, whichmayvaryfrom onecase toanother, wetakethe case ofaresonator ofthetype considered in 86,whose dimensions aresmallcompared with thewave-length.Thetheoryissimplest when thefrequencyofthesource isverynearly equaltobhenaturalfrequencyoftheresonator, asdetermined by86(9),sothat theforced vibration inthe latter isatitsstrongest.Itwillperhaps make thematter clearer ifweimagineinthe first instance that theresonator hasashortcylindrical neck inwhich athin massless disk, almostexactly fitting it,canbemade tomove toand froby asuitableapplicationofforce.Suppose then that thedisk ismade toexecute avibration such that thevolumeswept overbyitoutwardsuptotime tis q~0cosnt; (1) and lettheextraneous force which must beappliedtothe disk tocompensatethedifference oftheair-pressuresonthe twosides bedenotedby Acosnt +Bsmnt, (2) thisexpression being (say) positive when theforce isoutwards. Thecomponent Acosntwhichkeeps stepwith thedisplace- ment isrequiredtocontrol theinertia ofthe air.From the general theoryofforced vibrations(9,12)itappearsthatthe coefficient Acanbemade tohave onesignortheother byadjustingthevalue ofn,thesignbeingthesame asthat ofCwhen theimposedvibration isrelatively slow, andthe oppositewhen itisrelatively rapid. Wemaytherefore supposentobesoadjustedthatA=0.Thecircumstances arethenverynearlythose ofafreevibration, andtherequired value ofnisgiven by n*=Kc*/Q, (3) very approximately.Thesecondcomponentoftheforce(2), which keeps stepwith thevelocity (q),isrequiredtomaintain theemission ofenergy outwards, whichis,by76(15), (4) 276 DYNAMICAL THEOEY OFSOUND Thismust beequaltothemean value ofpq,where pisthe pressureattheouter face ofthedisk. Hence bycomparison wefindthatpmust have theform Y$Q\Jp=p+Dcosnt~sinW............. (5) r4<7rcx/ Thecorresponding pressure ontheinner face willbe p=p+2)cosnt, ..................... (6) simply,since nowork isdone, onthewhole, onthe air contained intheresonator. Wemaynowinvoke theaction oftheexternal source. Ifthisbesuch aswould producethepressure (7) atthemouth oftheresonator ifthediskwere atrest,then inthemotion which iscompoundedofthatduetothesource andthatduetothedisknoextraneous force willberequired, andthediskmaytherefore beannihilated withoutcausing any appreciable changeinthe conditions. If <2bethe velocity-potentialdue tothesource alone, atthemouth of theresonator, wemust have, inthis case, .....................(8) since("7)mustbeidentical withpp Thehypothesisofarigiddiskvibratinginacylindrical space wasmerelyintroduced forfacilityofconception, and isinnowayessential totheargument. Thediskmay,if weplease,bereplaced byaflexible andextensible membrane enclosingtheapertureoftheresonator, andabuttingonthe external wall intheregionofdivergingwaves. Comparing (1)with(8)weseethat toadisturbing potential whose value atthemouth is (nt e) ..................(9) willcorrespondavibration PIPES ANDEESONATOES 277 under thecondition ofmaximum resonance, when nisgiven by(3)approximately.Thecorrespondingflux is q= ysin(nt- e) (11) fc The emission ofenergyisbest calculated from acon- sideration ofthecircumstances atagreatdistance. The velocity-potentialwillbecompoundedofthatdue tothe originalsource andthatduetothefluxq,andunder certain conditions the lattercomponent may greatly preponderate. Theemission ofenergyisthen W=2<rpcJz , (12) approximately, by76(15). Thus if$2heduetoasimplesourceAcosJcctatadistance b from theaperture,wehave ^ </>2=T-Tcos Tc(ct b) (13) Hence /=A/4nrb, ^and q=TTsink(ct b) (14) This isequivalenttoasource whoseamplitudeistothat of theprimarysource intheratiol/kb.Ifbbesmallcompared with X/27Tthis ratio islarge;and theemission ofenergy exceeds thatdue totheoriginalsource inthe ratio~lfk'bz . Inthecase ofadouble source Bcoskct wemay write, ifJcbbesmall,D <2=7racosacos Ic(ct b), (15) by76(23),ifadenote theanglewhich theaxisofthesource makes with the linedrawn from ittotheaperture. Hence J==BCOBOL/4i-rrbz ,andtheemission, asgiven by(12),is F=pcacosa a/87r&* (16) Theemission due totheoriginal (double) source alone would bep/c*c.S2 /247r,by7G(26). The ratio inwhich theemission isincreased istherefore 3cos2 a/Mb*. Since themean value ofcos2ais^,themean value ofthis ratio, for alldirections oftheaxis ofthedouble source, isl//t"'i4 .That the ratio 278 DYNAMICAL THEOBY OFSOUND should, under thegivenconditions, besomuchgreater than intheprecedingcase isduetotherelativelysmallerefficiency ofadouble source, ascomparedwithasimple one,inpropagating energyoutwards(80). Itmaybewell toinsist againthat theincreasedoutput ofenergyisanindirect consequenceofthepresence ofthe resonator, which itself doesnowork. Thewholeenergyis supplied bytheoriginal source, where themotion takesplace againstanaugmented componentofpressureinthesamephase with thevelocity.Thevelocity-potential due tothefluxq outwards from theresonator, asgiven by(11),is </>i=- tsin(nt-Jcr- e), ...............(IT) andtheresultantpressureis kre)..........(18) Inthecase ofasimple primarysource wehadJ=A/4i7rb> ekb', hence, puttingr=b,wefind that theconsequent pressureintheneighbourhoodofthissource is cos(n$-2fc&) .............(19) Since theimposed outward flux isAcosnt,themean rate of workagainstthispartofthepressureis Theoutputisthereforegreaterthan itwould beinthe absence oftheresonator, intheratio cos2/ci/7c262 .Thisagrees with theformer result, obtained onthehypothesis that Jobis small. Theenergystored intheresonator under theconditions of maximum vibrationis,by86(15), E=S'ir*pc*J*/n'Q=8'napQJ*/K*..........(21) This variesdirectlyasthecapacity Q,and isforapertures of similar forminversely proportionaltothearea. The effect ofaresonator under theinfluence ofadistant PIPESANDRESONATORS 279 source inunison with itmaybesufficientlyillustrated onthe assumptionthattheincident waves areplane.If <2=Jcosk(ct~ai), (22) theratio oftheenergyscattered bytheresonator, which is given by(12),totheenergy-fluxintheprimary waves, viz. %pk*cJ2 ,is47T/&2 ,orXa /7r.Theenergydivertedpersecond, atitsmaximum, istherefore equalto'318 ofthatwhich in theprimarywaves istransmitted across asquareareawhose side isthewave-length. Whenapproximate agreementbetween thefrequency (n/2-Tr)ofthesource andthenaturalfrequency (n/27r)ofthe resonator isnolonger assumed, theexternalpressure which is requiredtomaintain asteadyvibration (1)throughtheaperture willconsist oftwoparts.Inthe firstplacewehave acomponent keeping stepwith thedisplacement,which isrequiredinorder tocontrol theinertia ofthe air.This iseasily foundbyan extension ofthemethod of 86. Iffadenote thevelocity- potentialintheinterior oftheresonator,<j&2that atashort distance outside theaperture,intheregionofapproximately spherical waves, wehave q^K(fa-fa} (23) inaccordance with theelectricalanalogy. Intheinterior we have 5=q/Q,c2s=fa,asbefore. Hence q+nfq^-Kfa, (24) where n<?=Kc*JQ (25) Thisgives,fortheexternalpressure, jj2 |j2 pPo=p^>2= j~-pCcoant (26) Thesecondpart,which isinthesamephaseasq,isneeded in order thattheremayontheaveragebenogainorlossofenergy totheaircontained intheresonator, and isaccordingly given by(7).Hence wehave, altogether, 3\ sinni); ...(27) / 280 DYNAMICAL THEOEY OFSOUND andthecomplete expressionforthedisturbing velocity-potential nearthemouth mustbe ,nO Intheproblemasitactually presentsitself thevalue ofip2 atthemouth isprescribed, say fa=Jco8(nt-e);............... (29) andinorder toidentifythiswith(28)wemust have kKnO-r-.-==, 4-Tr jfiT 22T.A ??2\nG/0.NJsme= 1--- }-j=.(30)V n*jKv TTHenee Thisdetermines Cinterms ofJ.IfTdenote themodulus of decayoffreevibrations, asgiven by86(18), theformula may alsobewritten ,721_=jL. J[1_in fiS~VA T7"*% t1*- Exceptinthecase ofapproximate synchronism thesecond term within thebrackets willbesmall comparedwith the first. Hence foragivenvalue ofJ,thevalue ofnG(whichis theamplitudeofthe fluxq)willbegreatest whenn=n, approximately. Moreover, foragivendeviation oftheration/n fromunitytheintensityoftheresonance falls short ofthe maximum inagreater proportionthegreaterthevalue ofKOT, i.e.thegreatertheratio ofthemodulus ofdecaytothefree period.Inother words, thesmaller thedampingoffree vibrations, themoresharplydefined isthepitchofmaximum resonance. This isinaccordance with thegeneral theory of 13. The vibrations ofaresonator under theinfluence ofan internal source ofsound arediscussed in90withspecial reference tothetheoryofreed-pipes. 89.Mode ofAction ofanOrgan Pipe. Vibrations caused byHeat. Althoughthelossofenergyinasingle periodmaybesmall, the freevibrations ofthecolumn ofaircontained inanorgan PIPES ANDBESONATOKS 281 ?.84.pipearepractically dissipatedinafraction ofasecond;this is owingtothesmall inertia ascompared with that ofapiano- wire. Formusicalpurposessome device forsustainingthe note isrequired.Intheordinary"flutepipe," thelowerpartofwhich isshewn insection in Fig. 84,athinstream ofairisdriven bypressure from awind-chest soastostrikeagainstthe bevelledlipoftheaperture. Under these cir- cumstances avery slightcause willmake thejet passeither whollyinside orwhollyoutside the pipe.Theprecise mode ofaction isobscure, but there canhardly beanydoubt that initsmain features itisanalogoustothat ofaclock-escape- ment. Periodicimpulsesaregiven bythejet, alternately inwards andoutwards, totheairnear themouth, alwaysinthedirection inwhich the airistending ;whilst thevibrating column itself mainlydetermines theepochsatwhich theseimpulsesshall occur. The circumstances areaccurately periodic,sothat thedrivingforce canberesolved byFourier's theorem into aseries ofharmoniccomponents whosefrequenciesareas 1,2,3,....The relativeamplitudes with which these are reproducedinthevibrating column willdependonthe closeness oftheirfrequenciestothenaturalfrequencies. Thus ina"closed" pipe,i.e.one closed attheupper end, the harmonics ofoddorder arealone excited. Againthetheory of87indicates that inasufficiently widepipethenatural frequencies maydeviatesensiblyfrom theharmonic relation, inwhich case only thelower harmonics(after thefunda- mental)willbesensible; inparticular,awide closed pipe givesalmost apuretone.Ontheotherhand apipewhich is narrow incomparisonwiththelength maygiveanote richin harmonics. Indeed,ifsuch apipebeblown with sufficient force, thefundamental isnotsounded atall,theperiod becoming that ofthe firstharmonic; ifthestrengthoftheblast be further increased thenotemayjumptothenextmember of theseries, andsoon.Anexplanationisprobablytobefound inthesortofdynamical elasticity possessed bythejet. 282 DYNAMICAL THEORY OFSOUND Metal pipesarericher inharmonics thanwoodenpipesof thesame dimensions. Thismaybepartly duetothegreater fineness ofthelip,which introduces agreater degreeofabrupt- nessintheaction ofthejet,and sofavours theamplitudeof theterms ofhigherorder intheFourier series whichexpresses thedrivingforce. Another source ofthecontrast inquality maybefound inthesmallerrigidity andimperfect elasticity ofthewalls ofawoodenpipe,which maytend toabsorb the energy, especiallyinthecase ofthehigher harmonics. The"speaking"ofaresonator ofanykind,when ajetof airisblown across itsmouth, istobeexplained onthesame principles.Inaresonator oftheusualtypethenormal modes after thefirst arefarremoved inpitch from thefundamental, andarenobsensiblyexcited bytheessentially periodic impulse. Thenoteobtained istherefore apuretone. Thevibrations ofacolumn ofairmayalsobeexcitedbythe periodic applicationofheat, asinthewell-knownexperiment ofthe" singing flame," where ajetofhydrogenburns within an open cylindrical pipe.Forthemaintenance ofthevibration it isnecessarythat heatshould besuppliedatamoment ofcon- densation, orabstracted atamoment ofrarefaction. Toexplain howtheadjustmentiseffected, itwould benecessarytotake account ofthefactthatthevibrating systemincludes thegas contained inthesupplytube ofthejet,aswellasthecolumn ofairinthepipe. Thematter isthussomewhat intricate, butasatisfactory theory hasbeenmade out,which accounts clearlyfortheseveral conditions under which theexperiment isfound tosucceed ortofail*. 90.Theory ofReed-Pipes. Themechanism ofthereedstopsoftheorganisquite different. The current ofairissuingfrom thewind -chest is made intermittent byitspassage througharectangular apertureinametalplate, which isperiodically opened and closed (partially) byavibratingmetaltongue,or"reed." The periodisaccordinglydeterminedmainly bytheelasticity and inertia ofthetongueitself. Thevibrations ofthelatter were *Loid Rayleigh, Theory ofSound, 322h. PIPES ANDEESONATOBS 283 found byHelmholtz, bydirect observation, tobeofthesimple- harmonictype,butthefluctuations inthecurrent ofairare necessarilyofamorecomplex character. Iftheperiodic current beexpressed byaFourier series <7+G!cos(nt+O+Gzcos(2n<+ea)+ ...,...(1) the coefficients Gz,G3,...areusually bynomeans insensible ascomparedwithG1}andaccordinglyifthesound isheard directlyithasaveryharsh andnasal character. Inpractice, thereed isfitted with asuitable resonator, or" sound-pipe," whichspeciallyreinforces oneormore ofthelower elements in theharmonic series(1). Forthepurposesofmathematical treatment wemay idealize thequestion somewhat, andimaginethat atagiven pointintheinterior oftheresonator wehave asimplesource ofthetype correspondingtoone oftheterms in(1).It appearsfrom theelementary theoryof62that inthecase of acylindrical pipe,with thesource atoneend,thefrequencies ofmaximum resonance arevery approximately those ofthe free vibrations when thatend isclosed. Hence areed fitted with acylindrical sound-pipeofsuitablelengthwillemit aseries oftones whosefrequenciesarcproportionaltotheodd integers 1,3,5,....Inaconicalpipe,ontheother hand, with thesource near thevertex, wehave thecompleteseries of harmonics withfrequencies proportionalto1,2,3,4,...(see 84). But ineither case theharmonics ofhighorder are discouraged bytheincreasingdeviation ofthefrequenciesof maximum resonance from theharmonic relation which neces- sarilyholds intheexpressionfortheessentially periodiccurrent ofair. Asthequestionisinstructive invariouswaysitmaybe worth while toexamine more indetail thecase ofacylindrical sound-pipe (ofanyform ofsection), applyingthecorrection fortheopen end,andallowingforthedissipationduo tothe escapeofsound outwards. Theplanoftheinvestigationis similar tothat of 87,thedifferencebeingthatwenowhave asource Ceint(say)attheendx=LForsimplicity wewill 284 DYNAMICAL THEOBY OFSOUND assume thissource tobedistributeduniformlyoverthecross- section, sothat ^=5><[-q (2)9* toL J Letussupposeforamoment thatwehave aflux q=Acosnt (3) outwards from themouth. Thepressureat willconsist of twocomponents. Wehave firstthepartnecessarytocontrol theinertia oftheairnear themouth;thecorresponding part ofthevelocity-potential justinside is ,a.aA ... d>=-q=cosnt, (*) T0)*O) where ahasthesamemeaningasin87.Nextwehave the partwhich iseffective ingenerating divergingwaves outside. Ontheprinciplesof88this isfound tobe ^rAcosnt, (5) kA correspondingto(f>=-7sinnt, (6) since k=n/c.The totalvelocity-potentialat0,corresponding to(3),istherefore (ak \-cosnt+-r-sinrailA (7) ft> 4?r /^ Generalizing this,wemaysaythat toaflux q=Aeint(8) (nI/*ft\\A a -T}eint , (9)47T/ O) theexpression (7)beinginfacttherealpartof(9),whenA isreal. Thecorrespondencewillholdeven ifAbecomplex, since this ismerely equivalenttoachangeintheoriginoft. Wenowassume, fortheregionofplane waves, =,{cosk(l-a:)-0ink(I-x)}eint 3...(10)KCi) PIPES ANDRESONATORS 285 where theconstants havebeenadjustedsoastosatisfy (2). Comparingwith(8)and(9)wefind Bsinkl+CcosIdA,\ n J7n 11 (ilik-(O\A ?......(11)BcosklGsmkl=[kct-.}A. I V 47r;J Hence .77. ,.mkl+ka--7 coskl**' ......(12).n A( 7 fi ik'(O\. .,)G=A\coskl ka. TsmId ( \ 4>7TJ} The latterequation givesAinterms ofG.Considering only absolute values wehave =(coskl-kasinkl)*+ sin2JM....(13) Since k2coisusuallyasmall fraction, theemission ofenergy, which varies as |Az \,willbegreatestforagivensource Ocosnt when cosklkasinkl, ..................(14) nearly,i.e.when theimposed frequency approximatestothat ofoneofthenormal modes ofthepipewhen closed atx=I, asdetermined by8"7(11). Inthecase ofthereed-pipe, therefore, thetones which arespeciallyreinforced consist of thefundamental andtheharmonics ofodd order. When(14)issatisfied, wehaveby(12) G ik'2o) .,, ._ ._= amkl ...................... (15)A 4>7T^ ' Thisdetermines therelation between thefluxoutwards atthe mouth, andthatconstitutingthesource. Theformer nowgreatly exceeds thelatter inamplitude,andthefactor ishews that it differs inphase byaquarter-period. Again,from(10)and(11)wehave \sin/bU1+ko.coskx .coskx[eint . ...(IG) (4-7TjV' 286 DYNAMICAL THEORY OFSOUND When (14) holds, thisreduces to .A{,., ,ik-u) .77 <_- .rJcos fcn-.x\ sm jficos rkctismkl\^ J4-7T ,4nriC ( ,,, xi&2&> .77 7),/ /-i>r\ =-7, , o77icos &(J-#) -3sm 2cos/fo^e1'1 *.(17) 7cjco2sin2kl{^ '4vrjvx The realpart gives 47r<7f./7 x. .&co .7 J <P=nr~ -> <.77{cosK(Ix)sm T?i^sinklcos&#cosnt>. ^^co'5sin2/[x4vrJ .........(18) correspondingtoasource (7cos?ii. The variablepartofthe pressureatthesource(sc=Z)is ,- 4>irpcCf,,l?u> .77 ,7.A/in \PPo=P<f>= 7---"--Tf -7cosnt+-rsmAtcosklsmni .(19) j-rurr/c^^sm2^V 4?r / The firstpartofthis isbyfarthemore considerable;itis, moreover, theonlypartwhich iseffective indoingwork. The mean rate ofwork done atthesource, i.e.themean value of pOcosnt,is w.*=eOL................... (20)k-io*sm2kl^ ' Itmay easily beverified that this isequaltotheworkspentin generatingwaves atthemouth, where, by(15), 4-77(7 rj-.smki(21)^ ' Itappearsfurther from(19)that themaximum ofpros- sure atx=lsynchronisesalmost with themaximum influx ofair,followingithowever byashort interval. There is therefore atendency slightlytolower thepitchofthereed, which is,intheinstruments here referred to,ofthe"in- beating typo,"i.e.thepassageisopened when thereedswings inwards, towards thewind-chest. The factthattheresultant forceonthereed isapproximatelyinthesamephasewith the displacementindicates that thereed isvibrating with a frequency somewhat lessthan that natural toit(12). PIPESANDRESONATORS 287 Thereed-stopsofanorganfallinpitchasthetemperature rises,owingtothediminishedelasticityofthemetaltongues; this istheoppositeofwhathappenswithregardtotheflute- pipes (62).Areed-pipeistunedbyacontrivance which alters theeffectivelengthofthevibrating tongue. Itshould bementioned that there isanother class of instruments inwhich the"reed" hasamuch smallerelasticity and ismainlycontrolledbythereaction oftheresonant chamber, itsown naturalfrequency being relativelylow.The reed isthen ofthe"out-beating type,"theaperture being widest when thereedswings outwards,i.e.with thewind. Thehumanlarynx comesessentially under this class. 90a.Multiple Resonance. Some other forms ofresonator besides thesimple type described in86maybenoticed. Inthe firstplace suppose wehave aresonator with two aperturessofarapartasnottointerfere withoneanother. The fluxoutwardsthroughthesebeing denoted byq1andqz,and theconductivities byKI,K^,wehave =#!<, q9=K9<j),d*s= (j>y=-(ft+2a)/<2, (1) bythesamereasoningasin86.Hence Kc2 Kc2 &+--(2i +ffa)=0&+-g-(2i+fc)=0.(%) One solution isevidently ft+q$=0,whichrepresents merelya current throughtheresonator. Thealternative solution is |L= .p.=(?cos<>+), (3) /ij.jg where n2=(K,+IQc2/Q (4) Again,takethecase ofa"double resonator," consistingof two cavities connected byanarrowopening.Letthevolumes bedenoted byQ,Q'andthefluxes outwards from thesebyq,q, andtheconductivities ofthetwoapertures byK,K'. Wehavethen withanobvious notation q=K(f>, q'=K'(qb'-</>),C2s=<, C2s'= cj>', ...(5) and s=-(-q'}JQ,s'=-q'IQ' (6) 288 DYNAMICAL THEORY OFSOUND Hence ..K# Kc* , Assumingthat<?and' varyascos(nt+a),andeliminating theratioq'/q,weobtain thefollowing quadraticinw2 : Ifweputn2=or=oo,theexpressionontheleft-hand side is positive,whilst itisnegativeifn2=Kc~/Q,or=K'tf/Q'. Hence oneroot liesbetween zeroandthesmaller ofthetwo latter quantities,whilst theother isgreaterthan either.Moreover, since thefirstoftheequations (7)gives itappearsthat inthecase ofthelower rootqandq'have the samesign,whilst thesignsareoppositeinthehigher mode. 90b.TheHot-Wire Resonator. Increased sensitiveness intheuseofresonators oftheHelm- holtztype (86)fortheanalysisofcomplexsounds canhe securedbyelectrical means. Inthe"selective hot-wire micro- phone"asitiscalled, anelectricallyheatedplatinum gridis placedintheneck ofacylindricalHelrnholtz resonator which iscapableofbeing tuned byasliding piston. Thesubjectof observation isthechangeofresistance ofthegridowingtothe coolingeffect oftheairflowingtoand frointheneck. This changeconsists ofasteady andaperiodic part,either ofwhich canbemade thesubjectofmeasurement;thesteady part,for instance, bymeans ofaWheatstonebridgeinonearm ofwhich thegridisinserted. The resonator itself must becalibrated independently, e.g.bymeans ofaseries oftuning-forksofknown pitch. Themethod hasrecently beenappliedtotheanalysisof thevariousperiodsofvibration ofanairscrew*. *A.Page, Proc. Roy. Soc. vol.cvn.p.451(1925). CHAPTER X PHYSIOLOGICAL ACOUSTICS 91.Analysis ofSound Sensations. Musical Notes. The vibrations ofelastic bodies andthepropagationof waves throughtheatmospherearesubjecttowell-ascertained mechanical laws,andtheinferences drawn from these canbe controlled bymore orless decisiveexperiments. Butwhen weapproachthefieldwhere thehuman mechanism comes into play,wearemetbythepeculiardifficulties which areinherent intheobservation andstudyofsubjective phenomena.In particular,whenweendeavour toanalyseafamiliarcomplex sensation into itselements, weareattemptingatask forwhich theexperienceofdailylifehaspeculiarlyunfitted us.Thus wemayhave beenaccustomed tointerpretthesensation in questionasindicatingthepresenceofaparticular object ;or theoccurrence ofaparticularkind ofevent, inaparticular place. Theelements ofwhich itismadeupgive individually little ornoinformation; itisthecombination which issignificant, andattention tothedetails wouldonlydistract fromwhat isof immediatepracticalinterest. Tousearough andindeed an utterly inadequate illustration, itisasifwewere toinsistupon spelling every wordweread. Thetheoryofsense-perception, especiallyinrelation to opticsandacoustics, isafascinating subject, but itcannot be dealt with here. Thestudent who isunversed initmaybe referred tothewritingsofHelmholbz*. *Thetheoryisexplained initsacousticalbearingsintheTonempfindungcn, already cited(p.3).Itisalsodiscussed from theoptical point ofviewmhis Handbuch derphysiologtschen Optik, 2nded.,Hamburg andLeipzig, 1890. Elementary expositions willbofound inthetwovolumes ofhieVortrtige und Reden, Brunswick, 1884, ofwhich there isanEnglish translation byE.Atkinson, with thetitle: Popular LedarcsonScientific Subjects, 2nd ed.,London, 1803. firstpointonwhich thestudent shouldsatisfy himself isthat thevarious simple-harmonicvibrations which areasarule combined intheproductionofamusical note arereally represented byindependentelements intheresultingsensation; thatthelatter caninfactberesolved intoafundamental tone andaseries ofharmonics. Forthisaslight course ofeducation isnecessary. Aseries ofresonators ofthetypeshewn in Fig. 80,p.261,tuned totheovertones which itisdesired to detect, areofgreatservice forthispurpose*. But such assistance isnotindispensable,andagooddealcanbeeffected with thepianoormonochord. Take forinstance thenotec, whose harmonics arec',g',c",e",g",.... Ifonthepiano one ofthese, say g',begently sounded, andthekeythenreleased, sothatthevibration isstopped,and ifimmediately afterwards thenote cbestruck with fullintensity,itisnot difficult to recognizeinthecompoundsensation thepresenceofthe element previouslyheard. This isoften moreperceptibleas thesound diesaway,theovertonesbeing apparently extinguished moreslowlythan thefundamental. Amoreimmediately convincingseries ofexperimentscanbemade with the monochord, orwith apiano whosestringsarehorizontal and thereforeeasilyaccessible. Ifastringbesetinto vibration whilst dampedatanodalpointofoneoftheharmonicsby contact with ahair-pencil,thefundamental toneand allthe harmonics oflower rankmaybereduced inintensityor altogether extinguished, accordingtothedegree andduration ofthepressure applied.Inthiswayawhole series oftypes ofvibration canbeproducedinwhich theharmonic inquestion isaccompanied byavarying admixture ofthefundamental, &c. Theoccurrencethroughoutofthecorresponding sensation as anindependentelement intheresulting sound isinthisway easily appreciated. Thepianoalso lends itselfreadilytothe *Itmay benoted that theexternalear-cavityisitself aresonator, responding most intensely toacertain tone, which varies for different individuals but isusuallyintheneighbourhood ofeiv r/lv .Theaperture being relatively large, thedamping andconsequently therange ofresonance isconsiderable. analysisofcompound notesbyresor- example,befreed from itsdaropp whilst cissounded foramomeni andcontinued bythefirst-menti hand, thestringcbefreefrom it; foramoment, thetone c'istake lower string.Thesesimple expei arerecommended byHelmholtz, many ways. Again, when theear partialtones inacomplex note,it ofaparticulartone oftheseries wl harmonic vibration isnotexcited, isstruck atitsmiddlepoint,the wanting (26). 92. Influence ofOvertone; Thequalityofamusical not number and relative intensities composeit.Thekind ofinfluence w ranks exercise onthequalityissi somewhat asfollows: 1.Pure tones likethose oftuningforks with resonai. boxes, orofwidestopped organ pipes,aresoftandpleasin, smooth, butwantinginpower. 2.Notes which contain aseries ofovertones uptothe fifth orsixth inrank arericher andmore musical, and are perfectlysmooth solongasnohigher overtones aresensible. Thenotes ofthepianoandofopen organ pipesareexamples, whilst those oftheflute,and oftheflute-stops ontheorgan whensoftly played, approximate more tothecharacter ofpure tones. Inthe"mixture" stopsoftheorganthe lower harmonics areexpressly providedingreater intensity by auxiliary pipeswhich areplayed automatically alongwith that whichgivesitsname tothenote. 3.When theharmonics ofeven order areabsent, asinthe case ofastopped organ pipe,orapiano stringstruck atthe middlepoint,thenotehasahollow, andeven anasal character, iftheoddharmonics arenumerous. 292 DYNAMICAL THEORY OFSOUND 4".Thesound mayfurther bedescribed as"full," ifthe fundamental tonebepredominant,and as"empty"ifitbe relativelyfeeble. This isexemplifiedinthe difference of qualitybetween thesound ofapiano-wire when struck with asoftorahardhammer, respectively (26,38). 5.When harmonics beyondthe sixth orseventh are presentinconsiderableintensity,thesound isharsh and rough, owingtothediscords which thesehigher overtones make with oneanother. If,however, thehigher harmonics, though present,arerelatively weak, asinthecase ofthe stringedinstruments ofthe orchestra, reed-pipes, and the human voice, theyareuseful asgivingcharacter andexpression tothesound. Brass instruments, ontheother hand, with theirlongseries ofpowerful overtones, areasaruleonly tolerable incombination with others, orforthesake of particulareffects. Theanalysisofthesounds ofthehuman voice isnaturally amore difficult matter. Inparticular,theconstitution ofthe vowel sounds hasbeenmuch debated, without anyvery definite conclusion. Thesame vowelmaybesungonawiderangeof notes, butpreservesitspeculiarcharacterthroughout;andthe question arises, does thisspecial quality depend solely onthe relative intensities ofthevariouspartial tones, oronthe predominanceofone ormore overtones of,ornearto,a particular pitch?Itwillberemembered that thevibration of thelarynxisperiodic,andthatparticular harmonicsmaybe reinforced bytheresonance ofthemouth-cavity,asinthecase ofareed-pipe (90).Thebalance ofauthority appearsto incline, thoughnotvery decisively,tothe"fixed-pitch" theory, which isthesecond ofthetwoalternatives above stated. A review ofthesubject down totheyear1890 willbefound in theconcluding chapterofLordRayleigh'streatise. 93. Interference ofPure Tones. Influence onthe Definition ofIntervals. Ithas sofarbeen assumed that thesensations duo to twocoexistentsimple-harmonicvibrations uroproduced quite independentlyofoneanother. Thisappearstobeinfactthe PHYSIOLOGICAL ACOUSTICS293 casewhen theinterval hetween thetwotones issufficiently great;butwhen theinterval issmallwehave" interference," asweshouldexpect from theanalysisof10,andthesensation isinwhole orinpartintermittent. Thephenomenon of"beats" hardly needsdescription ;itisoftenmetwith inmistunedpairs ofpiano-wires,inthevibrations offinger-bowls, andsoon.For methodical studytwopure tones arerequiredofequal intensity, ase.g.fromtwotuningforks(with resonators),ortwostopped organ pipes,which canbemade todiffer inpitch byavariable amount. Asunison isdeparted from, thebeats(whose frequencyisalways equaltothedifference ofthefrequencies oftheprimary tones)areatfirstslowandeasilycounted. As theinterval widenstheybecome morerapid, andasensation of roughnessordiscord isexperienced;moreover, theprimary tones arenowheardalongwith thebeats.Finally,asthe interval iscontinually increased, thebeats andtheconsequent roughness graduallycease tobeperceptible. The intervals atwhich roughness begins andceases, varyin differentpartsofthescale. Forthesame interval therough- ness isless,thehigherthepitch;ontheotherhand foragiven number ofbeatspersecond theroughnessisgreaterinthe higheroctaves. Inthecase oftwo(ormore) compoundmusical notes,we mayhave beats andeventual roughness betweenanyconstituent tones which aresufficiently near inthe scale.Wemayeven have interference between thehigherovertones ofthesame note;and itisforthisreason thatharmonics ofhigherorder thanthesixth areprejudicialtogoodmusicalquality. Itisthroughtheinterference ofpairsofovertones that deviations from theconsonant intervals(3)usually make themselves fell;.Thus inthe cu.se oftheOctave cc'wehave tones with thefrequeneioH c=132, "2(>4, 396, 528, 660, 792, ..., c'= 20-i, 528, 792, ..., and ifthisbemistuned ;illtheharmonics of c'areinterfered withbytheevenharmonica of o. 294 DYNAMICAL THEORY OFSOUND InthecaseoftheFifth egwehave c=132, 264, 396, 528, 660, 792, ..., g= 198, 396, 594, 792, ..., and ifthisbemistimed thesecond tone ofgbeats with the third tone ofc,and soon.When theratio ofthevibration numbers ofthefundamentals islesssimple,theharmonics which caninterfere areofhigherorder. Thus inthecase of theMajor Third, where theratio is4:5, the firstpairof interferingovertones consists ofthefifthtone ofthelower note, andthefourth ofthehigher.Since inmanymusical instruments thefifthtone isvery feeble, thisconsonance islesswell defined than theprecedingones.Ontheotherhand thefundamentals may fall,inthelowerpartsofthescale, withinbeating distance (forexamplec=132, e-165),sothat thisconsonance istobe reckoned also aslessperfectthan theformer ones. Similar remarksapplywithgreaterforce tosuch cases astheMinor Third (5:6) andtheMinor Sixth (5:8). 94.Helmholtz Theory ofAudition. The connection between primarysensations andsimple- harmonic vibrations has still tobeaccounted for.Theproblem isaphysiological one; butthetheory which Helmholtz has framed toexplain Ohm's law, sofarasitholds, andthevarious deviations fromit,isinitsessentials sosimple, and isso successful inbinding togetherthe facts ofaudition intoa coherentsystem,thatabriefstatement ofitmaybeattempted. Initssimplestform thetheory postulatestheexistence, somewhere intheinternal ear,ofaseries ofstructures each ofwhich hasanaturalperiodofvibration, and isconnected with adistinctnerve-ending. Forbrevity wewillspeakof these structures as"resonators," since that istheirproper function. Aparticularresonator isexcited whenever a vibration ofsuitablefrequency impingesonthe ear; the appropriatenerve isstimulated; andthesensation iscom- municated tothe brain. Inthiswaythe resolution ofa musical note into itsconstituent tones isatonce accounted for. Itisnecessarytosupposethat theresonators aresubject PHYSIOLOGICAL ACOUSTICS 295 toaconsiderable amount ofdamping.Ifitwere not so, eachresonator would goonvibrating, andthecorresponding sensation wouldpersist,foranappreciable time after the excitingcause hadceased. Asimilar interval oftimewould elapsebefore thesensation reached itsfullintensity when the cause first sets in.The effect would bethat thesensations duetoasufficiently rapidsuccession ofdistinct notes would notbealtogetherdetached from oneanother inpointoftime. From considerations ofthiskind Helmholtz estimated that thedegreeofdampingmust besuch that theintensity (as measured bytheenergy)ofafree vibration would sink to one-tenth ofitsinitial value inabout tencomplete vibrations. Itfollows, asexplainedin 13,that each resonator will respondtoacertainrangeoffrequenciesoneach side of theonewhich hasmaximum effect. Itisassumed, further, thatthedifference ofpitchofadjacentresonators issosmall thatthesamesimple-harmonicvibration will excite awhole group,theintensity fallingofffrom thecentre oneither side. This isillustrated bytheannexedfigure, repeatedfrom 13, whichmaynow serve toexhibit thedistribution ofintensity overacontinuous series ofresonators under theinfluence of agiven simple-harmonicvibration. The abscissa isp/n l, wherepisnowtaken torepresentthenatural frequencyofa resonator, andnthatoftheimposed vibration. Thehorizontal scaledepends onthevalue of/3,orI/TIT, where Tisthe modulus ofdecayofafree vibration. Ontheabove estimate ofHelmholtz weshallhave __~~ 1(5> whence /9='018. Theintensityistherefore one-half the maximum for |=1-018. Itwillbeobserved thatontheabove viewweoughtin strictness tospeakof"simplest"rather than of"simple" sensations ofsound, absolutely simple sensations, inthe strict physiological meaning, being impossibletoexcite. When twosimple-harmonic vibrations, sufficientlyfarapart inthe scale, areinoperation,thetwogroupsofresonators which areaffected willbepractically independent,andthe two sensations (ofpure tones)will coexist. Butwhen the interval between thefrequenciesissufficiently small, thetwo groupswilloverlap,and theenergyofvibration ofthose resonators which arecommon tothem will fluctuate inthe mannerexplainedin10.Theexcitation ofthecorresponding nerve-endingswilltherefore beintermittent, with afrequency equaltothedifference ofthose ofthetwooriginatingvibrations. Thisis,onthetheory,theexplanationofbeats. Astheinterval isincreased, thebeats become morerapid.The"roughness" which isultimately perceived,inspiteofthediminishing amplitudeofthefluctuations, hasamore remotephysiological explanation. AccordingtoHelmholtz, there ishereananalogy with thepainfuleffectproduced byaflickering light,and in other caseswhere anerve isstimulated repeatedlyatintervals oftimewhich areneither toogreatnortoosmall. When the intervals aresufficiently long,thenerve hastime torecover itsinitialsensibility,andsoexperiencesthefulleffect ofeach recurringstimulus. When ontheother hand the intervals aresufficiently short, thesensation tends tobecome continuous. Itisforthisreason "that beatsexceeding, say,132persecond cease toproducethesensation ofroughness,evenalthough theinterval between thebeatingtones besuch aswould bo perceptiblydiscordant inalowerpartofthe scale. PHYSIOLOGICAL ACOUSTICS 297 Thestudent ofdynamicscannot failtoadmire thebeauty ofatheory which lends itself readilytotheexplanation ofso many complicated relations; but itiswith thephysiologist andtheanatomist that inthe last resort itlies todecide whether amechanism ofthekindpostulatedisreallytobe found intheinternal ear. Intheoriginalform ofthetheory (1862) theresonators were identified with thestructures known as"Corti's rods," which arefoundarranged, some 3000 in number, alongthe basilar membrane inthespiral cavity ofthecochlea. Adisturbing discovery byHasse that these structures donotoccur intheears ofbirds, towhom wecan hardly denytheperceptionofpitch,ledtoamodified form ofthetheor3r.Inthethird edition oftheTonempfindungen (1870) Helmholtzpropoundedtheview that theresonators consist ofthevariouspartsofthebasilar membrane itself. Thismembrane varies inbreadth from oneend toanother, likeaveryacute-angled triangle,andthetensionappearsto beverymuch less inthedirection oflengththan inthat ofbreadth. Onthisview the differentpartscould beset intosympathetic vibration, much asinthecase ofaseries ofstringsofvariablelength placedsidebyside,except that theindependenceofadjacent parts would beapproximate instead ofabsolute. Forafulldescriptionofthecomplicated structure oftheinternal ear,and forfurtherspeculationsas tothefunctions performed byitsvariousparts,wemust refer tobooks onphysiology. 95.Combination-Tones. Inoneimportant respectthetheoryassofardeveloped isinadequate. Theexplanationofconsonant intervals outlined in93assumes thatoneatleast, andgenerally both, ofthe notes concerned iscomplex,andcontains oneormore overtones inaddition tothefundamental. Itwas infactthroughthe interference oftwotones, oneatleast ofwhich isanovertone, thatdeparturefrom theexact relation ofpitchwasstated to make itself manifest. When both tones arepurethismeans ofdefinition iswanting,andonthetheoryofaudition sketched intheprecedingsection thereappearstobe110reason why 298 DYNAMICAL THEORY OFSOUND theoctave(forexample)should bedistinguished byany character ofsmoothness fromadjacentintervals oneither side, thetwogroupsofsensationsbeinginanycasequite independent.Since themore consonant intervals atallevents areasamatter offacteasily recognized bytheear,even in thecase ofapparently pure tones, and arethoroughly well defined, thedifficultyisaserious one.Tomeet it,Helmholtz developedhistheoryof"combination-tones," which areassumed tosupplythefunction ofthemissingovertones. Inmost ofourinvestigationsithasbeenassumed thatthe amplitudeofthevibrations maybetreated asinfinitely small, sothat disturbances due todifferent sources maybesuper- posed bymere addition. Inthetheory now inquestionthis assumptionisabandoned; the vibrations areregarded as small, butnotasinfinitely small, andtheinteraction ofthe disturbances duetodifferent causesis,toacertaindegreeof approximation, investigated. Wehavealreadyhadanindication in63ofthemanner inwhich twoimposed simple-harmonic disturbingforces of small butfiniteamplitude,withfrequencies Nl}J\r 2respectively, may generateinthe airother simple-harmonic vibrations whose frequenciesare 2N,,2N2,Nt-Ns, #,+# andwhoseamplitudesinvolve thesquaresorproductofthe amplitudesofthetwoprimaries.Iftheapproximation were continued weshould meet with further vibrations whose fre- quenciesareofthetypepi&\ p^z, where pltp,2areintegers. Inacoustical language, twosimple-harmonicvibrations can, if ofsufficient intensity, giverisenotonlytothepuretones usuallyassociated withthem, butalso toaseries ofotherpure tones ofhigherorder. The factthatasingle harmonic vibration canbyitselfgiverisetoapuretonetogetherwith itsoctave, &c. isitself ofsomeimportance,butthemostinterestingresult is due totheinteraction, viz.the"difference -tone" (Nj.N2). The existence ofdifference-tones wasobserved, apart from alltheory, bySorge (1745) andTartini(1754). The"sum- mation-tone"(JVj+JV2)ismore difficult tohear, and its PHYSIOLOGICAL ACOUSTICS 299 existence hasevenbeen denied. Ithashowever beenobjectively demonstrated byRiicker andEdser*, byitseffect onatuning forkofthesamefrequency. Difference-tones duetothecausesjustconsidered aremost easily perceptible where wehaveamass ofairwhich issubject tothejoint andvigorousaction oftheprimary vibrations, as intheharmonium andthesiren; theycanthen, likeother tones, bereinforced bysuitable resonators. There ishowever awayinwhich combination tonesmay conceivably beoriginatedintheear itself. Toexplainthis itisnecessary brieflytoconsider theforced vibrations ofan unsymmetrical system. When aparticle,oranysystem having virtuallyonedegreeoffreedom, receives adisplacement x,the force(intrinsic tothesystem)which tends torestoreequilibrium isafunction ofoc,andmaybesupposed expressed,forsmall values ofx,byaseries fjix+ax*+/3x3+ (1) Anexampleisfurnished bythecommon pendulum,where theforce ofrestitution isproportionaltogsin8,or but here, onaccount ofthesymmetrywithrespecttothe vertical, the forcechanges signwith 6,sothat onlyoddpowersof6occur. The correction for small finiteamplitudes dependstherefore onthe term ofthethird order in9.But ifthesystem beunsymmetricaljasinthecase ofapendulum hangingfrom thecircumference ofahorizontal cylinder-]-,theterm ofthesecond order comes in, andthecorrection ismoreimportant.Helmholtz Fig- 8G- laysstress onthe fact that intheslightly *Phil.Mag. (5),vol.xxxix.(1895). tIfabetheradius, and Ithelength ofthefreeportionofthestring when vertical, thepotential energyis__~2 I 6P"" where sisthearcdescribed bythebobfrom thelowest position.Therestoring force istherefore dV_m()1mya2 - s~~"4'"" funnel-shaped tympanic membrane and itsconnections we havepreciselysuch anunsymmetrical system, therestoring forcebeing somewhatgreaterforinward than foroutward displacementsofthesame extent*. Ifwekeep onlythe first twoterms in(1),theequationofmotion isofthetj7pe x+px~-ax*+X,..................(2) whereXrepresentsthedisturbing forcef. Thejointaction oftwosimple-harmonicforces willberepresented by X=/icosnj+/2cosnzt................(3) Neglecting,forafirstapproximation,thesquareofx,wehave f f< co~ ,cosn^t -i--^cosnt, ......... (4) IM-n? yu,-n22"' ^J theterms whichrepresentthefreevibrationsbeing omitted, since these arerapidly destroyed bydissipation.Ifwesub- stitute thisvalue ofxontherighthand of(2),andwrite for shortness fif(p-n1*)= ffi,/2/(>-n22 )=#2.......... (5) weobtain thedifferentialequation x+^x=X\a.((/f+#22 )\&g\cos2n:\ag.? cos2n2t -ag!g2cos(nt-wa)*-ag^cos(^+n.^)t,...(6) correct tothesecond order of/i,/2.Theterms written infull on"therighthandmayberegardedasacorrection tothe disturbingforceX.The solution of(6)gives,inaddition to (4),theterms 2 2 cos(Wl-,)*--T^^.cos ' The firsttermmerelyindicates ashift ofthemeanposition *Itmaybenoted that thesame element ofasymmetryispresent inthe investigationof63.When \veproceedtothesecond order ofsmall quantities, thechanges ofpressure due tocondensations =tsarenolonger eijual in amount. )Itisunnecessary totakeaccount ofthevariabilityofinertia, since tin's canbegotridofbyaproper choice ofthecoordinate x.Inany case itwillnotalter thegeneral character oftheresults obtained inthesecond approximation. about which theoscillations takeplace. Fortherest,wehave octaves oftheprimary tones, together with adifference- and asummation-tone. Iftheapproximation were continued we should obtain combination-tones ofhigher order, asinthe former case. When, asinthecase ofthetympanic membrane, the freeperiod 2-Tr/A/yu,isrelatively long,themostimportant combination-tone isthedifference-tone (n^?72),onaccount of therelative smallness ofthecorresponding denominator in(7). Thetheoryofcombination-tones herereproducedhasnot beenaccepted withoutquestion. The difference-tones, as already mentioned, wereknown asafact since thetime of Tartini, and aplausible explanation hadbeengiven by Thomas Young (1800). Accordingtothisview thebeats between thetwo tones, astheinterval increases, ultimately blend, as ifthey were somany separate impulses,into acontinuous tonehavingthefrequencyofthebeats. The difficultyofthisexplanationisthat theactualimpulses duringabeat areasmuchpositiveasnegative,sothat itdoes notappear howany appreciableresidual effect in either direction could beproduced,ifthevibrating system besymmetrical.Itistrue that ifweturn tothefigureon. p.23, itisapparently periodic,with theperiodofthe in- termittence;butfrom thepointofview ofFourier's theorem thelower harmonics areallwanting, andtheonlytwowhich are presentarepreciselythetwowhich areused inconstructingthe figure. OntheHelmholtztheoryofaudition theintermittent excitation ofaparticularresonator mtimes asecond isawholly different phenomenonfrom theexcitation ofanaltogether distinct resonator whose naturalfrequencyisin.Young's viewappearsindeed tobeinadmissible onanydynamical theoryofaudition, atleast inthecase ofinfinitelysmall vibrations. Ontheother hand itistrue, aswehave seen, thatgivenafiniteamplitude,andanunsymmetrical system, avibration ofthetypeshewn inFig. 10, p.23,doesactually generate (among others)avibration whoseperiod corresponds tothefluctuations there shewn. The distinction between the two theories might therefore, fromamerely practical pointof view, beheld tobealmost verbal, were itnotthatYoung's theoryfails togiveanexplanationofcombination-tones other than the first difference-tone. 96. Influence ofCombination-Tones onMusical In- tervals. Abrief indication ofthewayinwhich combination-tones mayassist indefiningtheconsonant intervals isallthatcanbe attemptedhere. Take firstthecase of(primarily) puretones. Inthe case ofaslightlymistuned Octave, sayNI=100, N~2= 201,wehaveNzNI=101,whichgives adifference-tone making1beatpersecond withN^. FortheFifth, letN,=200,N2=301. Wehave N2-N,=101, 2J^-N2=99, givingcombination-tones with 2beatspersecond. FortheFourth, letN^=300,N=401. Then 2J\r !~^2=199, 2Ar a-22^=202, andthecorrespondingtonesmake 3beatspersecond. FortheMajor Third, let^a=400,Na=*5Ql. Wehave 2^-2^ =202,3^-2^=198, giving 4beatspersecond. Wemight proceedfurther inthelist,but itwillalready have been remarked that combination-tones ofincreasingly o */ highorder arebeinginvoked. This isquiteinconformity with theobserved factthat thebeats are,inallcases after the octave, veryfaint unless theprimaries beespecially vigorous.Amore effectivepartisplayed bythecombination-tones when thenotes concerned haveoneortwoovertones, butnota sufficient rangeofthem toaccount forthedefinition onthe principlesof 93.Take forinstance thecase oftheFifth, when each note hasafirst harmonic inaddition tothe fundamental. Iftheinterval beslightly mistuned, wehavesay theprimarytones :200,400;301, 602. Thesegivethetwo difference-tones 301-200=101,400-301=99,which inter- ferewithoneanother. Thecombination-tones haveaninfluenceagain,inthecase PHYSIOLOGICAL ACOUSTICS 303 ofconsonant triads, especiallyofsimple tones, butenough hasbeen said toshew theirimportancefrom themusicalpoint ofview. Forfurtherdevelopmentsreference mustbemade to thework ofHelmholtz*. 97.Perception ofDirection ofSound. Oneimportant questionofphysiologicalacoustics inwhich dynamical principlesareinvolved remains tobementioned. Anobserver, evenwhen blindfolded, andwithnoadventitious circumstances toguide him, isingeneralable toindicate with great accuracythedirection fromwhich asoundproceeds.In thecase ofpuretones thediscrimination between backand front isindeed lost, aswas tobeexpected, consideringthe symmetry withrespecttothemedialplaneofthehead, but right and leftareclearly distinguished. Fortones ofsmall wave-lengththismay beaccounted forbythedifference of intensityofthesensation inthetwoears, since thehead acts tosome extent asascreen, asregardsthefurther ear.But when thewave-lengthofthesound much exceeds theperi- meter ofthehead theinvestigation givenneartheendof81 shews that this differenc themost recentinvesti; terpretation dependsont~~ reach thetwo ears,adifference u. beingeffective. Hefound that different channels tothetwo ears, and. allextrajucuuo dis- turbances beexcluded, thesound canbemade toappearto come from therightorleftatwill,byadjustingtherelative phase. Theoriginofthesound wasalwaysattributed tothat sideonwhich thephaseisinadvance(bylessthan halfa period).The result, which hasbeen arrived atindependently byother observers,isatpresent unexplained.Ithasbeen suggestedthat thephenomena may reallybedue toadiffer- ence ofintensity. Afraction ofthesoundmaybetransmitted fromeach side totheoppositeinternal ear,throughthebones of *SeealsoSedley Taylor, Sound andMusic, London, 1873. +Phil.Mag. (6),vol.xni. (1907). thehead, inwhich casetheoriginaldifference ofphase would produceaslightdifference ofintensity onthetwosidesowing tointerference between thedirect andtransmitted vibrations*. Bayleigh'sresults havefound apractical applicationtodirec- tion-findinginsubmarine audition. Two receivers atafixed horizontal distance dapart communicate with theearsofthe observer bytwochannels, oneofwhich isvariable inlength. Theobservation consists inadjustingthelengthuntil thesource ofsoundappearstobestraightahead. If6bethetrueazimuth ofthesource, theexcess oflengthoftheadjustablechannel will bedsinQ,accordingtothesideonwhich thesource lies.The instrument carries agraduation which enables 6toberead off directly. *Myers andWilson, Proc. Roy. Sac. vol.LXXS. A,p.260(1908).This hypothesisisdiscussed byLord Eayleigh, Proc.Roy. Soc. vol.LXXXIII. A,p.61 (1909). INDEX [Thenumerals refer tothepages] Absorptionofsound, 199 Acoustic propertiesofbuildings, 225 Adiabatic lines, 161 Aeolian tones, 88 Air-waves, general theory of,207 seealsoSound waves Amplitude, minimum audible, 170 Analysisofsound sensations, 2,289 Anticlastic curvature ofaflatbar,154 Approximatesolution ofperiod- equations, 83,86,127, 129,260 Audibility, rangeoffrequency for,3 leastamplitude for,170 Audition, Helmholtz theory of,294 Bars, longitudinal vibrations, 116 Hexural vibrations, 122, 126,129 Beats, 23,134,140 relation of,todissonance, 293 Bells, 159 Bessel's functions, 85,147, 149,263 Blackburn's pendulum, 35 Chain, vibrations ofhanging, 84,88 Circular vibrations, 55 'Circulation' defined, 206 Clamped-free bar,transverse vibrations ofa,129 Combination-tones, 184, 297,299 Communication ofvibrations toagas, 241 'Condensation' defined, 163 Conduction ofheat, effect of,onsound waves, 190 'Conductivity'ofanaperture, 249 Conical pipe,normal modes ofa,2G1 Consonant intervals, 3,292,302 Cosine-series, 94 Curved sheila, vibrations of,158 Cylindrical vessel, normal modes ofa, 263 Dampingofvibrations, 25,27,57 effect of,onresonance, 32 ofair-waves byviscosity, 186, 188, 193 ofaresonator, 268 ofanorgan pipe, 273 Degreesoffreedom ofadynamical system, 12,34 Diatonic scale, 5Diffraction ofsound, 241, 248,252 'Dilatation' denned, 109 Direction ofsound, perception of, 303 Discontinuity, waves of,184 Dissipation ofenergy byfriction, 27, 188 Dissipation (apparent), bygeneration ofair-waves, 169,238, 241, 269,273 'Divergence' denned, 202 Doppler's principle, 226 Double pendulum, 38 Double resonator, 287 'Double source' ofsound, 218,230 Elasticity, elementary theory of,108 coefficients of,112,115 ofgases, 162 Elliptic vibrations, 49 Emission ofenergy, byasimple source, 229 byadouble source, 231 byaresonator, 269 byanopen pipe, 273 Energy,ofasimple-harmonic vibra- tion, 15 ofastring,60 ofanelastic solid, 116 ofabar,125 ofamembrane, 143 ofabentplate, 154 ofair-waves, 166,208 'Extension' denned, 109 Extensional vibrations ofarod,116 ofacircularring, 138 Finite amplitude, air-waves of,177 Flexure, uniform, ofabar,123 ofaplate, 152 Flexural vibrations, ofabar,122 ofaring, 139 ofaplate, 154 'Flux' defined, 202 'Flux ofenergy,' 168,228 Forced oscillations, 16,20,47 effect offriction on,28,57 Fork, tuning, 133 Fourier's theorem, 89,94 influence ofdiscontinuities in,94 lawofconvergence ofcoefficients in, 90 306 INDEX [Thenumerals refer tothepages'] Freedom, degrees of,12,34 Free-free bar,transverse vibrations of a,126 Free oscillations, 12 with friction, 24 general theory of,44 Frequency, range of,foraudibility,3 Friction. SeeDissipation Gas, elasticityofa,160 isothermal andadiabatic lines ofa, 160,161 Graphical solution ofperiod-equations, 83,127, 130,260,262 Grating, transmission ofsound bya, 251 Harmonicanalysis, 103 Harmonics, 5 Heat, vibrations causedby,282 Heat-conduction, effect of,ousound waves, 190 Hooke's lawofelasticity, 11,112 Hot-wire resonator, 288 Huygens' principle,253 Imaginaries, use of,53 Impact, vibrations ofastring dueto, 73,101 Indicator diagram, 160 Inertia,coefficients of,42 Interference ofsimple-harmonic vibra- tions, 23 ofpure tones, 292 Intervals, musical, 5,292 degreeofdefinition of,302 'Irrotational' motion defined, 206 Isothermal lines, 160 Laplace's equation, 208 Leslie's experiment, 241 Lines ofmotion, 235 Lissajous' figures, 49 Loaded string, normal modes ofa,36, 37,82 Local periodic force, effectof,ina gaseous medium, 239 Longitudinal vibrations, ofbars, 116 ofcolumns ofair,173,270 Loops, onavibrating string, 70 inapipe, 174 Membrane, transverse vibrations ofa, 141 normal modes ofarectangular, 144 ofacircular, 146 Mersenne's laws, 70 'Modulus ofdecay' defined, 25Modulus ofair-waves, 188 ofavibrating sphere, '239 ofaresonator, 269 ofapipe,274 Modulus, Young's, 113 Multiple resonance, 287 Multiple system, equationsofmotion ofa,41,44 normal modes ofa,44 forced vibrations ofa,47 Nodal lines ofamembrane, 144,145, 147,151 ofaplate, 155,157 Nodes, inavibrating string, 70 inabar,118,129 inapipe,174 'Normal functions,' 103,132 Normal modes ofvibration, 44 Notes, musical, 3,289 Ohm's law, 2,290,294 Organ pipe,normal modes of,174 corrected theory of,270 mode ofaction of,281 Overtones, 5 influence of,onquality, 291 onthe definition ofconsonant intervals, 293 Pendulum, 8,16 Blackburn's, 35 double, 38 Period-equations, graphical solution of,83,127,130, 260,262 Permanency oftype, condition for,in air-waves, 178 Pipe,normal modes ofa,174 modulus ofdecay ofa,274 velocity ofsound inanarrow, 196 Plane waves inanelastic medium, 120 inair,163, 177,228 Plate, transverse vibrations ofacir- cular, 155;ofasquare, 157 Pluckedstring, theory of,66,100 Poiseuille's law,198 Poisson's ratio, 113 Propagation inwater, 234 'Quality' ofmusical notes, 4 influence ofovertones on,291 Reciprocity, principle of,47,81 Bectangular -vessel, normal mades of a,258 Eeed-pipes, theory of,282 Reflection ofwaves, 64,171, 218. 232, 271 INDEX 307 [Thenumerals refertothepages'] Befraction ofsound, duetovariation oftemperature, 219 towind, 222 Besonance, 18,20,22,32,274 Besonator, 265,287 free-vibrations ofa,267 forced vibrations ofa,274 Bing, normal modes ofa,135 Scattering ofsound waves byobstacles, 244 Sensations, analysis of,2,289 Shearing strain, 110 stress, 111 Shells, vibrations ofcurved, 158 Simple-harmonic vibrations, 2,9 energy of,15 superposition of,22,48 'Simple source' ofsound, 217,228 Sine-series, 89 Sound, velocity of,inair,163,165 inwater, 165 Sound waves, plane, 163 spherical, 208,228 general, 207,215,217 offinite amplitude, 177 Sounding board, function of,68,81 Source ofsound, simple, 217,228 double, 218,230 'Speed' ofasimple vibration, 10 Sphere, waves produced byoscillating, 234 vibrations ofanelastic, 159 Spherical vessel, normal modes ofa, 259,262 Stability,coefficients of,43 Stationary propertyofnormal modes, 45 Stiffness ofpiano-wire,effect of,82, 135 Strains, 108 Stresses, 110 String excited byplucking, 66,72,100 byimpact, 73,101String excited bybowing, 75,100 String, transverse vibrations ofa,59 waves ona,61,64 normal modes ofafinite, 68 forced vibrations ofa,80 Submarine signalling, 157, 218,304 Superpositionofvibrations, 22,48 Temperament, equal, 7 Temperature,effect ofunequal, on propagationofsound, 219 Tension, effect ofpermanent onthe vibrations ofabar,134 Tones, pure,1 interference of,292 Transmission ofsound byanaperture, 248 byagrating,251 Transverse vibrations, ofstrings, 59 ofbars, 122 ofmembranes, 141 ofplates,154 Tuning fork, 133 Velocityofsound, 163,165 inanarrowpipe,196 'Velocity-potential,' 204 Violin-string, 75,100 Viscosity, 186 effect of,onair-waves, 188,189 onwaves inanarrow pipe, 193 Water, velocityofsound in,165 vibrations ofacolumn of,175 Watt's indicator diagram, 160 Waves, onastring, 61,64 inatar, 117,125 inanelastic medium, 120. 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Partial Contents: Thermodynamic sys- tems, 1stand2nd laws, potentials; Entropy, phase rule,- Reversible electric cells; Gaseous reactions; Van't Hoff reaction box, principle ofLeChatelier; Thermodynamics ofdilute solutions: osmotic, vapor pressures; boiling, freezing point; Entropy constant. 25problems. 24illustrations, x+160pp.5% x8. S361 Paperbound $1.75 FOUNDATIONS OFPOTENTIAL THEORY, 0.D.Kellogg. Based oncourses given atHarvard, suitable forboth advanced and beginning mathematicians, Proofs rigorous, much material here notgenerally available elsewhere. Partial contents: gravity, fields offorce, divergence theorem, properties ofNewtonian potentials atpointsoffree space, potentials assolutions ofLaPlace's equation, harmonic functions, electrostatics, electric images, logarithmic po- tential, etc. ix+384pp. 53/8x8. S144 Paperbound $1.98 DIALOGUES CONCERNING TWONEWSCIENCES, Galileo Galilei. Classic ofexperimental science, mechanics, engineering, asenjoyable as itisimportant. Characterized byauthor as"superior toeverything else ofmine." Offers alively exposition ofdynamics, elasticity, sound, ballistics, strength ofmaterials, scientific method. Translated byH.Grew, A.deSalvio. 126'diagrams, xxi+288pp.5%x8. S99Paperbound $1.65 THEORETICAL MECHANICS; ANINTRODUCTION TOMATHEMATICAL PHYSICS, J.S.Ames,F.D. Murnaghan. Amathematically rigorous development foradvanced students, with constant practical applications. Used inhundreds ofadvanced courses. Unusually thorough coverage ofgyroscopic baryscopic material, detailed analyses ofCorilis acceleration, applications of Lagrange's equations, motion ofdouble pendulum, Hamilton-Jacob! partial differential equa- tions, group velocity, dispersion, etc. Special relativity included. 159problems. 44figures. ix+462pp. b3/a x8. S461 Paperbound $2.00 STATICS ANDTHEDYNAMICS OFAPARTICLE, W.D.MacMillan. This isPartOne of"Theoret- icalMechanics." Forover 3decades aself-contained, extremely comprehensive advanced undergraduate text inmathematical physics, physics, astronomy, deeper foundations of engineering. Early sections require only aknowledge ofgeometry; later, aworking knowledge ofcalculus. Hundreds ofbasic problems including projectiles tomoon, harmonic motion, ballistics, transmission ofpower, stress and strain, elasticity, astronomical problems. 340 practice problems, many fully worked outexamples. 200 figures, xvii+430pp. 5% x8. S467 Paperbound ?2.00 THETHEORY OFTHEPOTENTIAL, W.D.MacMillan. This isPartTwo of"Theoretical Mechan- ics." Comprehensive, well-balanced presentation, serving both asintroduction and reference with regard tospecific problems, for physicists and mathematicians. Assumes noprior knowledge ofintegral relations, allmath isdeveloped asneeded. Includes: Attraction of Finite Bodies; Newtonian Potential Function; Vector Fields, Green and Gauss Theorems; Two-la/er Surfaces; Spherical Harmonics; etc."The great number ofparticular cases . . . should make thebook valuable togeo-physicists and others actively engaged inpractical applications ofthepotential theory," Review ofScientific Instruments, xii+469pp.5% x8. S486 Paperbound $2.25 DYNAMICS OFASYSTEM OFRIGID BODIES (Advanced Section),E.J.Routh. Revised 6th edi- tion ofaclassic reference aid. Partial contents: moving axes, relative motion, oscillations aboutequijibrium,motion. Motion ofabody under noforces, anyforces. Nature ofmotion given bylinear equations andconditions ofstability. Free, forced vibrations, constants of integration, calculus offinite differences, variations, procession and mutation, motion of themoon, motion ofstring, chain, membranes. 64figures. 498pp.5% x8. S229 Paperbound $2.35 THEDYNAMICS OFPARTICLES ANDOFRIGID, ELASTIC, ANDFLUID BODIES: BEING LECTURES ONMATHEMATICAL PHYSICS, A.G.Webster. Reissuing ofclassic fillsneed forcomprehensive work ondynamics. Covers wide range inunusually great depth, applying ordinary, partial differential equations. Partial contents: laws ofmotion, methods applicable tosystems of allsorts; oscillation, resonance, cyclic systems; dynamics ofrigid bodies; potential theory; stress and strain; gyrostatics; wave, vortex motion; kinematics ofapoint; Lagrange's equa- tions; Hamilton's principle; vectors; deformable bodies; much more not easily found to- gether inone volume. Unabridged reprinting of2nd edition. 20pages ondifferential equations, higher analysis. 203 illustrations, xi+588pp.5% x8. S522 Paperbound $2.35 CATALOGUE OF PRINCIPLES OFMECHANICS, Heinrich Hertz. Aclassic ofgreat interest inlogic ofscience. Lastwork bygreat 19th century physicist, created newsystem ofmechanics based upon space, time, mass; returns toaxiomatic analysis, understanding offormal, structural aspects ofscience, taking into account logic, observation, apriori elements. Ofgreat historical importance toPoincare', Carnap, Einstein, Milne. 20page introduction byR.S. Cohen, Wesleyan U.,analyzes implications ofHertz's thought and logic ofscience. 13page introduction byHelmholtz. xlii+274pp.5% x8. S316 Clothbound $3.50 S317 Paperbound $1.75 MATHEMATICAL FOUNDATIONS OFSTATISTICAL MECHANICS, A. I.Khinchin. Athoroughly up-to-date introduction, offering aprecise and mathematically rigorous formulation ofthe problems ofstatistical mechanics. Provides analytical tools toreplace many commonly used cumbersome concepts and devices. Partial contents: Geometry, kinematics ofphase space; ergodic problem; theory ofprobability; central limit theorem; ideal monatomic gas; foundation ofthermodynamics; dispersion, distribution ofsum functions; etc. "Excellent introduction . ..clear, concise, rigorous," Quarterly ofApplied Mathematics, viii+179pp. 53/ax8. S146 Clothbound $2.95 S147 Paperbound $1.35 MECHANICS OFTHEGYROSCOPE, THEDYNAMICS OFROTATION, R.F.Deimel, Prof, ofMe- chanical Engineering, Stevens Inst. ofTech. Elementary, general treatment ofdynamics of rotation, with special applicationofgyroscopic phenomena. Noknowledge ofvectors needed. Velocity ofamoving curve, acceleration toapoint, general equations ofmotion, gyroscopic horizon, free gyro, motion ofdiscs, thedamped gyro, 103 similar topics. Exer- cises. 75figures. 208pp.5% x8. S66Paperbound $1.65 MECHANICS VIATHECALCULUS, P.W.Norris, W.S.Legge. Wide coverage, from linear motion tovector analysis; equations determining motion, linear methods, compounding ofsimple harmonic motions, Newton's laws ofmotion, Hooke's law, thesimple pendulum, motion of aparticle in1plane, centers ofgravity, virtual work, friction, kinetic energy ofrotating bodies, equilibrium ofstrings, hydrostatics, sheering stresses, elasticity, etc.Many worked- outexamples. 550problems. 3rdrevised edition, xii+367pp. S207 Clothbound $3.95 ATREATISE ONTHEMATHEMATICAL THEORY OFELASTICITY, A.E.H.Love. Anindispensable reference work forengineers, mathematicians, physicists, themost complete, authoritative treatment ofclassical elasticity inonevolume. Proceeds from elementary notions ofexten- sion totypes ofstrain, cubical dilatation, general theory ofstrains. Covers relation between mathematical theory ofelasticity and technical mechanics; equilibrium ofisotropic elastic solids and aelotropic solid bodies; nature offorce transmission, Volterra's theory of dislocations; theory ofelastic spheres inrelation totidal, rotational, gravitational effects onearth; general theory ofbending; deformation ofcurved plates; buckling effects; much more. "The standard treatise onelasticity," American Math. Monthly. 4th revised edition. 76figures, xviii+643pp. 6Vsx9V4. S174 Paperbound $2.95 NUCLEAR PHYSICS, QUANTUM THEORY, RELATIVITY MESON PHYSICS, R.E.Marshak. Presents basic theory, and results ofexperiments withem- phasis ontheoretical significance. Phenomena involving mesons as virtual transitions avoided, eliminating some ofleast satisfactory predictions ofmeson theory. Includes pro- duction study ofTTmesons atnonrelativistic nucleon energies contracts between TTandu mesons, phenomena associated with nuclear interaction ofifmesons, etc. Presents early evidence fornew classes ofparticles, indicates theoretical difficulties created bydiscovery ofheavy mesons andhyperons. viii+378pp.5% x8. S500 Paperbound $1.95 THEFUNDAMENTAL PRINCIPLES OFQUANTUM MECHANICS, WITH ELEMENTARY APPLICATIONS, E.C.Kemble. Inductive presentation, forgraduate student, specialists inother branches of physics. Apparatus necessary beyond differential equations andadvanced calculus developed asneeded. Though general exposition ofprinciples, hundreds ofindividual problems fully treated. "Excellent book... ofgreat value toevery student . . .rigorous anddetailed mathematical discussion ...hassucceeded inkeeping hispresentation clear andunder- standable," Dr.Linus Pauling, J.ofAmerican Chemical Society. Appendices: calculus of variations, math, notes, etc. 611pp. 5s/a x8%. T472 Paperbound $2.95 WAVE PROPAGATION INPERIODIC STRUCTURES, L.Brillouin. General method, application to different problems: pure physics scattering ofX-rays incrystals, thermal vibration in crystal lattices, electronic motion inmetals; problems inelectrical engineering. Partial contents: elastic waves along 1-dimensional lattices ofpoint masses. Propagation ofwaves along 1-dimensional lattices. Energy flow. 2,3dirnensionaF lattices. Mathieu's equation. Matrices andpropagation ofwaves along anelectric line. Continuous electric lines. 131 illustrations, xii+253pp.5% x8. S34Paperbound $1.85 HEAT HLorenteW HTSA $PLCTION T0THEPHENOMENA OFLIGHT ANDRADIANT historical cover/LM *delivered atColumbia Univ., byNobel laureate. Unabridged, form ODtical nhennrn!8 !9^l^eeelectrons, motion, absorption ofheat, Zeemlneffect,tE 9PfS"wn T/'ngbdles'etc '109pages notes exP|ainmoreadvanced sec^lions. 9figures.352pp. 53/8x8.S173 Paperbound $1.85 MDPrwhirh oM- hHELECTRODYNAMICS, edited by J.Schwinger. Facsimiles of ofllrwrthpn ^h ?h Kd ,quanjt ,4melectrodynamics, beginning topresent position aspart nirar rLnny c"^rokPublication inanylanguage ofcollected papers ofBethe, Bloch, Tnmnnn w/V J"11 ;^/nman 'Heisenberg, Kusch, Lamb, Oppenheimer, Pauli, Schwinger iTn?$in u-sl iop?' ,Wlgner 'etc -34papers: 29 inEnS|ish >1^^ench, 3inGerman1inItalian. Historical commentary byeditor, xvii+423pp. 6Vs x9V*. S444 Paperbound $2.45 FOUNDATIONS OPNUCLEAR PHYSICS, edited byR.T.Beyer. 13ofthemost important papersonnuclear physics reproduced infacsimile intheoriginal languages; thepapers most oftencited infootnotes, bibliographies. Anderson, Curie, Joliot, Chadwick, Fermi, Lawrence, Cock-ron Harm, Yukawa. Unparalleled bibliography: 122 double columned pages, over 4,000articles, books, classified. 57figures. 288pp. 6Vs x9V4. S19Paperbound $1.75 THETHEORY OFGROUPS ANDQUANTUM MECHANICS, H.Weyl. Schroedinger's wave equation,deBrogue swaves ofaparticle, Jordon-Hoelder theorem, Lie's continuous groups oftrans- formations,Pauli exclusionprinciple, quantization ofMawell-Dirac field equations, etc. Unitary geometry, quantum theory, groups, application ofgroups toquantum mechanics,symmetry permutation group, algebra ofsymmetric transformations, etc.2nd revised edi- tion. xxii+422pp. 53/8 x8. S268 Clothbound $4.50 S269 Paperbound $1.95 PHYSICAL PRINCIPLES OFTHEQUANTUM THEORY, Werner Heisenberg. Nobel laureate dis-cusses quantum theory; hisown work, Compton, Schroedinger, Wilson, Einstein, manyothers. Forphysicists, chemists, notspecialists inquantum theory. Only elementary formulae considered intext; mathematical appendix for specialists. Profound without sacrificing clarity. Translated byC.Eckart,F.Hoyt. 18figures. 192pp.5% x8. S113 Paperbound $1.25 INVESTIGATIONS ONTHETHEORY OFTHEBROWNIAN MOVEMENT, Albert Einstein. Reprintsfrom rare European journals, translated into English. 5basic papers, including Elementary Theory oftheBrownian Movement, written atrequest ofLorentz toprovide asimple explanation. Translated byA.D.Cowper. Annotated, edited byR.Fiirth. 33pp. ofnotes elucidate, give history ofprevious investigations. 62footnotes. 124p'p. 53/ax8. S304 Paperbound $1.25 THEPRINCIPLE OFRELATIVITY, E.Einstein, H.Lorentz, M.Minkowski, H.Weyl. The11basic papers thatfounded thegeneral andspecial theories ofrelativity, translated into English. 2papers byLorentz ontheMichelson experiment, electromagnetic phenomena. Minkowski's "Space andTime," andWeyl's "Gravitation and Electricity." 7epoch-making papers byEin- stein-. "Electromagnetics ofMoving Bodies," "Influence ofGravitation inPropagation of Light," "Cosmological Considerations," "General Theory," 3others. 7diagrams. Special notes byA.Sommerfeld. 224pp.5% x8. S93Paperbound $1.75 STATISTICS ELEMENTARY STATISTICS, WITH APPLICATIONS INMEDICINE ANDTHEBIOLOGICAL SCIENCES, F.E.Croxton. Based primarily onbiological sciences, butcanbeused byanyone desiring introduction tostatistics. Assumes noprior acquaintance, requires onlymodest knowledge ofmath. Allbasic formulas carefully explained, illustrated;allnecessary reference tables included. From basic terms and concepts, proceedstofrequency distribution, linear, non- linear, multiple correlation, etc. Contains concrete examples from medicine, biology. 101 charts. 57tables. 14appendices.Iv+376pp. 53/8x8. S506 Paperbound $1.95 ANALYSIS^AND DESIGN OFEXPERIMENTS, H.B.Mann. Offers method forgrasping analysis of variance, variance design quickly. Partial contents: Chi-square distribution, analysis of variance distribution, matrices, quadratic forms, likelihood ration tests, test oflinear hypotheses, power ofanalysis, Galois fields, non-orthogonal data, interblock estimates, etc. 15pp. pfuseful tables, x+195pp. 5x7%. S180 Paperbound $1.45 FREQUENCY CURVES AND CORRELATION, W. P.Elderton. 4th revised edition ofstandard work onclassical statistics. Practical, one offewbooks constantly referred toforclear presentation ofbasic material. Partial contents: Frequency Distributions; Pearsons Fre- quency Curves- Theoretical Distributions; Standard Errors; Correlation Ratio Contingency; Corrections forMoments, Beta,Gamma Functions; etc.Key toterms, symbols 25examples 40tables. 16figures,xi+272pp. 5Vi x8Vi. Clothbound $1.49 5 CATALOGUE OF HYDRODYNAMICS, ETC. HYDRODYNAMICS, Horace Lamb. Standard reference work ondynamics ofliquids andgases. Fundamental theorems, equations, methods, solutions, background for classical hydrody- namics. Chapters: Equations ofMotion, Integration ofEquationsinSpecial Gases, Vortex Motion, Tidal Waves, Rotating Masses ofLiquids, etc. Excellently planned, arranged, Clear, lucid presentation. 6thenlarged, revised edition. Over 900 footnotes, mostly bibliograph- ical. 119 figures, xv+738pp. 6Vs x9V4. S256 Paperbound $2.95 HYDRODYNAMICS, ASTUDY OFLOGIC, FACT, ANDSIMILITUDE, Garrett Birkhoff. Astimulating application ofpure mathematics toanapplied problem. Emphasis isoncorrelation of theory and deduction with experiment. Examines recently discovered paradoxes, theory of modelling anddimensional analysis, paradox and error inflows and free boundary theory. Classical theory ofvirtual mass derived from homogenous spaces; group theory applied tofluid mechanics. 20figures, 3plates, xiii+186pp.5% x8. S22Paperbound $1.85 HYDRODYNAMICS, H.Dryden, F.Murhaghan, H.Bateman. Published byNational Research Council, 1932. Complete coverage ofclassical hydrodynamics, encyclopedic inquality. Partial contents-, physics offluids, motion, turbulent flow, compressible fluids, motion in 1,2,3dimensions; laminar motion, resistance ofmotion through viscous fluid, eddy viscosity, discharge ofgases, flow past obstacles, etc. Over 2900-item bibliography. 23 figures. 634pp.5% x8. S303 Paperbound $2.75 ACOUSTICS ANDOPTICS PRINCIPLES OFPHYSICAL OPTICS, Ernst Mach. Classical examination ofpropagation oflight, color, polarization, etc. Historical, philosophical treatment unequalled forbreadth and readability. Contents: Rectilinear propagation, reflection, refraction, dioptrics, composition oflight, periodicity, theory ofinterference, polarization, mathematical representation of properties, etc,279 illustrations. 10portraits. 324pp.5%x8. S170 Paperbound ?1.7'5 THETHEORY OFSOUND, Lord Rayleigh. Written byNobel laureate, classical methods here willcover most vibrating systems likely tobeencountered inpractice. Complete coverage ofexperimental, mathematical aspects. Partial contents: Harmonic motions, lateral vibra- tions ofbars, curved plates orshells, applications ofLaplace's functions toacoustical problems, fluid friction, etc. First low-priced edition ofthis great reference-study work. Historical introduction byR.B.Lindsay. 1040pp. 97figures. 53/s x8. S292, S293, Twovolume set,paperbound $4.00 THEORY OFVIBRATIONS, N.W.McLachlan. Based onexceptionally successful graduate course, Brown University. Discusses linear systems having 1degree offreedom, forced vibrations ofsimple linear systems, vibration offlexible strings, transverse vibrations of barsandtubes, ofcircular plate, sound waves offinite amplitude, etc.99diagrams. 160pp. 53/8 x8. S190 Paperbound $1.35 APPLIED OPTICS ANDOPTICAL DESIGN, A.E.Conrady. Thorough systematic presentation of physical and mathematical aspects, limitedmpstlyto"real optics." Stresses practical problem ofmaximum aberration permissible without affecting performance. Ordinary ray tracing methods; complete theory raytracing methods, primary aberrations; enough higher aberration todesign telescopes, lowpowered microscopes, photographic equipment. Covers fundamental equations, extra-axial image points, transverse chromatic aberration, angular magnification, similar topics. Tables offunctions ofN.Over 150 diagrams, x+518pp.5%x85/ a. S366 Paperbound $2.98 RAYLEIGH'S PRINCIPLE AND ITSAPPLICATIONS TOENGINEERING, G.Temple, W.Bickley. Rayleigh's principle developed toprovide upper, lower estimates oftrue value offunda- mental period ofvibrating system, orcondition ofstability ofelastic system. Examples, rigorous proofs. Partial contents: Energy method ofdiscussing vibrations, stability. Per- turbation theory, whirling ofuniform shafts. Proof, accuracy, successive approximations, applications ofRayleigh's theory. Numerical, graphical methods. Ritz's method. 22figures, ix+156pp. 53/a x8. S307 Paperbound $1.50 OPTICKS, SirIsaac Newton. Initsdiscussion oflight, reflection, color, refraction, theories ofwave andcorpuscular theories oflight, thiswork ispacked with scores ofinsights and discoveries. Initsprecise and practical discussions ofconstruction ofoptical apparatus, contemporary understanding ofphenomena, it istruly fascinating tomodern scientists. Foreword byAlbert Einstein. Preface byI.B.Cohen, Harvard. 7pages ofportraits, facsimile pages, letters, etc. cxvi+414pp.5% x8. S205 Paperbound $2.00 DOVER SCIENCE BOOKS ONTHESENSATIONS OFTONE, Hermann Helmholtz. Using acoustical physics, physiology, experiment, history ofmusic, covers entire gamut ofmusical tone: relation ofmusic science toacoustics, phys :al vs. physiological acoustics, vibration, resonance, tonality, progression ofparts, etc. .-3appendixes onvarious aspects ofsound, physics, acoustics, music, etc.Translated by(J.Ellis.New introduction byH.Margenau, Yale. 68figures. 43 musical passages analyzed. Over 100 tables, xix+576pp. 6Vs x91/4. S114 Clothbound $4.95 ELECTROMAGNETICS, ENGINEERING, TECHNOLOGY INTRODUCTION TORELAXATION METHODS, F.S.Shaw. Describes almost allmanipulative re- sources ofvalue insolution ofdifferential equations. Treatment ismathematical rather than physical. Extends general computational process toInclude almost allbranches of applied math and physics. Approximate numerical methods aredemonstrated, although high accuracy isobtainable without undue expenditure oftime. 48pp. oftables forcomputing irregular star first andsecond derivatives, irregular star coefficients forsecond order equations, forfourth order equations. "Useful. . . .expositionisclear, simple ... no previous acquaintance with numerical methods isassumed," Science Progress. 253 dia- grams. 72tables. 400pp. 5Va x8. S244 Paperbound $2.45 THEELECTROMAGNETIC FIELD, M.Mason, W..Weaver. Used constantly bygraduate engineers. Vector methods exclusively; detailed treatment ofelectrostatics, expansion methods, with tables converting any quantity into absolute electromagnetic, absolute electrostatic, prac- tical units. Discrete charges, ponderable bodies. Maxwell field equations, etc. 416pp. 53/8 x8. S185 Paperbound $2.00 ELASTICITY, PLASTICITY ANDSTRUCTURE OFMATTER, R.Houwink. Standard treatise on Theological aspects ofdifferent technically important solids: crystals, resins, textiles, rubber, clay, etc. Investigates general laws fordeformations; determines divergences. Covers gen- eral physical andmathematical aspects ofplasticity, elasticity, viscos'ity. Detailed examina- tion ofdeformations, internal structure ofmatter inrelation toelastic, plastic behaviour, formation ofsolid matter from afluid, etc. Treats glass, asphalt, balata, proteins, baker's dough, others. 2nd revised, enlarged edition. Extensive revised bibliography inover 500 footnotes. 214 figures, xvii+368pp. 6x9V4. S385 Paperbound $2.45 DESIGN ANDUSEOFINSTRUMENTS ANDACCURATE MECHANISM, T.N.Whitehead. Forthe instrument designer, engineer; how tocombine necessary mathematical Abstractions with independent observations ofactual facts. Partial contents: instruments and their parts, theory oferrors, systematic errors, probability, short period errors, erratic errors, design precision, kinematic, sernikinematic design, stiffness, planning ofaninstrument, human factor, etc.85photos, diagrams, xii+288pp. 5Va x8. S270 Paperbound $1.95 APPLIED HYDRO- ANDAEROMECHANICS, L.Prantltl,0.G.Tietjens. Presents, formost part, methods valuable toengineers. Flow inpipes, boundary layers, airfoil theory, entry condi- tions, turbulent flow, boundary layer, determining drag from pressure and velocity, etc. "Will bewelcomed by allstudents ofaerodynamics," Nature. Unabridged, unaltered. An Engineering Society Monograph, 1934. Index. 226 figures. 28photographic plates illustrating flow patterns, xvi -I-311pp. 53/a x8. S375 Paperbound $1.85 FUNDAMENTALS OFHYDRO- ANDAEROMECHANICS, L.Prandtl, 0.G.Tietjens. Standard work, based onPrandtl's lectures atGoettingen. Wherever possible hydrodynamics theory is referred topractical considerations inhydraulics, unifying theory and experience. Presenta- tion extremely clear. Though primarily physical, proofs arerigorous andusevector analysis toagreat extent. AnEngineering Society Monograph, 1934. "Still recommended asan excellent introduction tothis area," Physikalische Blatter. 186 figures, xvi+270pp. 53/8x8. S374 Paperbound $1.85 GASEOUS CONDUCTORS: THEORY ANDENGINEERING APPLICATIONS,J.D.Cobine. Indispensable text, reference, togaseous conduction phenomena, with engineering viewpoint prevailing throughout. Studies kinetic theory ofgases, ionization, emission phenomena; gasbreakdown, spark characteristics, glow, discharges; engineering applications incircuit interrupters, recti- fiers, etc. Detailed treatment ofhigh pressure arcs (Suits); lowpressure arcs (Langmuir, Tonks). Much more. "Well organized, clear, straightforward," Tonks, Review ofScientific Instruments. 83practice problems. Over 600 figures. 58tables, xx+606pp. 53/3 x8. S442 Paperbound $2.75 PHOTOELASTICITY: PRINCIPLES ANDMETHODS, H.T.Jessop, f.C.Harris. Forengineer, spe- cific problems ofstress analysis. Latest time-saving methods ofchecking calculations in 2-dimensional design problems, new techniques forstresses in3dimensions, lucid descrip- tion ofoptical systems used inpractical photoelectricity. Useful suggestions, hints based onon-the-job experience included. Partial contents: strain, stress-strain relations, circular disc under thrust along diameter, rectangular block with square hold under vertical thrust, simply supported rectangular beam under central concentrated load, etc. Theory held to minimum, noadvanced mathematical training needed. 164 illustrations, viii+184pp. 6Va x91/4. S137 Clothbound $3.75 CATALOGUE OF HYDRODYNAMICS, ETC. HYDRODYNAMICS, Horace Lamb. Standard reference work ondynamics ofliquids and gases. Fundamental theorems, equations, methods, solutions, background for classical hydrody- namics. Chapters: Equations ofMotion, Integration ofEquationsinSpecial Gases, Vortex Motion, Tidal Waves, Rotating Masses ofLiquids, etc. Excellently planned, arranged, Clear, lucid presentation. 6thenlarged, revised edition. Over 900 footnotes, mostly bibliograph- ical.119 figures, xv+738pp. 6Vex9V4. S256 Paperbound $2.95 HYDRODYNAMICS, ASTUDY OFLOGIC, FACT, ANDSIMILITUDE, Garrett Birkhoff. Astimulating application ofpure mathematics toanapplied problem. Emphasis isoncorrelation of theory anddeduction with experiment. Examines recently discovered paradoxes, theory of modelling anddimensional analysis, paradox and error inflows and free boundary theory. Classical theory ofvirtual mass derived from homogenous spaces; group theory applied tofluid mechanics. 20figures, 3plates, xiii+186pp. 53/8x8. S22Paperbound $1.85 HYDRODYNAMICS, H.Dryden, F.Murhaglian, H.Bateman. Published byNational Research Council, 1932. Complete coverage ofclassical hydrodynamics, encyclopedicinquality. Partial contents: physics offluids, motion, turbulent flow, compressible fluids, motion in 1,2,3dimensions; laminar motion, resistance ofmotion through viscous fluid, eddy viscosity, discharge ofgases, flow past obstacles, etc. Over 2900-item bibliography. 23 figures. 634pp.5% x8. S303 Paperbound $2.75 ACOUSTICS ANDOPTICS PRINCIPLES OFPHYSICAL OPTICS, Ernst Mach. Classical examination ofpropagation oflight, color, polarization, etc. Historical, philosophical treatment unequalled forbreadth and readability. Contents: Rectilinear propagation, reflection, refraction, dioptrics, composition oflight, periodicity, theory ofinterference, polarization, mathematical representation of properties, etc..279 illustrations. 10portraits. 324pp.5% x8. S170 Paperbound $1.75 THETHEORY OFSOUND, Lord Rayleigh. Written byNobel laureate, classical methods here willcover most vibrating systems likely tobeencountered inpractice. Complete coverage ofexperimental, mathematical aspects. Partial contents: Harmonic motions, lateral vibra- tions ofbars, curved plates orshells, applications ofLaplace's functions toacoustical problems, fluid friction, etc. First low-priced edition ofthis great reference-study work. Historical introduction byR.B.Lindsay. 1040pp. 97figures. 5% x8. S292, S293, Twovolume set,paperbound $4.00 THEORY OFVIBRATIONS, N.W.McLachlan. Based onexceptionally successful graduate course, Brown University. Discusses linear systems having 1degree offreedom, forced vibrations ofsimple linear systems, vibration offlexible strings, transverse vibrations of barsandtubes, ofcircular plate, sound waves offinite amplitude, etc.99diagrams. 160pp. 53/8x8. S190 Paperbound $1.35 APPLIED OPTICS ANDOPTICAL DESIGN, A.E.Conrady. Thorough systematic presentation of physical and mathematical aspects, limited mostly to"real optics." Stresses practical problem ofmaximum aberration permissible without affecting performance. Ordinary ray tracing methods; complete theory raytracing methods, primary aberrations; enough higher aberration todesign telescopes, lowpowered microscopes, photographic equipment. Covers fundamental equations, extra-axial image points, transverse chromatic aberration, angular magnification, similar topics. Tables offunctions ofN.Over 150diagrams, x+518pp. 53/sx85/8. S366 Paperbound $2.98 RAYLEIGH'S PRINCIPLE AND ITSAPPLICATIONS TOENGINEERING, G.Temple, W.Bickley. Rayleigh's principle developed toprovide upper, lower estimates oftrue value offunda- mental period ofvibrating system, orcondition ofstability ofelastic system. Examples, rigorous proofs. Partial contents; Energy method ofdiscussing vibrations, stability. Per- turbation theory, whirling ofuniform shafts. Proof, accuracy, successive approximations, applications ofRayleigh's theory. Numerical, graphical methods. Ritz's method. 22figures. ix+156pp.5% x8. S307 Paperbound $1.50 OPTICKS, SirIsaac Newton. Initsdiscussion oflight, reflection, color, refraction, theories ofwave andcorpuscular theories oflight, thiswork ispacked with scores ofinsights and discoveries. Initsprecise and practical discussions ofconstruction ofoptical apparatus, contemporary understanding ofphenomena,itistruly fascinating tomodern scientists. Foreword byAlbert Einstein. Preface byI.B.Cohen, Harvard. 7pages ofportraits, facsimile pages, letters, etc. cxvi+414pp.5% x8. S205 Paperbound $2.00 DOVER SCIENCE BOOKS ONTHESENSATIONS OFTONE, Hermann Helmholtz. Using acoustical physics, physiology,experiment, mstory ofmusic, covers entire gamut ofmusical tone: relation ofmusic science toacoustics, phys :al vs.physiological acoustics, vibration, resonance, tonality, progressionorparts, etc. :-3appendixes onvarious aspects ofsound, physics, acoustics, music,etc.Translated by f.j.Ellis.New introduction byH.Margenau, Yale. 68figures. 43 musical passages analyzed. Over 100 tables, xix+576pp. 6Vs x9V*. S114 Clothbound $4.95 ELECTROMAGNETICS, ENGINEERING, TECHNOLOGY INTRODUCTION TORELAXATION METHODS, F.S.Shaw. Describes almost allmanipulative re- sources ofvalue insolution ofdifferential equations. Treatment ismathematical rather than physical. Extends general computational process toinclude almost allbranches of applied math and physics. Approximate numerical methods aredemonstrated, although highaccuracy isobtainable without undue expenditure oftime. 48pp. oftables forcomputing irregular star first andsecond derivatives, irregular star coefficients forsecond order equations, forfourth order equations. "Useful. . . .exposition isclear, simple... no previous acquaintance with numerical methods isassumed," Science Progress. 253 dia- grams.72tables. 400pp. 53/ax8. S244 Paperbound $2.45 THEELECTROMAGNETIC FIELD, M.Mason, W..Weaver. Used constantly bygraduate engineers. Vector methodsexclusively; detailed treatment ofelectrostatics, expansion methods, with tables converting any quantity into absolute electromagnetic, absolute electrostatic, prac- tical units. Discretecharges, ponderable bodies. Maxwell field equations, etc. 416pp.5% x8. S185 Paperbound $2.00 ELASTICITY, PLASTICITY ANDSTRUCTURE OFMATTER, R.Houwink. Standard treatise on Theological aspects ofdifferent technically important solids: crystals, resins, textiles, rubber, ciay, etc. Investigates general laws fordeformations; determines divergences. Covers gen- eral physical andmathematical aspects ofplasticity, elasticity, viscosity. Detailed examina- tion ofdeformations, internal structure ofmatter inrelation toelastic, plastic behaviour, formation ofsolid matter from afluid, etc. Treats glass, asphalt, balata, proteins, baker's dough, others. 2ndrevised, enlarged edition. Extensive revised bibliography inover 500 footnotes. 214 figures, xvii+368pp.6x9V*. S385 Paperbound $2.45 DESIGN ANDUSEOFINSTRUMENTS ANDACCURATE MECHANISM, T.N.Whitehead. Forthe instrument designer, engineer; how tocombine necessary mathematical abstractions with independent observations ofactual facts. Partial contents: instruments and their parts, theory oferrors, systematic errors, probability, short period errors, erratic errors, design precision, kinematic, semikinematic design, stiffness, planning ofaninstrument, human factor, etc. 85photos, diagrams, xii+288pp.5% x8. S270 Paperbound $1.95 APPLIED HYDRO- ANDAEROMECHANICS,L.Prandtl, 0.G.Tietjens. Presents, formost part, methods valuable toengineers. Flow inpipes, boundary layers, airfoil theory, entry condi- tions, turbulent flow, boundary layer, determining drag from pressure and velocity, etc. "Will bewelcomed by allstudents ofaerodynamics'," Nature. Unabridged, unaltered. An Engineering Society Monograph, 1934. Index. 226 figures. 28photographic plates illustrating flow patterns, xvi+311pp.5% x8. S375 Paperbound $1.85 FUNDAMENTALS OFHYDRO- ANDAEROMECHANICS, L.Prandtl, 0.G.Tietjens. Standard work, based onPrandtl's lectures atGoettingen. Wherever possible hydrodynamics theoryis referred topractical considerations inhydraulics, unifying theory and experience. Presenta- tion extremely Clear. ThOUgh primar;l> '"hwcioal nrnnfc ?!<, rinnrr.no onH ..co ,,o,-tr,r on^l-.cir toagreat extent. AnEngineeri excellent introduction tothis a 53/8 x8. GASEOUS CONDUCTORS: THEORY ANDENGINEERINi,.. . text, reference, togaseous conduction phenomena, with engineering vi throughout. Studies kinetic theory ofgases, ionization, emission phenomen, spark characteristics, glow, discharges; engineering applicationsincircuit interrupters, recti- fiers, etc. Detailed treatment ofhigh pressure arcs(Suits); lowpressure arcs (Langmuir, Tonks). Much more. "Well organized, clear, straightforward," Tonks, Review ofScientific Instruments. 83practice problems. Over 600 figures. 58tables, xx+606pp. 53/s x8. S442 Paperbound $2.75 PHOTOELASTICITY: PRINCIPLES ANDMETHODS, H.T.Jessop, F.C.Harris. Forengineer, spe- cific problems ofstress analysis. 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SI15Paperbound $1.95 TREATISE ONTHERMODYNAMICS, Max Planck. Classic based onhisoriginal papers. Brilliant concepts ofNobel laureate make noassumptions regarding nature ofheat, rejects earlier approaches ofHelmholtz, Maxwell, tooffer uniform point ofview forentire field. Seminal work byfounder ofquantum theory, deducing new physical, chemical laws. Astandard text, anexcellent introduction tofield forstudents withknowledge ofelementary chemistry, physics, calculus. 3rdEnglish edition, xvi -f297pp.5% x8. S219 Paperbound $1.75 DOVER SCIENCE BOOKS KINETIC THEORY OFLIQUIDS, J.Frenkel. Regards kinetic theory ofliquids asgeneralization, extension oftheory ofsolid bodies, covers alltypes ofarrangements ofsolids; thermal displacements ofatoms; interstitial atoms, ions; orientational, rotational motion ofmole- cules; transition between states ofmatter. Mathematical theory developed close tophysical subject matter. 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Discusses thermodynamic laws from Caratheodory's axiomatic standpoint; adiabatic, polytropic laws; work ofRitter, Emden, Kelvin, etc.; Stroemgren envelopes asstarter fortheory ofgaseous stars; Gibbs statistical mechanics (quantum); degenerate stellar configuration, theory ofwhite dwarfs; elc. "Highest level ofscientific merit," Bulletin. Amer. Math. Soc. 33figures. 509pp.5%x8. S413 Paperbound $2.75 LESMETHODES NOVELLES DELAMECANIGUJE CELESTE, H.Poincare". Complete French text ofone ofPoincarfi's most important works. Revolutionized celestial mechanics: first use of integral invariants, first major application oflinear differential equations, study ofperiodic orbits, lunar motion and Jupiter's satellites, three body problem, andmany other important topics. "Started anew era... soextremely modern that even today fewhave mastered hisweapons," E.T.Bell. 3volumes. Total 1282pp. &Ve x9V4. Vol. 1S401 Paperbound $2.75 Vol. 2S402 Paperbound $2.75 Vol. 3S403 Paperbound $2.75 The set$7.50 THEREALM OFTHENEBULAE, E.Hubble. One ofthegreat astronomers ofourtime presents hisconcept of"island universes," anddescribes itseffect onastronomy. Covers velocity- distance relation; classification, nature, distances, general field ofnebulae; cosmological theories; nebulae intheneighborhood oftheMilky way ;etc.39photos, including velocity- distance relations shown byspectrum comparison. "One ofthemost progressive lines ofastronomical research," TheTimes, London. New Introduction byA.Sandage. 55illustra- tions, xxiv+201pp. 53/8 x8. S455 Paperbound $1.50 HOW TOMAKE ATELESCOPE, Jean Texereau. Design, build anf/6 orf/8Newtonian type reflecting telescope, with altazimuth Couder mounting, suitable forplanetary, lunar, and stellar observation. Covers every operation step-by-step, every piece ofequipment. Dis- cusses basic principles ofgeometric and physical optics (unnecessary toconstruction), comparative merits ofreflectors, refractors. Athorough discussion ofeyepieces, finders, grinding, installation, testing, etc.241 figures, 38photos, show almost every operation and tool. Potential errors areanticipated. Foreword byA.Couder. Sources ofsupply,xiii f191pp. 6V4 x10. T464 Clothbound $3.50 BIOLOGICAL SCIENCES THEBIOLOGY OFTHEAMPHIBIA, G.K.Noble, Late Curator ofHerpetology atAm.Mus. of Nat. Hist. Probably most used text onamphibia, most comprehensive, clear, detailed. 19 chapters, 85page supplement: development; heredity; life history; speciation; adaptation; sex, integument, respiratory, circulatory, digestive, muscular, nervous systems; instinct, intelligence, habits, economic value classification, environment relationships, etc. "Nothing comparable toit"C.H.Pope, curator ofAmphibia, Chicago Mus. ofNat. Hist. 1047 item bibliography. 174 illustrations. 600pp. 53/8x8. S206 Paperbound $2.98 THEORIGIN OFLIFE, A. I.Oparln. Aclassic ofbiology. This isthe firstmodern statement oftheory ofgradual evolution oflifefrom nitrocarbon compounds. Abrand-new evaluation ofOparin's theory inlight oflater research, byDr. S.Margulis, University ofNebraska, xxv+270pp. 53/8x8. S213 Paperbound $1.75 CATALOGUE OF THEBIOLOGY OFTHELABORATORY MOUSE, edited byG.D.Snell. Prepared in1941 bystaff ofRoscoe B.Jackson Memorial Laboratory,still the standard treatise onthemouse, assembling enormous amount ofmaterial forwhich otherwise youspend hours ofresearch. Embryology, reproduction, histology, spontaneous neoplasms, gene andchromosomes muta- tions, genetics ofspontaneous tumor formations, oftumor transplantation, endocrine secre- tionandtumor formation, milk influence andtumor formation, inbred, hybrid animals, parasites, infectious diseases, care and recording. "Awealth ofinformation ofvital con- cern. . . .recommended toallwho could useabook onsuch asubject," Nature. Classified bibliography of1122 items. 172 figures, including 128 photos, ix+497pp. 6Va x9V4. S248 Clothbound $6.00 THETRAVELS OFWILLIAM BARTRAM, edited byMark Van Doran. Famous source-book of American anthropology, natural history, geography,isrecord kept byBartram in1770's on travels through wilderness ofFlorida, Georgia, Carolinas. Containing accurate, beautiful descriptions ofIndians, settlers, fauna, flora, itisone offinest pieces ofAmericana ever written. 13original illustrations. 448pp. 53/e x8. T13Paperbound $2.00 BEHAVIOUR AND SOCIAL LIFE OFTHEHONEYBEE, Ronald Ribbands. Outstanding scientific study; acompendium ofpractically everything known ofsocial life ofhoneybee. Stresses behaviour ofindividual bees infield, hive. Extends von Frisch's experiments oncommuni- cation among bees. Covers perception oftemperature, gravity, distance, vibration,- sound production; glands; structural differences; wax production; temperature regulation; recogni- tion, communication; drifting, mating behaviour, other highly interesting topics. "This valuable work issure ofacordial reception bylaymen, beekeepers and scientists," Prof. Karlvon Frisch, Brit. J.ofAnimal Behaviour. Bibliography of690 references. 127diagrams, graphs, sections ofbeeanatomy, finephotographs. 352pp. S410 Clothbound $4.50 ELEMENTS OFMATHEMATICAL BIOLOGY, A. J.Lotka. Pioneer classic, 1stmajor attempt to apply modern mathematical techniques onlarge scale tophenomena ofbiology, biochem- istry, psychology, ecology, similar life sciences. Partial contents: Statistical meaning of irreversibility; Evolution asredistribution; Equations ofkinetics ofevolving systems; Chem- ical, inter-species equilibrium; parameters ofstate; Energy transformers ofnature, etc. Canberead with profit byeven those having noadvanced math; unsurpassed asstudy- reference. Formerly titled "Elements ofPhysical Biology." 72figures, xxx+460pp.5% x8. S346 Paperbound $2.45 TREES OFTHEEASTERN ANDCENTRAL UNITED STATES ANDCANADA, W.M.Harlow. Serious middle-level text covering more than 140 native trees, important escapes, with informa- tion ongeneral appearance, growth habit, leaf forms, flowers, fruit, bark, commercial use, distribution, habitat, woodlore, etc. Keys within text enable you tolocate various species easily, toknow which have edible fruit, much more useful, interesting information. "Well illustrated tomake identification very easy," Standard Cat. forPublic Libraries. Over 600 photographs, figures, xiii+288pp. 55/a x6V2. T395 Paperbound $1.35 FRUIT KEYANDTWIG KEYTOTREES ANDSHRUBS (Fruit keytoNortheastern Trees, Twig key toDeciduous Woody Plants ofEastern North America), W.M.Harlow. Only guides with photo- graphs ofevery twig, fruit described. Especially valuable tonovice. Fruit key(both deciduous trees, evergreens) has introduction onseeding, organsinvojyed,types, habits. Twig key introduction treats growth, morphology. Inkeys proper, identification isalmost automatic. Exceptional work, widely used inuniversity courses, especially useful for identification in winter, orfrom fruit orseed only. Over350 photos, upto3times natural size. Index of common, scientific names, ineach key. xvii+125pp.5% x8%. T511 Paperbound $1.25 INSECT LIFEANDINSECT NATURAL HISTORY, S.W.Frost. Unusual foremphasizing habits, social life, ecological relations ofinsects rather than more academic aspects ofclassification, morphology. Prof. Frost's enthusiasm andknowledge areeverywhere evident ashediscusses insect associations, specialized habits like leaf-rolling, leaf mining, case-making, the gall insects, boring insects, etc.Examines matters notusually covered ingeneral works: insects ashuman food; insect music, musicians; insect response toradio waves; use ofinsects in art, literature. "Distinctly different, possesses anindividualityall itsown," Journal of Forestry. Over 700 illustrations. Extensive bibliography, x+524pp. 5% x8. T519 Paperbound $2.49 AWAY OFLIFE, ANDOTHER SELECTED WRITINGS, SirWilliam Osier. Physician, humanist, Osier discusses brilliantly Thomas Browne, Gui Patin,Robert Burton, Michael Servetus, William Beaumont, Laennec. Includes such favorite writing astitle essay, "The OldHuman- ities andtheNew Science," "Books andMen," "The Student Life," 6more ofhisbest discussions ofphilosophy, literature, religion. "The sweep ofhismind and interests em- braced every phase ofhumanactivity,"G. L.Keynes, 5photographs. Introduction byG.L Keynes, M.D., F.R.C.S. xx+278pp.5% x8. T488 Paperbound $1.50 THEGENETICAL THEORY OFNATURAL SELECTION, R.A.Fisher. 2nd revised edition ofvital reviewing ofDarwin's Selection Theory interms ofparticulate inheritance, byone of greatest authorities onexperimental, theoretical genetics. Theory stated inmathematical form. Special features ofparticulate inheritance areexamined-, evolution ofdominance, main- tenance ofspecific variability, mimicry, sexual selection, etc. 5chapters onman's special circumstances asasocial animal. 16photographs, x+310pp.5% x8. S466 Paperbound $1.85 10 DOVER SCIENCE BOOKS THEAUTOBIOGRAPHY OFCHARLES DARWIN, ANDSELECTED LETTERS edited byFrancis Darwin. Darwin sown record ofearly life; historic voyage aboard "Beagle;" furore surround- Lng ,6 wanS'r I-rePiieS!J^"1111'""" 5" fhisson. Letters toHenslow, Lyell, Hooker,Huxley, Wallace, Kmgsley, etc., andthoughts onreligion, vivisection. Weseehow herevo- lutionized geology with concepts ofoceansubsidence; how hisgreat books onvariation ?-fplaria un nrna ,S"l!l? 'tcprim tIVVman'?xPrslonofetionamong primates, plant fertiliza- tion, carnivorous plants, protective coloration, etc.,came into being. 365pp. 53/8x8. T479 Paperbound $1.65 ANIMALS INMOTION, EadweardMuybridge. Largest, most comprehensive selection ofMuy- bridgesfamous action photos ofanimals, from his"Animal Locomotion." 3919 high-speed shots of34different animals, birds, in123 types ofaction; horses, mules, oxen, pigs, goats, camels, eephants, dogs, catsguanacos, sloths, lions, tigers, jaguars, raccoons, baboons, deer, elk, gnus, kangaroos, many others, walking, running, flying, leaping. Horse alone inover 40ways. Photos taken against ruled backgrounds; most actions taken from 3angles atonce: 90,60, rear. Most plates original size. Ofconsiderable interest to scientists asbiology classic, records ofactual facts ofnatural history, physiology. "Really marvelous series ofplates," Nature. "Monumental work," Waldemar Kaempffert. Edited by L.S.Brown, 74page introduction onmechanics ofmotion. 340pp. ofplates. 3919 photo- graphs. 416pp. Deluxe binding, paper. (Weight: 41/2 Ibs.) 7Va x10%. T203 Clothbound $10.00 THEHUMAN FIGURE INMOTION, Eadweard Muybridge. New edition ofgreat classic inhistory ofscience and photography, largest selection evermade from original Muybridge photos of human action: 4789 photographs, illustrating 163types ofmotion: walking, running, lifting, etc. intime-exposure sequence photos atspeeds uptol/6000th ofasecond. Men,women, children, mostly undraped, showing bone, muscle positions against ruled backgrounds, mostly taken at3angles atonce. Notonlywas this agreat work ofphotography, acclaimed bycontemporary critics aswork ofgenius, but itwas also agreat 19th century landmark inbiological research. Historical introduction byProf. Robert Taft, U.ofKansas. Plates original size, full ofdetail. Over 500 action strips. 407pp.7% x10%. Deluxe edition. 7204 Clothbound $10.00 ANINTRODUCTION TOTHESTUDY OFEXPERIMENTAL MEDICINE, Claude Bernard. 90-year old classic ofmedical science, only major work ofBernard available inEnglish, records his efforts totransform physiology into exact science. Principles ofscientific research illus- trated byspecified case histories from hiswork; roles ofchance, error, preliminary false conclusion,inleading eventually toscientific truth; use ofhypothesis. Much ofmodern applicationofmathematics tobiology rests onfoundation setdown here. "The presentation Ispolished. . .reading iseasy," Revue desquestions scientifiques. New foreword byProf. I.B.Cohen, Harvard U.xxv+266pp.5% x8. T400 Paperbound $1.50 STUDIES ONTHESTRUCTURE ANDDEVELOPMENT OFVERTEBRATES, E.S.Goodrich. Definitive study bygreatest modern comparative anatomist. Exhaustive morphological, phylogenetic expositionsofskeleton, fins, limbs, skeletal visceral arches, labial cartilages, visceral clefts, gills, vascular, respiratory, excretory, periphal nervous systems, etc., from fish to higher mammals. "Formanyaday this will certainly bethestandard textbook onVertebrate MorphologyintheEnglish language," Journal ofAnatomy. 754 illustrations. 69page bio- graphical study byC.C.Hardy. Bibliographyof1186 references. Twovolumes, total 906pp. 53^ x8. Two vol. setS449, 450Paperbound $5.00 EARTH SCIENCES THEEVOLUTION OFIGNEOUS BOOKS, N.L.Bowen. Invaluable serious introduction applies techniques ofphysics, chemistry toexplain inneous rock diversity interms ofchemical composition, fractional crystalli/atinn. Discusses liquid immiscibility insilicate magmas, crystal sorting, liquid lines ofdescent, fractional resorption ofcomplex minerals, petrogen, etc'. Ofprime;' importancetounionists, mining engineers; physicists, chemists working with liiHh temperature, pressures;. "Most important," Times, London. 263 bibliographic notes. 82fifiurcs. xviii -I-334pp. iP/n xH. S311 Paperbound $1.85 GEOGRAPHICAL ESSAYS, M.Davis. Modern |',e<)Rr:i|)"y, Rcomorphology rest onfundamental work ofthis scientist. ?(>fiimous essays present most important theories, field researches. Partial contents: Gi>ogr;ipliir;il Cycle; Plains ofMarine, Subnerial Denudation; ThePeneplain; Rivers Valleys(ifPennsylvania; Outline ofCnpe Cod; Sculpture ofMountains byGlaciers; etc'"Lone tin:leader and guide,"Economic Geography. "Part ofthevery texture ofgeoR- ranhy models ofclear thought," Geographic Review. 130 figures,vi+777pp. 53/ x8. 1'S383 Paperbound $2.95 URANIUM PROSPECTING, H. L.Barnes. Forimmediate practical use, professional geologist considers uranium cues, ceuUii'ical occurrences,field conditions, allaspects ofhighly profitable occupation. "Helpful information . . .easy-to-use, easy-to-find style,'1Geotimes X+117pp.5% xH.T309 Paperbound $1.00 11 CATALOGUE OF DEREMETALLICA, Georgius Agricola. 400year oldclassic translated, annotated byformer President Herbert Hoover. 1st scientific study ofmineralogy, mining, forover 200years after itsappearance in1556 thestandard treatise. 12books, exhaustively annotated, discuss history ofmining, selection ofsites, types ofdeposits, making pits, shafts, ventilating, pumps, crushing machinery; assaying, smelting, refining metals; also saltalum, nitre, glass making. Definitive edition, with all289 16th century woodcuts oforiginal. Biographical, historical introductions. Bibliography, survey ofancient authors. Indexes. Afascinating book foranyone interested inart, history ofscience, geology, etc.Deluxe Edition. 289 illustra- tions. 672pp. 6% x10.Library cloth. S6Clothbound ?10.00 INTERNAL CONSTITUTION OFTHEEARTH, edited byBeno Gutenberg. Prepared forNational Research Council, this isacomplete, thorough coverage ofearth origins, continent forma- tion, nature andbehaviour ofearth's core, petrology ofcrust, cooling forces incore, seismic and earthquake material, gravity, elastic constants, strain characteristics, similar topics. "One isfilled with admiration ... ahigh standard . . .there isnoreader who will notlearn something from thisbook," London, Edinburgh, Dublin, Philosophic Magazine. Largest Bibliography inprint: 1127 classified items. Table ofconstants. 43diagrams. 439pp. 6Vs x9V4. S414 Paperbouno* $2.45 THEBIRTH ANDDEVELOPMENT OFTHEGEOLOGICAL SCIENCES, F.D.Adams. Most thorough history ofearth sciences ever written. Geological thought from earliest times toend of 19th century, covering over300 early thinkers andsystems; fossils and their explanation, vulcanistsys.neptunists, figured stones and paleo_ntology, generation ofstones, dozens of similar topics. 91illustrations, including Medieval, Renaissance woodcuts, etc.632footnotes, mostly bibliographical. 511pp. 53/e x8. T5Paperbound $2.00 HYDROLOGY, edited by0.E.Meinzer, prepared fortheNational Research Council. Detailed, complete reference library onprecipitation, evaporation, snow, snow surveying, glaciers, lakes, infiltration, soil moisture, ground water, runoff, drought, physical changes produced bywater hydrology oflimestone terranes, etc. Practical inapplication, especially valuable forengjneers.24experts have created "the most up-to-date, most complete treatment of thesubject," Am.Assoc. ofPetroleum Geologists. 165 illustrations, xi+712pp. 6Va x91/4. 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What, when to'tip.5thyear ofpublication. 128pp. 3Vz x5V4. T260 Paperbound 60$ NEW RUSSIAN-ENGLISH ANDENGLISH-RUSSIAN DICTIONARY, M.A.O'Brien. Unusually com- prehensive guide toreading, speaking, writing Russian, forboth advanced, beginning stu- dents. Over 70,000 entries inneworthography, fullinformation onaccentuation, grammatical classifications. Shades ofmeaning, idiomatic uses, colloquialisms, tables ofirregular verbs forboth languages. Individual entries indicate stems, transitiveness, perfective, imper- fective aspects, conjugation, sound changes, accent, etc. Includes pronunciation instruction. Used atHarvard, Yale, Cornell, etc.738pp.5% x8. T208 Paperbound ?2.00 PHRASE ANDSENTENCE DICTIONARY OFSPOKEN RUSSIAN, English-Russian, Russian-English. Based onphrases, complete sentences, not isolated words recognized asone ofbest methods oflearning idiomatic speech. Over 11,500 entries, indexed bysingle words, over 32,000 English, Russian sentences, phrases, inimmediately useable form. Shows accent changes inconjugation, declension; irregular forms listed both alphabetically, under main form ofword. 15,000 word introduction covers Russian sounds, writing, grammar, syntax. 15page appendix ofgeographical names, money, important signs, given names, foods, special Soviet terms, etc. Originally published asU.S. Gov't Manual TM30-944. iv+573pp. 53/8x8. T496 Paperbound $2.75 PHRASE ANDSENTENCE DICTIONARY OFSPOKEN SPANISH, Spanish-English, English-Spanish. Compiled from spoken Spanish, based onphrases, complete sentences rather than isolated words notanordinary dictionary. Over 16,000 entries indexed under single words, both Castilian, Latin-American. Language inimmediately useable form. 25page introduction provides rapid survey ofsounds, grammar, syntax, full consideration ofirregular verbs. Especially apt inmodern treatment ofphrases, structure. 17page glossary gives translations ofgeographical names, money values, numbers, national holidays, important street signs, useful expressionsofhigh frequency, plus unique 7page glossary ofSpanish, Spanish- American foods. Originally published asU.S. Gov't Manual TM30-900. iv+513pp. 55/ex8%. T495 Paperbound $1.75 SAY ITCORRECTLY language record sets The best inexpensive pronunciation aids onthemarket. Spoken bynative linguists asso- ciated with major American universities, each record contains: 14minutes ofspeech 12minutes ofnormal, relatively slow speech, 2minutes of normal conversational speed. 120basic phrases, sentences, covering nearly every aspect ofeveryday life, travel introducing yourself, travel in'autos, buses, taxis, etc., walking, sightseeing, hotels, restaurants, money, shopping, etc. 32page booklet containing everything onrecord plus English translations easy-to-follow phonetic guide. Clear, high-fidelity recordings. Unique bracketing systems, selection ofbasic sentences enabling you toexpand useof SAY ITCORRECTLY records with adictionary, tofitthousands ofadditional situations. Use this record tosupplement anycourse ortext. Allsounds ineach language illustrated perfectly imitate speaker inpause which follows each foreign phrase inslow section, andbeamazed atincreased ease, accuracy ofpronounciation. Available, onelanguage per record for French Spanish German Italian Dutch Modern Greek Japanese Russian Portuguese Polish Swedish Hebrew English (forGerman-speaking people) English (forSpanish-speaking people) 7"(331/3rpm) record, album, booklet. $1.00 each. SPEAK MYLANGUAGE: SPANISH FORYOUNG BEGINNERS, M.Ahlman, Z.Gilbert. Records pro- vide one ofthe best, most entertaining methods ofintroducing aforeign language to children. Within framework oftrain tripfrom Portugal toSpain, anEnglish-speaking child isintroduced toSpanish bynative companion. (Adapted from successful radio program of N.Y. State Educational Department.) Adozen different categories ofexpressions,, including greeting, numbers, time, weather, food, clothes, family members, etc. Drill iscombined with poetry and contextual use. Authentic background music. Accompanying book enables areader tofollow records, includes vocabulary ofover 350 recorded expressions. Two 10"331/3 records, total of40minutes. Book. 40illustrations. 69pp. 5V* xIQVfc. T890 Theset$4.95 13 CATALOGUE OF LISTEN &LEARN language record sets LISTEN &LEARN istheonly extensive language record course designed especially tomeet your travel andeveryday needs. Separate sets foreach language, each containing three 331/3 rpm long-playing records 11/2 hours ofrecorded speech byeminent native speakerswho areprofessors atColumbia, New York U.,Queens College. Check thefollowing features found only inLISTEN &LEARN: Dual language recording. 812selected phrases, sentences, over3200 words, spoken first inEnglish, then foreign equivalent. Pause after each foreign phrase allows time to repeat expression. 128-page manual (196 page forRussian) everything onrecords, plus simple transcrip- tion. Indexed forconvenience. Only setonthemarket completely indexed. Practical. Notime wasted onmaterial youcan find inanygrammar. Nodead words. Covers central core material with phrase approach. Ideal forperson with limited time. Living, modern expressions, notfound inother courses. Hygienic products, modern equipment, shopping, "air-conditioned," etc. Everything isimmediately useable. High-fidelity recording, equal inclarity toanycosting upto$6perrecord. "Excellent . . .impress measbeing among thevery best onthemarket," Prof. Mario Pei, Dept. ofRomance Languages, Columbia U."Inexpensive and well done . . .ideal present," Chicago Sunday Tribune. "More genuinely helpful than anything of itskind," Sidney Clark, well-known author of"All theBest" travel books. UNCONDITIONALGUARANTEE. TryLISTEN &LEARN, then return itwithin 10days for full refund,ifyouarenotsatisfied. Itisguaranteed after youactually use it. 6modern languages FRENCH, SPANISH, GERMAN, ITALIAN, RUSSIAN, orJAPANESE*one language toeach set of3records (331/3 rpm). 128page manual. Album. Spanish theset$4.95 German theset$4.95 Japanese* theset$5.95 French theset$4.95 Italian theset$4.95 Russian theset$5.95 *Available Oct. 1959. TRUBNER COLLOQUIAL SERIES These unusual books aremembers ofthefamous Trubner series ofcolloquial manuals. Theyhave been written toprovide adults with asound colloquial knowledge ofaforeign lan- guage, andaresuited foreither class useorself-study. Each book isacomplete course in itself, with progressive, easy tofollowIess9ns. Phonetics, grammar, andsyntax arecovered,while hundreds ofphrases andidioms, reading texts, exercises, andvocabulary areincluded. These books areunusual inbeing neither skimpy noroverdetailed ingrammatical matters,and inpresenting up-to-date, colloquial, and practical phrase material. Bilingual presentation isstressed, tomake thorough self-study easier forthe reader. COLLOQUIAL HINDUSTANI, A.H.Harley, formerly Nizam's Reader inUrdu, U.ofLondon. 30 pages onphonetics and scripts (devanagari &Arabic-Persian) arefollowed by29lessons, including material onEnglish andArabic-Persian influences. Key toallexercises. Vocabufary.5x7V2. 147pp. Clothbound $1.75 COLLOQUIAL ARABIC, DeLacy O'Leary. Foremost Islamic scholar covers language ofEgypt, Syria, Palestine, &Northern Arabia. Extremely clear coverage ofcomplex Arabic verbs &noun plurals; also cultural aspects oflanguage. Vocabulary, xviii+192pp. 5x71/2. Clothbound $1.75 COLLOQUIAL GERMAN, P.F.During. Intensive thorough coverage ofgrammar ineasily-followedform. Excellent forbrush-up, with hundreds ofcolloquial phrases. 34pages ofbilingual texts. 224pp. 5x7Vz. Clothbound $1.75 COLLOQUIAL SPANISH, W. R.Patterson. Castilian grammar and colloquial language loaded with bilingual phrases andcolloquialisms. Excellent forreview orself-study. 164pp. 5x71/2 Clothbound $1.75 COLLOQUIAL FRENCH, W.R.Patterson. 16th revised edition ofthisextremely popular manualGrammar explained with model clarity, andhundreds ofuseful expressions and phrases- exercises, reading texts, etc.Appendixes ofnew and useful words and phrases. 223pp"5x71/2-Clothbound $1.75 CQLLOaUIAL PERSIAN,L.P.Elwell-Sutton. Best introduction tomodern Persian, with 90page grammaticalsection followed byconversations, 35page vocabulary. 139pp. Clothbound $1.75 COLLOQUIAL CZECH, J.Schwarz, former headmaster ofLingua Institute, Prague. Full easily followed coverage ofgrammar, hundreds ofimmediately useable phrases, texts. Perhaps the best Czech grammarinprint. "An absolutely successful textbook," JOURNAL OFCZECHO- SLOVAK FORCES INGREAT BRITAIN. 252pp. 5x71/2. Clothbound $2.50 COLLOQUIAL RUMANIAN, G.Nandris, Professor ofUniversity ofLondon. Extremely thorough coverage ofphonetics, grammar, syntax; also included 70page reader, and70page vocabulary. Probably thebestgrammar forthis increasingly important language. 340pp. 5x71/2. Clothbound $2.50 COLLOQUIAL ITALIAN, A. L.Hayward. Excellent self-study course ingrammar, vocabulary, idioms, andreading. Easy progressive lessons will give agood working knowledge ofItalian intheshortest possible time. 5x7V2. Clothbound $1.75 MISCELLANEOUS TREASURY OFTHEWORLD'S COINS, Fred Reinfeld. Finest general introduction tonumis- matics; non-technical, thorough, always fascinating. Coins ofGreece, Rome, modern coun- tries ofevery continent, primitive societies, such oddities as200-lb stone money ofYap, nailcoinage ofNew England;allmirror man's economy, customs, religion, politics, philos- ophyart. Entertaining, absorbing study-, novel view ofhistory. Over 750 illustrations. Table ofvalue ofcoins illustrated. List ofU.S. coin clubs. 224pp. 61/2 x91/4. T433 Paperbound $1.75 ILLUSIONS ANDDELUSIONS OFTHESUPERNATURAL ANDTHEOCCULT, C.H.Rawcliffe. Ra- tionally examines hundreds ofpersistent delusions including witchcraft, trances, mental healing peyotl, poltergeists, stigmata, lycanthropy, live burial, auras, Indian rope trick, spiritualism, dowsing, telepathy, ghosts, ESP, etc. Explains, exposes mental, physical de- ceptions involved, making this notonly anexposS ofsupernatural phenomena, but avaluable exposition ofcharacteristic types ofabnormal psychology. Originally "The Psychology of theOccult." Introduction byJulian Huxley. 14illustrations. 551pp.5% x8. T503 Paperbound $2.00 HOAXES, C.D.MacDougall. Shows how art, science, history, journalism can beperverted forprivate purposes. Hours ofdelightful entertainment, awork ofscholarly value, often shocking. Examines nonsense news, Cardiff giant, Shakespeare forgeries, Loch Ness monster, biblical frauds, political schemes, literary hoaxers like Chatterton, Ossian, disumbrationist school ofpainting, lady inblack atValentino's tomb, over 250 others. Will probably reveal truth about few things you've believed, will help you spot more easily the editorial "gander" orplanted publicity release. "Astupendous collection ... andshrewd analysis," New Yorker. New revised edition. 54photographs. 320pp.5% x8. T465 Paperbound $1.75 YOGA- ASCIENTIFIC EVALUATION, Kovoor T.Behanan. Book that for first time gave Western readers asane, scientific explanation, analysis ofyoga. Author draws onlaboratory experiments, personal records ofyear asdisciple ofyoga, toinvestigate yoga psychology, physiology "supernatural" phenomena, ability toplumb deepest human powers. Inthis study under auspicesofYale University Institute ofHuman Relations, strictest principles ofphysiological, psychological inquiry arefollowed. Foreword byW.A.Miles, Yale University. 17photographs, xx+270pp. 53/a x8. T505 Paperbound $1.65 Write forfree catalogs! Indicate your field ofinterest. Dover publishes books onphysics, earth sciences, mathematics, engineering, chemistry, astronomy, anthropol- ogy, biology, psychology, philosophy, religion, history, literature, math- ematical recreations, languages, crafts, art,graphic arts, etc. Write toDept. catr Dover Publications, Inc. Science B 180Varick St.,N.Y.14,N.Y. 15 DateDue Demco 38-297 (continued from inside front cover) Vector andTensorAnalysis, G.E.Hay $1.75 Theory ofFunctions ofaReal Variable andTheory ofFourier's Series, E.W.Hobson Twovolume set $6.00 Introduction toDifferential EquationsinPhysics,L.Hopf $125 TheContinuum andOtherTypes ofSerial Order, E.V.Huntington $1.00 Ordinary Differential Equatin^-" > $2.55 Table ofFit-'' ro-"^la'.00 Foundation Modern T\ Mathematical Fc Mathematical F< Led Inft Elements Diction Num$1.98 $1.45 nchin $135 High Fun Lectures o; Introc Collected ) ATabi 1 Elementi El PC(2) Hunt Library Carnegie-Mellon University Pittsburgh, Pennsylvania Available TF1,Do indicate fit yearonsc ByHorace Lamb This book isacomprehensive mathematical treatment ofthephysical aspects ofsound, covering thetheory ofvibrations, thegeneral theoryofsound, andtheequations of motion ofstrings, bars, membranes, pipes,and resonators. The author also includes chapters onplane waves, spherical waves, andsimple harmonic waves, "thusgivinga connected view of allthemore important branches inthedynamicsofsound. Acon- cluding chapter covering theanalysisofsound sensations, theinfluence ofovertones onquality, andtheHelmholtz Theory ofAudition extends thedomain ofthebook to include physiological acoustics.' Thestudent andspecialistwill find inthisbook acomplete andself-contained develop- ment ofthetheory ofsound. Allthefundamental differential equations areconstructed from thephysical conditions andsolved completely.Theauthor provides the specific mathematical details forsuch important phenomenaasharmonics, normal modes, forced vibrations ofstrings, plane and longitudinal waves inabar, vibrations ofacolumn of air, thetheory ofreed pipes, andmanysimilar topics. Relevant mathematical ideas which maynotbefamiliar tothereader, such asFourier series andconcepts ofvector analysis, areexplained inthe text. PARTIAL CONTENTS: THEORY OFVIBRATIONS: ThePendulum, Forced Iscillations, Selec- -tive Resonance, Effect ofDamping, TheDouble Pendulum, General Uiuations ofaMul- tiple System. STRINGS: Energy, Free Periods, Forced Vibrations, Aeolian Tone. FOURIER'S THEOREM: Sine andCosine Series, Complete Form, ApplicationtoViolin String. BARS: Theory ofElasticity, Effect ofPermanent Tensions, Theory ofDeformation. MEMBRANES AND PLATES: Energy, Normal Modes, Uniform Flexure. .PLANE WAVES OFSOUND: Velocity, Viscosity, Effect ofHeat Conduction. GENERAL THEORY OFSOUND WAVES': Flux, Diver- gence, Velocity-Potential, Spherical Waves, Reflection, Acoustic Properties. SIMPLE HARMONIC WAVES. DIFFRACTION: Point Sources, .Vibrating Sphere, Scattering, Trans- mission. PIPES ANDRESONATORS: Normal Modes, Free Vibrations, Dissipation,Multiple Resonance. PHYSIOLOGICAL ACOUSTICS: Musical Notes, Combination Tones, Perception ofDirection ofSound. Unabridged republication of2nd, revised, corrected edition. Index. Bibliographyinnotes. 86diagrams,viii+307pp.5% x8. S655 Paperbound .$1.50 THISDOVER EDITION ISDESIGNED FORYEARS OFUSE THEPAPER ischemically thesame qualityasyouwould find inbooks priced $5.00 ormore. Itdoes notdiscolor orbecome brittle with age.Not artificially bulked, either; this edition isanunabridged full-length book, but isstilleasy tohandle. THEBINDING: Thepagesinthisbook areSEWN insignatures,inthemethod traditionally used forthebest books. These books open flatforeasy reading andreference. Pagesdo notdrop out,thebinding does notcrack and split (as isthecase withmany paperbooks held together with elue).