Old Physics Notes Binder

\ ‘ | .oumtyH aq @MisePhysicsBinder 9.20.08 Most stuff isquite old,butafewthings arerecent. Mechanics ‘What makes tides Gut basics ofHamiltonian Mechanics Goldstein mechanics notes from Nov 1977 Physics ofSprinting paper with Igor, March 1981 Cars, torque, andbrake horsepower Acceleration Times based oncartorque Oct1985 TheoryofEffective Horsepower Oct1985 Computing Acceleration Times ‘Thermal Loss inHouses Thermal conduction Feb 1978 Heat lossthrough windows, etc(refs) Quantum Mechanics MessiahandthemeaningofbrasandketsApril1975, Operators inWavefunction Quantum Mechanics Clearing upActive vsPassive transformations Solution of2Dharmonicoscillator(Laguerre @Harmonic oscillator in3D Comments onQuantum Mechanics Special Relativity Relativistic mechanics andmassless particles Accelerating rockets E&M andspecial relativity ‘Thomas rotation effect Thomas precession andthe4factor Foucault pendulum comment. Some other mechanics/magnetism problem. Particle Physics dates when certain things were found Goldberger-Watson footnote forpage 347equation 12b PCAC andGT(Partially Conserved Axial Current andGoldberger Trieman relation Relating A"and V"tothegenerator currents, more Goldberger Trieman Chiral comments Anexperiment tomeasure g-2. Standard Model summary page (from web) Chew 2008 paper Cosmological Hilbert Space .| | Mechanics | | ;& — 4 4, Wak amoleoaspidge ee |0)Ce aoe a— :Oe eee Merttm|| RS Oo=See aa"C=) a Ra=O = MeHien—WeWs.aa ae ay =Gihe sve “(+s — TS 5h wandfoLaie amle akfoLindSoeeutksoalle=hnnsovclad.Key brosanWateAone{7 ~_o _ a_NOso crane MookResRnOLNaess a Cae aSy a ri oe Ss Ss Ques Elona)=_F/won Recose ; taal, Tie, 98co¢2007)hanopto <0 osad, a, ry aaenaasun oak~uae0ggikOnn———OBaie eeboys Ce >w= Gti . : C l= .—_ ar —__— en= ok Rose: De =Oma yanGleRoose _ 2)Quia_motiod qiaw. Ueuneanouada UneJaokOuck7Oma. Shighed cose {mrLontla conden 40,Orecensathoax~2anlA ankaite ca— — _Bown) =ECR) RewSS (MePMY p P| Ee 7 Qa. 4 _Youn BOMMWumiedid Woot kefordenrowed.Narea.Qescke up: =5.9gx/0" = 6.6)x/0 . =2.45% J0"k = 3%x10% A ee58= BY En aw =D x0Sov=3.84 x708m _ or P a a a a:iSSTaal =45ootinhea) o.(Run=6,0 lower, = 10B.6enimastod g Mm VE — @Wao,hakeey) hanA=ksTo , a aya a a 5 se = S16?at——_Lomgaue beRaatJouDaetaal; —@ Fachn theety=BPR isoffltrh2Sore 98—@Finally, ==3g=Yor180. a iad=3x/O°ak_%323xopKeee whotdahJoreriasmall,britomyfoeuMAdwaaswader | NO —@)_Qcstrnl edugtiledmesavonce AB aeney — im fa) eee eee eee Ors paae— ee NN \ La" ry ns ghj= $e_paegh — — ee . [Redo Spe Rds SNCs aapFlue)life 9snob >a 7 . 3 @‘lypradictemn GWG)AueoddsMowke _ 46 28En(etsRewes es ES Siege As —_ EL ETS ~ =65GS = ep 1s Saa(A eeee Tia Souda asawenolde fxDscmoredmnal Lae —__) Ganladeton dvauga a,oyuefe=WrAt=108 mjc’ so hinGland makaf hwyace,cosine. (ie,ss}10 WatGoreoreoflfecdeedoJens oarse dheAn 7 oeoGia.,Lowooacpom _LC9i,44), ne_soon_peadan, Stedman imdsnsoniabens, . sa § . 0 Qk=4Qu NeBS a% “ r ; h ead A las» =a de & 4 ax @Mawdaljest.aromiborinns HeSeg=. lnkicrMok Fs}pi)re=O 30adane.nastdopeom.g;**. Qu=-9L=-$. ound AY=+4, ou Ou OR 3t= ae2423Be=fea>t at==f Qu=4 dh=a4=->4 Eyae at at ot ®yLidoama.expdinkSuneDopomdouase funH=corsenved ' Nov\47) AReviewoOosrisa WMedomicas We Sy ietectedtyseeped etindan, |Veet, apoles nwJumekin A)Duar50UTiGqa).QhawyorSanwaeoman, agersnaSlychs Jonasad Qh-Wag. AniaJottothr Jnoenauarebae |jpDwar Jrwogobatad @Mashes Lew WaPicanWewremik einway! &(22)-@T) =QFZR.(@)ca34 Ba, y x 343 yakJonetenwillBekart WasaWacegreSamapriagthdowSanaa ein oomeanveteie T=KE. dyodsbepemeMate,Lenten, ageed ~@©iainsiormetrnl sieprre hyhefomaeTEA.theo oe virahe Bene yondogodrowBWwaratbee RteBW Ngarnmadey sd”onde gi. @Hfrereaeareerurcbein, DrewJfyordikesLapmnginn Let oanAV=\G@) +NGa4c)\Nomaloerath le ee _ _ A (aL) de LT ‘ ationia(=Be78‘yea“| eNedLELGG85BRDaundyperdfremaleded -2-— te Qua T=Suggest ="Meaction’|At @Howibrots Puwagre yade—ST=Owheeyorremyfe Tarergo, aanpotods YuiadGtdaben derdentheLagrange ©Wo*cewroriesomnmumbun” courageqe?gian=(35)| OmesayaOnskacondwele g,aeye ag=0, @WeYowdionam adakar oO .7 SEWEY)=a4~Uydest): hoklee ck OH=Pi~PK=OSeW*RG.) aa: :—db =-4.(U\- F8; Nolaalooaton#GE)Pa =4 Se aw=-& , Antgh=4 [ama Iiy oH244,|rors |aH=+3: & ak=-3t! eee |= ©meisterUCRat).WeyergeneaeAionQeray @crete corte+cormnical pom suekeMk Drscra”QonaeysbinyDaaarmpribering Grnye.Offoemmr Rowconyagale. beQaQi+satay ceorn sik, ak. -& dQi Qu ">AQceaaddad0commnica, BarGpithnanan, |K>KCRG) dnonsenforaCV,boxed, Warsucksukaggermmasbing prin Shee F.CqjQin)aepage240 @6peisbixtrwedun |iaSeeQEBEG) ok,YeeAaspree wean Amcammnenl Smmefonmuteor - fo)&%Qcondgyoatasecbadd bey0eeronieah, donakrmntin, QuanyorearseedOnskMesWpredoren seanteewWom: St. dca) =SF20RD & 3(ay) ‘ 3(4v) vat uvrdowels apodk aratwo-demausiowl surface UAage, OWTdarmemsind precrspace Ad.dindg:gu. HAftews ok @ Ove ce Te»Zigdqagdl=ddZLear]|fooaade|soho, @®Whur Wk: SMe)2BeBe_Pas en coBlur) TV3a:8p: 2—|guovduav ov av =US =_Sagange BenceAwendl Nakxpaiontaa Out LaaQ=Twoipenal Ras By TH @Varupd downSoaXRosserbrosleade: = Qa 2Baden©2(GPGR-east Tri agcaSesDueecb = Tre] =o . @Voosm: Poissonbudeshe orjndig:AWid(Rovere ally psa \conde nyouaeLadey =Carlee ®Easton Arwen:iyi)=2(GNGe -GRah)) =Bh5 5 SN NBSAKSl=demke .% &° . LeaWl=av=+6 @ . Skarcugaita2ampoee,woefea WWNo=VE(I-€). Physicsofsprinting dogh£eam Noe(\-€) : = é IgorAlexandrovandPhitipLucht Qee.Sh Kaoo=teroe Department ofPhysics,University ofUtah,SaltLakeCity,Utah84112 o Sprinting isdescribed byasimple physical model. The shodelisused-to predict thedifferences between therecorded timesforracesonasttittrackand-onacurve.Itisshownthatthechoiceoftherunninglanemakesanon-egligible difference. 1.INTRODUCTION d2x(0)_dotsingi SeGeI~ooo. © Inthepast fewyears there hasbeen agrowing interestintheapplication ofphysicalmodelstoproblemsofanimal_whprex(t)isthedistancemeasured fromthestartofthelocomotion ingeneral! andtohuman athletic performance _rade and(2) istheprinter's instantaneous velocity. The inparticular.>$ solftion tothisequation subject totheinitial conditions x(0) Aprobiem ofspecial interest istodevise amodel that =,0(0) =v9isrepresents withreasonable accuracy thepropulsive and - went)+ygenetresistiveforcesactingonahumanrunner. v(t)=Gayl—e-*4)+vee, “ Onecommon modelassumes thatthe sumtotalofre x(0)=(Jo)t+(fo?=ogfale“"=1).(5) sistveforcesF,actingonarunnercanberepresented bY1.wet,knownsolitonsdescribethemotionofparticle F,= ~oMo, (1) sedimenting under theaction ofaconstant force of“grav- igity”Mfand aviscous force —Mo;theparticle (sprinter) where Misthemassoftherunner, visthespeed, andais NY “ ?aparameterofthemodel,presumedconstantornearlyapproaches theterminalvelocityv,=Jfoasymptoti-|é Ycally. Saeetiyanabeswithcostoeach«formigeLetussupposethatthetimes1,andf2foranathlete's,thatitcanadequately represent humanrunnerswherethePerformances overtwodifferent straight-line sprintingforcesarerapidlyvaryingand,ingeneral,complex. eianeesxpandaeain‘TheneyOsbemsedNonetheless suchaformwasusedbyKeller?tomodel WebatsHakesihe!beatao(seewei)‘twodif. @‘competitive runninganddrawcertainconclusions regarding, wweBiveinPaleTitebestrandingtimesovertwoott optimalrunningstrategyandmaximalfutureperformances. feFentdistances forfourworld-class sprintersfromthe Onesignificant conclusion reachedbyKelleristhata{967-69period.1%(Wenotethattwoorraneareawell-conditioned athleteisabletosustaina maximumandQoCNouCancersOndforcachworlrecon.)e nearlyconstantmusculareffortforracesoverdistancesofComPuledvallesofoandforeachrunnerareshownin 290morless.Weshalldefineallsuchracesassprints. ‘columns6and7,respectively. wee 'Abasicquestionis:Whataretypicalvaluésofthepa_,,ThelastentryinTable|isTommieSmithwhohastherameter?Itistobeexpected thatowilldependonthe_“istinction ofhavingarecordedtimeforthestraight-trackFanner, Nisspeed, thetyberace, andpossibly onexternat 200-M raceandalsofortherarely runstraight-track conditions suchasthetracksurface, altitude, etc.Keller? 240-Yard raceforwhichheholdstheworld'srecord,Inusesavalueof«=0.44sec~!whileWhittandWilson’ addition heistheholderoftheworld’'srecordforthe200quotesimilarvalues.Thelatter,however, weremeasured Lenthecurvewhichheestablished inMexicoCityinRoe rentsrinecneeds8PeTEEXSISMOMER” ThecomputedvaluesoftheparametersfanddifferWepropose that[oragivensprinter ameasurement ofWidely fromrunner torunner. Wehavenowayofknowingrandthepropulsiveforceparameter(definedbelow)canooeee neaareneaaeneee, “beperformed inprinciplebymeasuring thatsprinter's thaluesxtaythestopevdethiareoadrunning timesfortwostraight lineracesofdifferent dis eyeeneneasetthecloseneesofvarprediction tances.Ourmodelwillthenallowustousethevaluesof|="4‘mraceonthecurve.Theclosenessofourprediction theseparameterstopredictthatrunner'sperformancefor‘otheactualrecordedtimewillbe'atestofthevalidityof| ‘asprintrunonthecurve. thebasicassumptions ofthetheoryembodied inEqs.(1) * and (2). 7 . IL MODEL ulUL. RUNNING THE CURVE Weassume asdoesKeller? thatasprinter ofmass Mis Inatypical 200-m racethestarts arestaggered sothat ‘subject totwo horizontal forces. One istheresistive force each runner runs thefirst 100 moftherace onthecurve. expressedbyEq.(1)andtheotherisaconstantpropulsiveHowever,theradiusofcurvatureofthecurvedportionof eforceF,whichwewriteas eachJaneisdifferent. ‘Thelanesare1.22mwidesothatthe F</M, ~ (a) innerradiusofthemthlaneis whereJistheforceperunitmassoftheathlete.*The Ra)=1O00/x+(nx—1)(1.22)m. equationofmotion forthesprinter isthen Lettherunner berunning with speed oonacurve of ~ ° 254 Am.J.Phys.49(3), March 1981 0002-950S/81/030254-04500.50 ©1981American Assocation ofPhysics Teachers 254 OGYamao7Gur(19)amd(i)andackRKSomeR=Wt(wri)¥(22 @we=F and€=4(FRE é3 @thawC12)YeagreAmotac GrZoompvce,("pattem curve, ‘able1.Computedvaluesoftheparametersfand¢forvariousrunners(indicatesacurrentworldrecordholder).Thetimesartherunners timesoverthegivendatanecn Name Contin *worldrsondholder).Th " btee Runner x1 tibseo) 2 tnfeee) _alseem") AN/ke)_ovin/see) fBe/el__[&F/A__[BoerlRumery tsee) tae) ascent)fN/K)oen/seed_Ho/el F/MSeo John Carlos yards 6.0 100yards 900667 81312193, BillGaines Gyards 5.9100 yards 93 1250 134510768,dimHines 100yards9.2100m 9905817101222 65%SI1aTommieSmith 100m 10.1220 yards(straight) _19.51__1.252__ 13.461075 19% TonnesSree0tO atiyardstage) TPS)2esense constant radius Randsubject totheforces (1)and(2). rameters thedifference between v,andv-(«) isoftheorder ,Now, however, thepropulsive force mustsupplyacompo- of0.3msco-',Thustherunneraccelerates slightlyupon nentthatgives risetothecentripetal acceleration ofmag- entering thestraightaway." nitude v2/R. Theathlete's equation ofmotion thenbe- Equations (10)and(11)allow ustocompute thetotalcomes timefrfortheraceinaclosed(approximate) form dofdt +ov=(2—v/R2)'", (6) tr=ton4100/0,+fae]MEHMH100 Equation (6)istobesolvedwithgivenparameters f,c,and =2(100)/o, +I/a+€[(100/0,)—13/(120)] vA Rsubject totheinitial conditions(0)=0,x(0)=0. +&[(100/0,) —25/(120)}+OC).(12)/" ‘Anapproximate analytical solution canbeobtained by Equation (12)thencanbeusedtopredict therunningnoticing thatfortypical situations thesecond termunder imeYorexample, forTommie Smithrunning a200-mracethesquarerootissmall,»'/R?<<,andbyexpanding the onthecurve. Weusetheparameters ofTableIandassignsquare rootweobtain himtolane3,thelaneinwhich heranforhisworld record, dofdtov=f~(1/2)0fR2. (7)ThenR=34.27m,€=0.0314,o,=10.75,Weobtain Asitstands, Eq.(7)isstill notsubject toasimpleana- 4100=°+sec, lyticalsolution.Wecan,however,solveitbytreatingthe 2100OAsee. m=9.33 terminv4asasmallperturbation (seeAppendix). The fimtomewelastfortherary6Bsolution soobtained subject totheinitial conditions v-(0) ‘total predicted time fortherace= see. =0,x-(0) =O-gives thespeed v.(t)andthedistance trav- IntheMexico CityOlympics on16October 1968intheelledalongthecurvex(t)as: men’s200-mfinal,TommieSmithrunninginlane3sctthe=(fol-Z-')- 19-292 still-current world record of19.83 sec. 2elt)=Fol—2")—Lfaay FR-\on7ym2). Inordertotesttheaccuracyofourcalculation wenu-e@(8) merically integrated theexactEq.(6)andcomputed thexe(t)=(f/o?Inz =1+Z!) timenecessarytocoverthefirst100mofthecourseonthe—(1/2)f/e2)fR-'0-2)2F(Z), (9)curve.Thecomparison ofthis calculation withtheresults CMGOAUR'ePFLZ).0)rereapproximate analyticalsolutionisshowninTable. whereZ=e%and Thetwoprocedures giveresults thatdiffer atmost by0.3%. - - 167-2 ‘This surprisingly close agreement lends supporttotheva- Fu(Z)=1+(10/3—41n2)Z~~62.2-3—1/3.z-4, _lidityoftheapproximations madeinthecalculations, The +23 132, closeagreement between our“predicted” andthemeasured F(Z) =~37/12 +InZ+(2/3 +41nZ)Z-t value ofTommie Smith's performance inthe200-m race+3Z-2—2/3Z-3 +1/12Z-4,onthecurvecomesassomewhat ofasurpriseconsideringthemeagerness ofthedataandthectudeness ofthe Theaccuracyofourperturbation proceduredepends—tmodel, fonthesmallness ofthedimensionless parameter ¢"Finally, weapply ourmodel tothefollowing problem.=(R~'0~2)?/2 which forthecaseofinterest toushas Since thevalueoftheradius ofcurvature Renters Eqs.valueof0.03139.Forrunningtimesontheorderof10sec, ()-(12) wewouldexpectthatathletesrunninginthetermsinZ~!andhigherpowers ofZ~'atenegligibleand outside lane(larie8)wouldenjoyanadvantage overthose‘canbedroppedfromtheexpressions forFandFo fhtheinsidelane.Withtheseapproximations wecanwriteexpressions for Infact,itiswellknown thatsprinters dislike running in x-(0) andv-(©), thelatter being theterminal speed ofthe runner onthecurve. Wenotethatwithourtypical value’of ok ‘ yea!‘athesprinter attains 98%ofhisterminal speedwithin ap- TableIComparisonofnumericalsalutionswithapproximateanalytical proximately3sec: P2136N]kgs0=1.252see"! v-() =(1=©0,. (10) ‘o(sec) —_troo(see) -or- - retical approximate difference H(t)oI==(W/o37/12).UY R(m)solutionsolution(8). Equations (10) and(11) areuseful inthat they allow ustp bringouttheeffectofcurvature of-thetrackrepresented 1 3183 10.361 10389 027% e bytheparameter ¢.Physically, 2¢isthesquare oftheratio 3 3427 10.327 10.348 0.20% ofthecentripetal acceleration v?/Rcalculated atterminal ® 40.3710. ee M4ne 0.11%speedtothe.propulsive forceperunitmass.Withourpa-Stfaighttrack =10.100 10.100 285 Am.J.Phys, Vol. 49,No.3, March 1981 |.Alexandrov and P.Lucht 255 x Me281)4|¥s28.0) Suede: k=Bert)rele-He]e|% 23]+0) : theinsidelanecomplainingthattheturnis“tootight."Ourtheirrespectivelanes.Wefinditsmagnitudetobenon- calculation bears thisout.Wefindthat, withtheparameters negligible. _weusedforourtestcase,thetotalrecordedtimefora_Clearlyourresultssuggestalingoffurtherinvestigation runner inlane1would have been 19.72 secwhereas forlane forthose interested inthebiomechanics ofsports. Tothe 8itwouldhavebeen19.60sec.Atfinishingspeedsofaboutextentthatasprintinghumancanbecharacterizedbypa- e11m/secthatdifference intimetranslates into.adistance rametersfandaitmightbeofinteresttomeasurethesedifference ofoverameter.Sincemanysprintracesarelost__parameters undercarefully controlled conditions andcor-‘orwonbymere centimeters theabove effect isquite sig- ‘relate them withthephysical characteristics ofthesprinters, nificant, suchasheight,weight, typeofbuild,reaction time,strength‘Curiously enough sprinters donotconsider lane8tobe _oflegmuscle, etc.Itisconceivable thatthese parameters themostadvantageous one.Because ofthestaggered start_fandomightalsodepend onexternal conditions suchasrunnersinlane8runforhalftheraceaheadoftheothertracksurface,altitude,andotherfactors.competitors. This, theyclaim, putsthem atthepsycholog- Finally, wewish tocomment thatithasbeen ourexpe- icaldisadvantage ofnotbeing abletoseewhat their com- _rience thatthere isafairly large audience among today’s petition isdoing. college students interested inlearning about thebiome- chanics ofsports from thepoint ofview offairly sophisti- cated physics. Such atrend iscertainly indicated bythe IV.ERRORS amount ofresearch done inmany countries inthearea of sportsmedicine andtechnique. ‘Wewishtomakeafewbriefcomments aboutthe“ex-perimental” errors. Since neither therecords books northesportpressincludeerrorbarswiththeirreportedresultsweAPPENDIXwill limit ourselves toafewcrude estimates. Weassume - that mostoftheerroristheresultofinaccurate timemea- _Inordertodetermine theparameters fandoforasurements. Many ofthetimesgiveninTable Iwereprob- vensprinter, letusassume thatthesprinter hasrecordedablytheaverages ofseveral hgnd-recorded times. Other timesf,and¢2overtwostraightaway distances oflengths timesmighthavebeenrecorded electronically andthen x1andxUsingEq.(5)twicewithvo=0andneglectingrounded offtothenearest tenth ofasecond. theexportential term wesolve forfandotoobtain Forthepurposes ofthisestimate, weassume that the = - - accuracy intherecorded timesis0.05sec.Wenoticethato>(yedttaaid~Ueleatc)~(feo.(13) the’resulting variation inthecalculated values offand S=07x2(oty—1). (4) arequite large. This isdueinalarge parttothefactthatthe i .calulation involvesadenominator whichisaference Brabham ranteRnaistd :tweentwonumbers numerically closetooneanother. One callwisthesameasEq.(4)withvo=0,consequence ofthiseffectisthatdataforracesoversimilar ‘a oe distances such as50and60yards or100yards and100m w(t) =(Jo) -e-*"). arepractically useless.However, asisapparent fromEq.(12)thetotaltimerpNewt,Wewritethesolution toEq.(7)asp=w&uwheredepends primarily onv,,sotosomedegree theerrorsinf °'"eatwasasmallperturbation. uthensatisfiesandotendtocanceloneanother.Theuncertainties included du/dt+ou=-1/2v4/(fR2). as)inTable Iwere calculated accordingtotheassumption Ar A mati A =—Ara=£0.05 sec.Assuming thiserrorinthemeasure Replacing oinEq.(15)byitsapproximation w,weinte~imentoftimetheresulting errorinthepredicted ¢forT, Batetwicetoobtain thesolutions (8)and(9)Saath is Forourtestcase(Tommie Smith) theparameters fand@aredirectlydeterminedfromEqs.(13)and(14)usingthe Atr =£0.054 sec. data'given inTable 1. - ‘ Equations (8)and(9)canbefurther simplified ifwe Alessconservative errorestimate ofAty=—Atz noticethatforourtestcaseo=1.25sec~!andforrunning -s hs = ing raeorslilorieneandobservedtrwithinthetimesoftheorderof10sec,Z~!~10-6sothatF,andFz- canbeapproximated by Fy21 Fre to 37/12. V.CONCLUSIONS ‘ThisgivesEqs.(10)-(12). Theseequations thenpredict the Insummary wehave found thattheforces acting ona __timetocomplete thecurve portion oftheracet190,thespeed sprinter attypical sprinting speeds canbereasonably well __attheendof100m,andthetotal time.modeled byEq.(3).Theuseofthatequation overtwodif- Forcomparison wefindthat,usingthesamevaluesoffferent distances candetermine thesprinter’s parameters andg,a200-m racerunentirely onthestraightaway would Fand o,Thevalues oftheparameters thatwecalculate for _have taken 19.40 secascompared withourlane3calculated ‘afewsample cases areinsome disagreement withthose time19.68 sec.Even forarunner inlane8thestraightaway‘commonly foundintheliterature. Ourmodel,extendedto__timeisfasterby0.20sec. includecentripetaleffects,thenallowsustopredictén eathlete’stimeforaracerunonthecurve.Thisinturncanbeusedtocompute thedifference inthetimesfor(Wo*QneaveofabsencefromDepartmentofPhysics-Astronomy, Californie ‘comparable athletes running indifferent lanes. This dif- State Unversity atLongBeach,elieprevdaea ference issolely duc tothedifferent radii ofcurvature of 1K.Schmidt-Nielsen, Science 177,222(1972). 256 Am. J.Phys, Vol. 49,No.3, March 1981 I.Alesandrovand P,Lucht 256 e ’Mechanies andEnergetics ofAnimal Locomotion, editedbyR.M. =N,f=N/kg,and@=sec“AlexanderandG.Goldspink(Halsted,NewYork,1977). *IN.McWhirter,GulnnessBookofWorldRecords(Sterling,NewYork, 34.B.Keller,Phys.Today269),42(1973) 1979). 4H,Lin,Am.J.Phys.46(1),(1978). ‘SportsWustrated(NewYork,NY)andTrackandFieldNews(Los SF.WhittandD,Wilson,BicyclingScience(MIT,Cambridge,MA,Altos,CA)miscellaneous issuesbetween1967and1969. e 1974), SPersonalobservation(1A),WeusemsnitsthroughoutsothatMf=kg,x=m,o=m/sec,F_!0Wewishtothankananonymouseferceforpointingthisouttous. |257Am.J.Phys,Vol49,No.3sMarch1981J.AlexandrovandP,Lucht257 j ConyamdVorgut.audRHP WinDes eee SS a Measuring, dows:a.ee (.B)-BE ee - Thistype-of device wasactually usedto-measure enginehorsepower at.onetime.You open the throttle all the way, then tighten the friction brake toget your*dasifed RPM:TheWyoureadyourforcescaleinpéunds. Atypical setofnumbersmightbeD=i.feetandti30.Ibs.This-means_the torque.is120lbefeet. tt/lbs. That is, the test fixture isapplying that much torque onthe flywheel tokeepwovangilar aeeelerattont; toflywheel OfCourse“Ts"WingOpposite andequaltorque.To get the brake horsepower,. you mltiply.by angular_velocoty_of-course. This-is. — DPI f. Since 1HP=33,000 pound-feet ofwork in1minute, you get little equation:TEsSinceBEB231000Powe . He=.(Qedx(W)At =DWh2eas»)~.64F -.33,90... .$252 \$7450 Theformila isfreshman physics: powerisdistance timesforcediyided bytime,with unitsconverted tohorsepower. The fofcexdistence factor isthe torque which is - invariant, whether measured atflywheel or-at- scatey Think ofthe-distarice that e apointontheflywheel goes,working against thefrictional farce. "—“Wormally torquedecreases alittleasRPMincrease. Iftorquewerecdnstant, theBHP itd sitiply inéréasé With RPM litiearly.” The point ofmax BHP occurs at——maxRPM.you.can.useengine at;.max torqueis-at-slowerRPM— = ~Acceleration 6fYourVehicle is“ofcolrse directly artected bytorque. Ie,F=ma :andtorque =.Ialpha. Caracceleration-a-islinear intorque, inverse in'weight. Example: Saabturbo does160HPat5500RPM. Butitdoes188ft-1bs at3000RPM. This torque-gives BHP=~188x3000/5250° 107HPat3000RPM.~7 1 eAcceleration Times (torque, and all that stuff 10.31.85 1,Car engine israted athorsepower asfunction ofRPM, and also torque. Torque isthe more constant ofthe two versus RPM solets use "average torque" delivered bythe engine asyou shift through the gears. - 2.Units. Lets use good old American car units. So: v ft/sec M slugs N lb-feet %seconds E ft-lbs P f£t-lb/sec 3.Now, lets say you want toaccelerate acar from 0tosome velocity V. Indoing so, you keep your engine operating ataverage torque ofNand average RPM of RPM. (ie, this isrelated toomega). Assume that energy output ofengine per second is: Power =q) xTU (recall vxFfor linear motion) The energy you put into your vehicle, assuming nofrictional losses, isthis times time. Set that equal toyour kinetic energy atspeed Vand you can solve for tt 1. 2e twv*=swk@ka$em(skys)-[v(S$t/ax)\Bar$(az)%(44-48) Nowconvertthistomoreconmoncarunits:co u 1.b=&[MED Jycor)x49;$e(isft):Vow) = qa> (°°Se = 17 Remy(Ft)Y~-— Beh(owt) 7(H4-8) 60 232 Sohere is our result forf acceleration with nofriction due toair resistance or transmisstion train losses (enginer losses are taken into account since weuse actual measured engine torque; shifting losses also not accounted for): 2 £=(32)w-v RPM. & Example: you are sitting ina1986 RX, sototal weight is2800 lbs. Engine is 138 ft-lbs at3500 RPM. Assume this product can bemaintained. Then wecould compute atheoretical 0-60 time: the result is6.7 seconds, the quote is8.0 seconds, 0 you add about 20% toyour computed time. |7 e@Horsepower. TheBHPiscorrectatitsratedRPM,buttheeffective HPoverdrivingspeeds maydiffer. Ifyou have atorque atanRPM, here isyour HP+ 1 P(Htbfard) =2e-f(ae!) (4H) P(eHA)x550=arjena)}.v 60 P(eHe) =9203(remr) AE =Fe 60-550, 33000 Lew P(GHP)=RMT5252 Example: ifyourunyour RXenginer at138ft-lbs and3500 RPM, youaredoing92horsepower. Ratingis146HPat6500RPM.(itsarotaryengine’. Ifyoucould get120ft-lbs at4500 RPM, that is103HP.Youreally need toknow the torque asfunction ofRPM. FortheRX,youknow that at6500 youhave toruqof—14,6*5252/6500 =118ft-lbs. Thus, fairly constant, characteristic of Example: consider the prelude enginer: 4000 RPM «107ft-lbs implies BHP=81.5 hp 5500 RPM 100 hp implies torque =951b-feet Again,fiarlyconstant torque. Foranyvehicle youaregetting twotorque,RPMpointsbyreading the two specs. o [0-30-85 e Atheoryof,"effective horsepower".1.Assume that acar's torque curve isaparabola. You can determine this curve because the torque spec gives you the peak ofthe curve, and the horsepower spec gives you another point onthe curve. Onee you have the curve, you can compute the average horsepower over some range ofRPM. Having computed this figure, you can compare different vehicles. ° z t R-&}JN =%-(B&%)- =%-(&%) oo R WeaWPSTAR Fe . aC(r(a)-R)dR & ZP>ALCO8)a=_4| [u-k(ea}}Rua] aRb~Re. 5152(RoR) Lam & Ke ToT 3rary (R,~ho)* wk=Ty(RR)=H2Lseee f(em at gt/ F(R)2hor(RB-Re)+Ro(RbRe) e.Z AJer@e)-0)(es)-go(R)+26(Ge) HO)=5054(RRs)lzone Rake LAIS < Atthis point Ineed acomputer. You would enter the various numbers and get ananswer. 2.Lets try asimpler model. ABartlett curve: ° ' RaeTian TeT~(TT)a4]. ‘ ‘ on AfsRe Arye Ln=qe\Ye~H(tojjeleee—<——_-——_Clq+Hhe)ERdh 2 343 Lat) = Grebe)t(fe-R)—85(enelt) e¥ a) ves a Again Ineed acomputer. So: e 3.Letsgoevensimpler. Letsassume thataverage horsepower isaverage RPMtimes average torque andusethelinear aagain: cy Ler=deSah=BR) [erent —Re (aorta) (Rofo)~K+(86-Fe) (be~ke) =(Core) —EK(Rah) dr) =Brk[e-F) = xe-e) fe)=tye =4(e-O 2Me). ae KyeMo AT\{Z-1se<P HMSO de LID=TOteed[R-8] R= [ere] AT=Tort R= >[lorS] Ra=5500 Gi: Tuo: Ro=3000 T=188 ec=35 Retow+sse0\=y250 Ry=5500 T= 153 OR=2500 £r)=Wer—35[vasoSood)=INO. Suey=13R.0. mw:R=315%) T=(0 oteto f=See§(32+Son)&=4250 Lo=ISO QiL=1000 =1395- 2x)=no-22[usos—sem) =47S Cue)=122 @Entseonpitin. oopalaation) pours— -—- -- - daiwa. Te gw BEE pteERE aead)=+AS ORT 7ty FP Yao PTR=PLN].28000Mette 7 . __PTRted =BME] fe=pleSee =See -@ Se [Pitter =SsoPIA PO oe ee +sn - +--+ - a Lite)NICSee —_ 7 eeee “glee oo—oo a —— Or GD aaShoom, Gd-oi)~ESae\sass _. Susdsfable0=60_ =[0,) To SL 0-60ae_.aadan 34_2AK0-0 (af =5 = Wh_at 0-6CF _£28_ ES wih) vows fe)He Sam] oo ORKprank,(0-6)=(res) (60) =DYacrde.oois >15SEU¥L620io-D .7 . ae . .° a. Corrugation ofRoads . Joseph A.Both and Daniel C.Hong Ss Physics,LewisLaboratory, LehighUniversity, Bethlehem, Pennsylvania, 18015,=] Douglas A.Kurtze, [email protected] 3 Department ofPhysics, NorthDakota StateUniversity, Fargo, NorthDakota 58015 Zz Abstract Ss Wepresentaonedimensional modelforthedevelopment: ofcor- rugationsinroadssubjectedtocompressive forcesfromafluxofcars. nNThe cars aremodeled asdamped harmonic oscillators translating with > constant horizontal velocity across thesurface, and theroad surface ° issubject todiffusive relaxation. Wederive dimensionless coupled = equations ofmotion forthepositions ofthecars and theroad surfacena} H(z,t), whichcontaintwophenomenological variables: aneffective = diffusion constant A(H) that characterizes therelaxation oftheroad Sg surface,andafunctiona(#)thatcharacterizestheplasticityorexodi- e@ S bility oftheroadbed.Linear stability analysis shows thatcorrugationsof growifthespeedofthecarsexceedsacriticalvalue,whichdecreases& ifthefluxofcarsisincreased. Modifying thémodel toenforce theao simplefactthatthenormalforceexertedbytheroadcanneverbe= negative seems olead torestabilized, quasi-steady road shapes, inSg whichthecorrugation amplitudeandphasevelocityremainfixed. Pd 1Introductiona s Itiscommonly observed that under theinfluence ofaflow oftraffic, dirt oads develop regular corrugations oflongitudinal “pitch,” orwavelength, . between 0.5and1m,andamplitude upto50mm [1].Atfirstglance such an * instability oftheroad surface might seem counterintuitive, asonemight guess that aflow oftraffic would exert downward forces onthesurface which tend to smooth and compact theroad bed, thereby suppressing pattern development rather than promoting it.Yet irregular, rough roads donot infact. heal themselves, but instead become progressively rougher and more corrugated under aflow ofcars. Similar phenomena also oceur, though sometimes on different length andtime scales, onpaved roads, onrailroad rails [2],and | | ontherollers used tocalendar paper [3].‘The purpose ofthis paper isto investigate thisphenomenon using asimple, tractable physical model,» .3~ ‘Anearly attempt atexplaining road corrugation isduetoRelton (4],- who proposed that theunderlying instability mechanism isa“relaxation oscillation,” caused essentially bystick-slip dynamics. According tothis, view, amoving wheel pushes grains ahead ofit.The grains pileupinfront “ ofthe wheel and form aheap. When the heap grows large enough, the wheel sticks momentarily, andthen slips, running over theheap andleaving itbehind asaridge. Foragiven uniform speed, thisstick-slip process will befairly periodic, andsowillgenerate equidistant ridges. , Adifferent picture wasprovided byMather [2],whoargued thattheorigin ‘7 oftheroad instability isthebouncing motion ofthewheel, caused byrandom irregularities ontheground. When thebouncing occurs, thecarisprojected upward along acertain angle andisairborne forabrief time. When itthen strikes theground, thecarcreates acrater andthemotion then repeats ‘ itself. According tothis picture, itisnot thepiling upofgrains ahead of . thevehicle that isresponsible fortheinstability, buttheimpact stress ofthe vehicle ontheground. ‘The wavelength oftheresulting corrugations will be e determined bythecompetition betweenthetypicaldistance thecarfliesover theground and thesize ofthecrater generated bytheimpact stress, which should inturn depend onthehardness oftheground and therelaxation time oftheejected grains. Mather’s picture issimilar toother surface instabilities involving granular materials, inparticular theripple patterns inwind blown _ sand [5,6,7,8],where ejected grains arecarried away bythewind and land inaplace farfrom theejection point. Insuch anonlocal model, what setsthewavelength oftheripple istheratio ofthefluxofthegrains tothe ° appropriately scaled saltation length, i.e.,thedistance thatanejected grain iscarried bythewind. ‘The model wepresent inthis paper builds onMather’s picture, butwe ignore anynonlocal transport ofgrains along theroad, andweassume that, ‘the cars, and more specifically their wheels, generally donot lose contact. ‘with thexoad. Instead, wemodel thecars simply asmasses attached to domped springs. Weassume that thedownward contact forces exerted on theroad bythewheels causes apermanent downward deflection oftheroad surface, byeither erosion orplastic deformation. Ineither case wemodel the effect ofthe contact force onthe road asasimple proportionality between the deformation atany point onthe road and the force exerted onthat point, with aphenomenological “softness” parameter astheproportionslity 2 . Genaewea Aw ae e # constant, Wealsoinclude adiffusive relaxationthat.tendstgsmooththeroad surface, which may come about, forinstance, asaresult ofrain. Wefindthat, the diffusive relaxation isastabilizing effect, asone would expect, but that itspresence orabsence generally hasnoqualitative effect. onthecorrugation phenomena. What isimportant, however, isthephenomenon ofhardening, Weexpect that thesoftness parameter, and also thediffusion coefficient ifit . ispresent, willdecrease astheroad compacts, sothat theroad becomes less susceptible tothepassage ofmore cars once thefirstseveral have compacted itand perhaps produced corrugations. This turns outtobeastabilizing cffoct, narrowing theparameter range inwhich corrugations occur Since wearemodeling thecars, which inreality arecomplicated mechan- icalsystems, bysimple damped harmonic oscillators, thequestion quickly arises astowhat. theappropriate natural frequency might be. Given the observed pitch ofroad corrugations of0.5to1m,andthepresumed speeds intherangeof,say,10to25m/softhecarsthatproduce them,weexpect Poa thattheimportant oscillation modes must havenatural frequencies ofabout pene 0.02 to0.1s.This isoneortwoorders ofmagnitude shorter than thefre. # quency ofthecarbody bouncing onitssuspension, sothat isunlikely tobe_ e therelevant oscillation; indeeddriversnormally adjusttotheroughness oftheroad they areonand slow down toavoid such bouncing. This suggests » that therelevant mode may bethat ofthewheel attached tothesuspension. . Alternatively. theimportant oscillation may beanelastic deformation ofthe’ wheel itself [9].‘This could account foranobserved difference inthepitch of+ corrugations produced byhaxdandsofttires[1]. 2 Model Imagine aflux ofcars traveling with some average horizontal velocity v. along aroad surface whose height above some arbitrary zero level isgiven as afunction oftime tandposition ¢along theroad byH(a, t);seeFig. 1.The cars aresupported onsprings, such that the natural angular frequency of vertical oscillation ofthecars iswy.Further, weassume that thesprings are damped withdamping constant 6.LetZ(:c,t)betheheightofamovingcar, relativetoazerolevelchosenaucthatH(x,t)— Z(2,t)istheamountby whichthespringsarecompressed Vinaframeofreferencemovinghorizontally with the car, the vertical component ofthecar's equation ofmotion comes | Hi % i]Me.bvya ch Zext* Sé Hex) x Figure 1:Aschematic picture ofthecorrugation ofroads. ‘The road surface is subjected toaflux ofcars with horizontal speed v.,and thecars aremodeled asdamped harmonic oscillators with mass M,spring constant Mu, and damping coefficient b.The heights ofthecars and thesurface areZ(s:,t) +¢ e andH(c,t)respectively, where¢istheequilibrium lengthofthespring. simply from Newton's second law, & a SU (a,t) +d£[2(2.t) — H(2.t)] =MEyA(at) +05aet)—H(s.t)] Mas[H(a,t)—Z(z,1)]. (1) ‘The time derivatives here aretotal derivatives: that is,therelevant value ofxistime-dependent, since this equation follows thevertical motion of@ single car. Wemay convert this into anequation fortheheight ofthecarat: afixed «byreplacing thetotal time derivative d/dt with 0/01 +ve(0/d2). ‘Thus inareference frame that isfixed totheroad bed, thevertical equation ofmotion is a,ay a.9a M(5+-2) a+6(Jang.) (2-H) +Mw}(Z —H)=0. (2) Note thatwehaveneglectedanypossiblevariationinthehorizontalvelocity vzofthecars. U/ | Wemust now write anevolution equation fortheheight H(,t) ofthe road surface. We assume first that the road surface sinks atarate which isproportional tothedownward force ontheroad. ‘This may bedue to compaction oftheroad bedunder thesurface ortoejection ofloose grains at thesurface asacarpasses; ineither case weexpect that theproportionality constant between thedownward force ontheroad and itsrate ofsinking will decrease asthe road sinks. That is,the road should harden asmore and more cars pass over it.Inaddition, weassume that there isa“diffusive” relaxation process which tends toeven outanyroughness intheroad surface. This can result from the action ofwind orrain, and may also contain a contribution from thepassage ofthecars, astelooselyconnectedgrainsat yw” thesurfacearefluidizedbytheflowofcars“Again, weexpecttheeffective Reeddiffusioncoefficient: todecreaseascarspassandtheroadhardens.Withyettheseassumptions, theequation ofmotion fortheroadheight reads yy” rp on en2 5 aoKe Fe=DUMae~att)[Mg+Mug(H—2), (3)yxwhereMgistheweightofacar.Weareneglectingthegeometricaldistinction e “DY—vetweenGH/0tandtherateofmotionoftheroadsurfacenormaltoitself,and also thefact that thecompressive force isnot really vertical when H isnotconstant, These aresatisfactory approximations provided thevertical scale ofthesurface corrugations ismuch smaller than thehorizontal scale. However, these effects should beincluded inanymodel ofroad corrugation which isnonlinear intheamplitude ofthecorrugation. ‘Theproportionality factor a(H) above represents thesoftness oftheroad, thatis,itsresponsiveness tocompsessing forces. Ttshould include theflux‘ofcars asamultiplicative factor? astherate ofroad sinking should also dependonthedensityofthetrafficitsustains.Thecompression termshesildbereplaced byzero whenever thequantity inbrackets is_negative”since ‘that represents the situation inwhich the cars areairborne, sothat the compressive force ontheroad would really bezero rather than negative Requiring thattheroadhardenasitcompacts meansthata(andprobably_alsoD)should decrease asHdecreases, sowewant a(#)tobepositiveand_anjncgasing function ofH.Aphysically acceptable form ofa(#) isshown Before proceeding, wefirst nondimensionalize theequations ofmotion for Zand H.Wechoose thetime scale tobe1/wo, thehorizontal length scale to bev./w, andthevertical length scale tobeg/u3. ‘The equation ofmotion ‘ | 0.01 oe 0.008 &00s 0.004 0.002=I “05 0 osH Figure 2:Aphysically plausible form forthesoftness orcompactivity func- tion a(H). Werequire atobeanincreasing function ofHwhich either approaches zero forH-»—co orvanishes forHbelow some fixed, finity”limit. v e forthecarsthenbecomes aay (8 a(5+3)2+Rtg)e-M+z-H=0, (4) and theequation fortheroad surface is oH eH op=Alge-CHL+A2), (5) where thenew dimensionless parameters aregiven by T=b/2Mup, —A(H)=(wo/v) D(H), a(H) =Mupa(H). (6) ‘The parameter I’isaproperty ofthecars alone; thecars’ springs areun- derdamped for P<1.Since the instability iscontrolled bythe interplay ofessentially two mechanisms, ejection ofgrains from thesurface orcom- paction oftheroad bytheflux ofcars and subsequent relaxation oftheroad surface due todiffusion, Pand Awill control the dynamics ofthe surface. Aswewill see, thehardening oftheroad also plays animportant role inthe development oftheinstability. . | 1 83Linear Stability Analysis Wemay solve thesystem ofpartial differential equations (4)and(5)numer- ically, and wewill discuss this below, but useful information regarding the behavior ofthesystem may also beextracted from anapproximate linearstabilityanalysis.‘Thefirststepintheanalysisistorecootize‘that.astime increases, thespatially averaged values ofZ(x,t) andH(e,t) willtendt decrease, reflecting thegradual settling oftheroad bedWe denote this spa- tially averaged, i.e.spatially independent, quiescent solution byH(t) and ok: Za(t).. Substituting this solution into theequations ofmotion, wefind that itsatisfies — y+ 20(Sy—Ho)+Zo—Ho=0. (2) and ; Hy=—a(Ho)(1 +Ho-20), (8) ‘The fullsolution may now bewritten as H(a,t) =Ho(t)+h(a,t), Zw, t)=Zo(t) +2(2,t). (9) e inwhichthefunctions h(x,#)and2(e,2)wilcarryinformation aboutpattern development. Substituting these forms ofH(2,t) andZ(z,t) into (4)and (5),using (7)and(8)forthetime evolution ofZandHo,andexpanding to first order inhand 2,weobtain thefollowing linearized equations ofmotion: a. a\ ee)(5+)242(Rtg)eMte-n=9, (10) Oh .®h Fes AG lh2)Bh, (11) where@andAareevaluated atH=Ho,and0isgivenby , dinaHe B=(1+Ho~Zo)ov(Ht)=alt), a2) ‘The lastequality here follows from (8). Note that #should bepositive, since weexpect that awill increase with Hand decrease with time. Ifaand Aare nontrivial functions ofH,then itisnot simple tosolve thelinearized equations (10) and (11), because thecoefficients inthelatter equation arefunctions oftime. However, wemay perform anapprozimate | | stability analysis byregarding a,8,and Aasconstants, atleast forshort time intervals, and calculating thelinear growth rates that. obtain when those parameters have their current values, ‘Thus wewill determine time depen- dent growth rates andphase velocities, provided wecanascertain thetime dependent forms ofaandA ‘Weproceed byassuming both handztobeproportional toexp(ikx-+ct), wherethelineargrowthrateaisingeneralcomplex. Asusual,Re{a]isthe exponential growth ordecay rate oftheamplitude ofaperturbation with wave number k,andIm{a] isrelated tothephase velocity ¢ofthat mode through ¢=—Jm{o]/k. This substitution yields (9+ik)?+2P(o+ik)+Uz=[1+2(o+ik) (13) and oh=—Ak*h —a(h—z)-Bh. (14) Eliminating hbetween these equations gives thestability relation ((o+ik)?+20(6+ik)+1](8+0+Ak*)+a(0+ik)?=0,(15) efromwhichwecanfiudthegrowth ratesandphasevelocisies ofspatiallysinusoidal perturbations interms oftheir wave numbers. Itispossible todetermine thestability boundary forthemodel, atleast parametrically, from (15). ‘That is,wecould determine thelocus inparameter space onwhich thereal part ofthelinear growth rate o,asafunction ofk,has aglobal maximum atheight Re{a] =0.However, itismuch more instructive tosimplify thealready approximate problem further bytaking aand Atobe small, asweexpect tobethecase once theroad hashad achance toharden sufficiently. Toset.thestage forthis calculation, imagine setting a=0in (15). The cubic equation for¢would then factor. Two ofthesolutions would always have negative real parts, since ¢+ikforthese two solutions would bearoot ofaquadratic with positive coefficients; these represent decaying perturbation modes. The third would be«=—8—Ak?, sothis mode too would always bestable unless 4isalso small, onthesame order asaor smaller. Note that small adoes notnecessarily imply that #must besmall, becausefisthelogarithmic derivative ofa.Since weareinterested infinding parameter ranges forwhich corrugations grow, wenow focus onthecase where a,,and Aareallsmall. Aswejust saw, only oneofthethree solutions forocanpossibly bepositive; toleading 8 3, ea Figure 3:‘The approximate stability diagram inthe(8/a)-(A/a) plane, obtained from (16), forvalues ofthedamping parameter I’,ranging from 0.3 (top) to0.9(bottom) insteps of0.1. Allstability boundaries ForP>0.5 pass through thepoint (0,1). The flatroad surface isstable above andtothe e rightoftheboundary eurvefortheappropriate °value, order inthesmall parameters itis given by 1—k?) —2k =6-AR+aiAoE)Ak u o=~8-ARbak ara (20) ‘Wemay locate anapproximate stability boundary byseeking combina- tions ofparameters forwhich thereal part ofthis has amaximum, as afunction ofwave number k,atRe[c) =0.Thus wesolve theequations Rela] =0andd(Re[o})/dk =0toobtain thecritical values ofB/oandA/a parametrically interms ofk.The results areshown forseveral values ofthe damping parameter I’inFig, 3.Asisclear from (16), both @andArepre- sent stabilizing effects, soforagiven I’,theflatroad surface isstable above and totheright ofthestability boundary curve forthat I,The stability boundaries move down and leftasT’isincreased, sodamping ofthecar's springs also tends tostabilize theflatroad surface. One canshow explicitly that thestability boundary reaches A=0at8/a =[4C(1+D)]"', and it reaches J=0atA/a =[4P(1 —D)}"* forP<1/2orA/a =1forP>1/2 Moreover, forsmall I-that is,when theoscillations ofthecarsprings are very slightly damped ~thestability boundary isgiven approximately bythe 9 Figure 4:The wave number oftheperturbation whose linear growth rate ismaximal, plotted asafunction ofA/a forvarious values ofthedamping parameter [ranging from 0.3(top) to0.9(bottom) insteps of0.1. ‘The dashed curve marks where thelocal maximum inRelo(k)] merges with a localminimumandvanishes.Betweenitandthedottedcurve,Refo}hasa r) localmaximum attheindicated k,buttheglobalmaximum isatk=0. line A+=a/40. ‘Tofind thewave numbers ofthemost important perturbations, wefirst note from (16)thet theparameter @only enters theexpression forthegrowth rate additively. Asaresult, thewave mumber kar atwhich therealpart of«hasitsmaximum isindependent of8;itisgivenexplicitlyby A_(=Bhgg)? =AThie 1el re on Fig. 4shows kinar 888function ofA/c forseveral values ofthedamping constant P. (Below and tothe right ofthe dotted curve, themaximum growth rate isnegative forany positive value of,50theflatroad surface canonly beunstable when A/alpha istotheleftofthedotted curve.) We seethat thewave number ofthemost rapidly growing mode decreases with increasing damping I;italso decreases with increasing A/a, although the dependence onA/ar isweak when Tissmall. Infact, forsmall Twealways have kina =1—T'-+O(T). InnocasedowefindKay tobeabove 1.Recall that kisthewave number oftheperturbation inunits ofwo/v,. Thus the 10 | ., . fact that knaz isalways less than 1means that thewavelength ofthemost rapidly growing perturbation isalways greater than (27/wp) te,thedistance acartravels inone period ofitsnatural oscillation. From theimaginary part of¢weobtainthedriftvelocitycof#pertu ationofwavenumber&(inunitsofvz), 2ar'k? ox~Inlol/k =Ga aa (18) Note thefactor a:since aissmall, thedrift velocity issmall compared tovz, thespeed ofthecars. Also, since acontains afactor ofthenumber ofcars passing perunit time, thedrift isactually afixed distance per passing car, rather than perunit time, The drift velocity ispositive, sothat corrugations arepushed inthedirection that traffic ismoving asthey develop. Itreaches itsmaximum value of@/2P atk=1. 4 Numerical Results e Sincethelinearstability analysis ofthepreceding sectionisonlyapproxi-mate, wehave found ituseful tocheck itsresults bycarrying outnumerical simulations ofthefullmodel given byEqns. (4)and (5). Wechoose a wave number k,choose asinusoidal form fortheinitial H(x) and Z(x) with small amplitudes, and then integrate theequations onaninterval oflength 2n/k with periodic boundary conditions. Wethemonitor theamplitudes and phase velocities ofHand Zostime passes. The logarithmic derivative oftheamplitudes with respect totime gives theinstantaneous exponential growth rate o(t). For thenon-constant acases weusethesimple ansatz a(H) =a9exp(eH), >0, (19) which has the qualitative form shown inFig. 2and ismathematically tractable. ‘The parameter €controls therate atwhich thesoilhardens. We may solve forthetime evolution ofthequiescent road level Hyprovided a issmall, forthen Zovaries onamuch shorter time scale than Ho,and soZo should always remain near Hy. Then (8)reduces to Hy=—a(Ho)=~a0 exp(eHlo). (20) u| | : > ‘This canbeintegrated immediately toyield exp(—¢Ho) =ecto(t +to),where toisaconstant ofintegration. From thisweget 1 O°try (21) and from thedefinition (12) of@wethen get a Bara (22) sothat thecombination /a that appears intheapproximate linear stability analysis isjust the constant ¢. Figs. 5aand 5bshow typical numerical and theoretical determinations of thegrowth rate Re(a(t)] andphase velocity c(t)=—Jmjo(t)]/k foragiven by(19) above and Aconstant, forvarious values ofthehardening parameter = G/a. Asexpected, foragiven value ofktheagreement isquitestrong; this continmes tohold over arange ofapand ¢. Itisclear from thethelinear stability analysis that ifthediffusion pa- rameterAremainsfiniteas«decreases tozero,ormoregenerally if¢/A e goes tozero astheroad compacts, then wewill eventually reach asituation inwhich diffusion dominates thedynamics. From that point ou,any corru- gation intheroad will decay. Ontheother hand, wecertainly expect that asthe road compacts and itssurface hardens, the hardening should inhibit thelateral transfer ofmaterial which wehave modeled asdiffusion, aswell as further compaction oftheroad. Wesee, then, that inorder togetnontrivial corrugations ontheroad surface, wemust have Adecrease atleast asrapidly asarasthe road compacts ‘Toinvestigate thepossibility ofgenerating corrugation patterns, weex- amine thesimplest nontrivial case, namely where Aand ahave thesame H-dependence, A(H) =Agexp(eH) anda(H) =agexp(el). Asabove, this ansatz leads toB/a being constant —specifically, equal to¢~and all three parameters @,8,andAdecreasing ast~)atlong times t.From (16) wethen seethat thelinear growth rate @isalso proportional to¢~!forlarge t.Figures 6aand Gbshow this slow decrease ofthereal and imaginary parts ofa(t) forafewsingle mode solutions, arrived atbynumerical solution and stability analysis. Since oisthelogarithmic time derivative oftheamplitude ofapertur- bation, weseethat theamplitude itself grows ordecays algebraically inthe 2 : . long-time limit: and=o=ite+Ax(t-+to)”. (23) Thus thegrowth rate ofthecorrugation amplitude becomes small atlong times inthismodel, butthecorrugation does notreach atrue steady state. Itisclearly ofinterest todetermine whether restabilized steady states exist. However, such states would depend onahost ofnonlinear effects which areomitted inthemodel embodied inEqns. (4)and (5). While we donotbelieve ourmodel aspresented iscapable ofpredicting steady states, wehave observed aremarkable feature insome ofour numerical calculations. ‘Wehave investigated cases inwhich aand Aareconstants. ‘These cases neglect thehardening effect which, aswesaw above, tends tostabilize the flatroad surface; theparameter 9vanishes. ‘The linear stability analysis is then exact, and weexpect tofind purely exponential growth ordecay ofA. andz,Our numerical calculations show that modes that. arestable according tothelinear stability analysis doindeed decay inamplitude. Modes which arelinearly unstable grow until their amplitudes aresolarge that wehave @ 1+H(c,t)—Z(@,t)<0forpartofthecycle.Whenthishappens, (5)suggests ~unphysically -that thecompression oftheroad isnegative. What ishappening here isthat thecars arelosing contact with theroad. ‘Toavoid havingtheroadsurface spontaneously isewhenanajborie carpassesoverit,weseta=0whenever weareinthis situation. Such adetachment infact does occur inreal situations, and has been termed_“bouncing” {9].Inalaboratory experiment involving tworotating disks that areincontact with each other under static compression, thetwo disks lose contact and bounce against each other when theamplitude ofthe corrugation along theperimeter ofthedisks exceeds about 1/3thestatic compression. Inthecase ofacarmoving onacorrugated surface, such bouncing will kick thecarinto theair, but thecarwill quickly land ina different place, amechanisin suggested byMather [2]. Such abouncing motion should involve alocal flux ofthe cars and asaltation function that relates thelanding position tothestart-off position, asinthecase ofwind- blown sand {5,6,7].Wenote thatsuch anonlocal behavior would bedifficult. toaccount forinthemodel completely, butourmechanism ofsetting@locally tozero captures some ofitsflavor. Ourexpectations arethat themodes that grow large enough tobesubject tothelocal removal ofthetheforce term in(5)willobviously have their growth rates reduced from those predicted 13 | : . bythelinear stability analysis intheabsence ofsuch aforce removal, and that thisreduction ingrowth ratemay besufficient togenerate steady states with finite amplitudes, ‘Totestthisintuition, wenumerically solve thesystem using single cycles ofsinusoidal modes k,inperiodic systems whose lengths areasingle wavelength. Theinitial disturbances arechosen with amplitudes large enough tocause 1+H(2,t) —Z(z,t) tobenegative onoccasion, Provided agiven mode isunstable according tothelinear stability anal- ysis, ournumerical analysis indicates that itevolves toward aquasi-steady state inwhich theamplitude and phase velocity coftheroad corrugations remain fixed, even while theaverage height oftheroad bed continues tosink Fig. 7shows thesteady state amplitudes ofAasafunction of&forT,A, anda asgiven inthefigure caption. Forfixed k,thesteady state amplitudes increase with increasing a.‘Thus weseeforthischoice ofparameters that softer roads (i.e., larger «)support larger disturbances inthesteady state than doharder ones. ‘This result isintuitively satisfying, asweexpect the caseofdeformation ofthematerial tocorrelate with theamplitude ofthedis, turbance itsupports, The phase velocities ofthequasi-steady states exhibit, particularly interesting behavior. Fig, 8ashows theobserved quasi-steady e statephasevelocities, plottedassymbols, andan“envelope” inthek—¢ plane. Ifthere were nostep function intheterm ina,thephase velocities would lieintheregion bounded bytheenvelope. The topcurve would bethe velocity curve forthelargest oused (0.0055) andthebottom curve would be thevelocity curve forthesmallest aused (0.0040). The step fimction inthe compactivity hasforced thephase velocities to“collapse” sothat they seem toliealong ornear asingle curve inthe k—cplane. Examination ofthe quasi-steady state velocities onafiner scale, asshown inFig. 8b,indicates apersistence ofthekind ofvariation ofphase velocity with softness wehavetypically soen(thatis,thesoftertheroad,orequivalently thelargerais,the greater thephase velocity, allother things being equal), though thisvariation exists onamuch finer scale inthesteady state case than inthecase inwhich thestep function isnotinvoked. Even though wehave identified interesting steady state behavior inthe solution ofthedifferential equations inthis limiting case ofabeing constant, wemust reiterate that setting atoaconstant iscertainly unphysical, and that many nonlinear effects have been omitted from ourmodel equations, Moreover, our model does not account carefully forwhat happens when con- tact with the road and cars islost; itmerely turns offthe coupling between, cars androad intheequation ofmotion forH(,2), andfurthermore as- . u : . sumes theevolution ofZ(2,t) remains well described byEq. (4).Thus we cannot expect that ournonlinear calculations willspeak tothebehavior of real roads. Nonetheless, wefind itremarkable that simply setting a=0 when 1+H(z.) —Z(2,t) isnegative seems tocontain thegrowth ofthe unstable modes. 5 Conclusions and Prospects Inthis work, wehave explored theorigin ofthecorrugation instability of dirt roads subjected toaconstant flux ofcars. Wehave presented asimple phenomenological model fortheevolution oftheroad surface and carried outalinear stability analysis touncover thegross features oftheinstability. {From ourapproximate stability analysis and itsnumerical verification, we find that thedynamical processes ofdiffusion andcompression inaroad bed aresufficient togenerate instabilities that, cangiverisetopattern formation. Small-amplitude corrugations ontheroad surface grow when thediffusion parameter A/c and thehardening parameter 8/a liebelow thestability boundary(fortheappropriate dampingcoefficientI’forthecars)shownin 6 Figure 3.From thedefinitions (6)ofaand Ainterms oftheoriginal, dimensional softness anddiffusion coefficients a(H) andD(H), weseethat thedimensionless combination A/a’is inversely proportional tothesquare of Uz,thehorizontal speed ofthecars across theroad. Thus wefind that there isacritical carspeed above which theflatroad surface isunstable. Also, since a(/) isproportional tothefluxofcars, there isacritical flux forany given speed, above which theflatroad isunstable. Both diffusive relaxation and hardening ofthesurface arestabilizing effects. The wave number ofthe most rapidly growing mode isgiven implicitly indimensionless terms by(17), which isplotted inFig. 4. Our model isclearly schematic; aquantitative theory ofroad corrugations would need toincorporate anumiber ofeffects weleftout:ofourcalculations. Wehave notattempted toaccount, forinstance, forthegeometric distinction between Gh/Gt and thenormal velocity oftheroad surface asitcompacts, norforthefact that theforce exerted byamoving caronacorrugated road surface isnot purely vertical. For these reasons alone wehave notcarried outany nonlinear analysis onourmodel ~themodel isexplicitly notvalid beyond linear order intheamplitude oftheincipient corrugation. Wehave also ignored alldetails ofthephysical processes bywhich theroad compacts, 15 50itisnotclear how well onrphenomenological picture ofcompaction being proportional toapplied vertical force orimpulse candescribe thedynainics ofareal road, Our model forthe cars isthe simplest possible, neglecting themultiplicity ofoscillation modes ofreal cars and theHertzian nature ofthecontact force between thetires and theroad. Inparticular, thefact that realvehicles have twoormore pairoftires separated byfixed distances may introduce anew, relevant length scale into theproblem. Finally, a quantitative theory would have toincorporate distributions ofvehicle sizes, weights, oscillation frequencies, speeds, andthelike, Despite thefact that most: nonlinear effects areomitted from ourmodel, wefind particularly interesting themunerical result that simply setting the contact force tozero when theequations would make itnegative canlead to anapparently steady state. Atleast this results inadrastic slowing ofthe dynamias once acertain road configuration hasbeen established. Continuing work will focus ondetermining theselection mechanism ofsuch ‘frozen’ states,andtheirresponsetolocalperturbations. / e References [1]J.Stoddart, R.Smith, and R.M.Carson, Transp. En.J.108, 376 (1982). 2]K.B.Mather, Civ.Eng. Public Works Rev. 57,617,781(1962): Seien- tificAmerican 206(1), 128(1963). [3]J.Papadopoulos, private communication. [4]F.BRelton, Roads andRoad Constr. 16,340(1938). (5]R.A. Bagnold, Proc. Roy. Soc.A187, 594(1936); SeealsoThePhysics ofBlown Sand andDesert Dunes (Morrow, NewYork, 1941). ° (6]K.Pye andH.Tsoar, Aeolian Sand andSend Dunes (Unwin Hyman, London, 1990) (7]H.Nishimori andN.Ouchi, Phys. Rev. Lett. 71,197(1993) [8]D.A.Kurtze,J.A.Both,D.C.Hong,Phys.Rev.E61,6750(2000). {9]R.M.Carson andK.L.Johnson, Wear, 17,59(1971) 16 a > won REGEereeee eee Sts oomoe “on os Re(o) si cn 0.004 as. 0.002 oot : time Figure 5:(a)Linear growth rate Re[a] and(b)phase velocity cvs.time for single mode solutions inthecase a=aoexp(eH) with k=0.90, ag=0.004, T=0.1, A= 0.01, €=0.5circle, 0.1square. 0.05 diamond, 0.01 triangle. Continuous lines arefrom numerical solution ofthe full equations, symbols from theapproximate linear stability analysis. 17 nos teeeee 090|e o.oat 5 300001500 time 008 —— e ° 3001000 F300 time Figure 6:(a)Linear growth rate Re{a] and(b)phase velocity cvs.time forsinglemodesolutions with«=aoexp(eH) andA=Apexp(¢H). Parameter values arek=0.85, ay=0.006, Ap=0.01, P=0.1, €=0.5circle, 0.1 square, 0.05 diamond, 0.01 triangle. Continuous lines arefrom numerical solution ofthefullequations, symbols from theapproximate linear stability analysis. 18 j : u 3 B zy ioswo e‘ <a a a k Figure 7: Quasi-steady state amplitude va. kinthecase ofconstant a, but setting a=0when 1+H~Z <0. Parameter values aroT=0.10, A=0.01, ce=0.0040 circle, 0.0045 square, 0.0050 diamond, 0.0055 triangle, 0.0060 plus. 19 | ate a 003 ———— cms oon © ons oot 00s ° rr a a k 3-08 e 20-045 é 1-04ae 40042 08 09 1k Figure 8: ()Quasi-steady state phase velocity (symbols) ¢vs.#inthe cases described inFig. 7.The upper andlower solid curves mark where ¢(k) would lieifthe force were not removed when 1+ H~Z<0, for«=0.0055 and0.0040 respectively. (b)‘Thefiner structure inthedata presented in(a) isvisible here. Weplot cays =c(a=0.0060) —c(a) vs.&,Parameters are c= 0.0055 circle, 0.0050 square, 0.0045 diamond, 0.0040 triangle, 20 _|hermo {+ -o——- ee en - a eS as S|ae te aBikaginepaodole i ae _oe ee PLT ime SE==eSTO aa 2adh plishedeee oe nanan 207 SG,Come ©7 we ee ie Senge Se oe oo =0wslts LAgaat meee]2Lyaha Ts tenBite, faaat28K . ~a SoS - - -:7 @ wick(dug)4x10 ©SE omer: ADPhowdoeh. onus. Bye CSC ape wee OC” Jew Bare ©sc _ : ee 7 Lgoats190/38 SSBe sued 6x COC vod awe OSS 7 ¥ain. CRE260/82 oC. olLexis” aan e walker ©65? a copes.“yori” : ompyen __Yoo BO Spawn. icsHA -sees, gleaa.fo. -wal $ - a luoedyoly Sno). A. -- weebuckle OS re - .ain, OB 2LL . fon ~2- ww-Codoudhime, te ee Soa re eeeOA \awxlon chywad 0.5can. _@ 86x AMER ct ahathwo akPa 0 aewmedshiteees - a rk nnSe eebeefate,~(Bograedge Top.ee Oo TR oT ed aff Dane Moddo MEQ0% Nasaane!aghyesks (7@Rete! eespallbyXowrdddada0)xy,4px40 7 Oe ee aaa onsa ~~Bunaiiss OiaWnomar tsgnodunsuledn jfounkincam.putaA”>Fom oo de See ae Bo nieJwand)dallyprducetomadoweee———-s-by .0,XO=doumdo ©walter... -.eo NOt lo. - eee coe Ae _ a oo OS,sibeinatte doledlouseheatLaevolt ly +fhe ~—S Ue [|[eet=setae oo :_ _=Yep.=t_6uO. +(100 =asoyh ooeke oe a Sodee |Ne iMiwth =5000 walls. 7 _ -Ce _ oe _- a ~Wasegs — — “St =Some = ayndunnS —AL,050 wate a Se1tye= RB —alors_clwes« ae _ _. SoyGhobec&¢Ciuc =\%b,000tule. ae See WE.ubei ut22°F outside. . fon -3- ____ Qocrineask :Noro walls=YokW tt Vien) aoaROPineesoeCF heedaoeC145.ohnact,2.alent20d]bewhamAucrssvetSeamerabodail.| \@Sinustdan ghdokmakjocon/aenataydeed a-Waorsifogome, Acouiosiooasnllssn sty. ee TS emapanine,aak.Sand denopting —oalegheglae — aSiresvenSenetNaybgtdietehe tatakReon Sr A ASND tathE kal IRACO OsOEW297I oo TOCasdasuie SeeeWDateede aaah)xTho "Going naaOka, soAan.anynidoeayefianhe TON ahalkFides Loi|roe a] ofaRewsoarWart»Whaoshatthe faracde cock|| @leiion 2093 swUSTLR a Cgeramen. YeWaka 30acolt Pom “AQ” =.ae Wot)=S107 T(4a),(looses -a CS BT aif dh jaeeaT=EREWE-BPPBAR BEwade7ChegeSk=e6e =3005-Ai¢=/h761s~BEFaded, a a e"_AASWYOTewattft _ . a7eaiosaaloaa7 Vinkdopudes =Sekeke Boon ee —Ue(SBP SWFSR _=OXSOeET Sk=Tow6HPeake- Siminine=Q5jooo BMJw eq].eee ) /SO OE SPORT Eek TTBS geoteTe __Ctacting pS GSkAreare REL LEYee RE EA TT RE DhcondCte, Paley/®K) R= ——Blaging ComAvis. k=0.96W/m]K (WOmintapha Wer We2.00 WinPK = EA). Dp =RAT =AAT ~ eTeees ==6WwKWOlomEfal.i aL © =Aer =OMe 1 Ww= de ee orSenLa 2Altfladgleqr. ABSem=10"awgayle,0.5mk |22.Atal,aowe] K-- &Ret=Gem=TvaibaKf,adhtnSCBuk2 TO 4 OoArn) 4- 0K we! REf+ Pee|we MRE))eo eee ha patilcaucchinaTackakefa an —Cutakim oak? A=hkAdTee — | BT= hea Teo ig SSS er = Qe WAT RET 6 Tino Badaime Leckless,aadhis, eee nee ew a ae JoeDarYd,BoKmere.thomIleyoo!7 yo —_—Sone_swinanang sguescbinia: ee oeOi Fh tohienanaleaA aaa Ping, EET tek SEAT, SIF ehee haea 1 ————F —3Ss GSS aeTema Uhmtaeagianixondachn Romsinca = (o7_ee =.= // _ ——exagly BTS(GCY=BFema fom p “Ar am)=15)(12)*:on=SI(3y ATa gs OO —..__ a -!rag JubTTX (OY=1 =YO —!J 0d [SecalffGadelsteelaofandogTE “pode [INapce. a a ol cncaiamt8daly ES ReEREKsZO Ceabin otvicar”RENTHERI =~Luh: areWs Adee PB. Wiis BSomR=xi Vl x60kz i panis—So — Vile a 2 fo ste 18ee 08 008-08 —a ay a eHeatFlowthroughWindows Dec28,1984 Istarted into this subject bylooking atthe Utah handbook and also atDave Andrews computer program. The first thing you learn isthe definition of"R" : . . gt og se—>Qe= Axste hor WRA R(AFRe QTu/HR, Insulating materials are rated bytheir Rvalue. “For example, awindow is“approximately Rel" while 3.5-inches ofbatting isR=11. You quickly learn that books and technical materials are inSIunits, soyou have another similar equation, Es mm. a 9 uit, >Q= Aeste~ae 2R am—°K—&("EKRWek The basic conversion factors are as follows: 2 «H-F\_ 527R(mr @twit=Bafa a(S =5 w -2 \m= 10.76 4 . . ave“Kare ap). pataor|Ale)lon),fer(en)37,(oonIK=3°F aie EsGameo|alata Inthe case that the thermal resistance iscaused simply bythe conduction through anon-convecting substance, you can relate Rtothe thermal conductivity k . as follows: ™ sg reCF=k(Wack )xAxAEtem of oe ki@ k= @ For example, consider the 3.5" batting asifitwere just air. Then you can compute asfollows: " ot a >0 R=355sg¥[BE=3.4K=IhoBara r126410 2S BM/ hx. Wecompute 18,buttheactual value is11.Presumably becuase thebatting contains solid material which conducts much better than air, airishumid, etcetc. Ball park OK. -2- Now,ifyoucomputetheconductionresistancethrougha0.25"thickwindow, t)yougetaverylowRvalue: i st O25mm125iow 2 on y R=Sa ee 004mK=033H-°F~awt Vanek LS w arn 30 Ifthis were the actual resistance of window, you would beinreal trouble. Lucklly the main resistance comes from the convection layers oneach side ofthe glass, in particular onthemorecontrolled inside layer. (Handbook waysRabout 1forwindow...) Convection Layers . According tothe quoted equations ofO'Callahans book, here are the equations for theeffective Rvalue (inmeters, K,watt units) ofavertical convection layer: Ly). as is R=(BF) fomiar funichn ckOT) =6 + = g(Ly R=.6Gz)> dwlosteat Thefirstthingyounoticeisthattheresistance decreases asthetemperature difference e increases! (Totally unlike conduction where R=indep oftemperature roughly). Luckily the power isaweak one. For 65Finside and 25F outside, you have dT=40F =25¢. Aso windows are about Imhigh (this isL). Thus, folow isturbulent which means slightly lower R.For25getdT**1/3 =2.9, for100get2.0socall it2.5, no3.0. Thus, yougetR=.6/3 =.2SIunits forasingle layer. InBritish units multiply by5.27 toget R=1,asclaimed! So,with aOF degree difference, you getRel for asingle window. You should regard this asresisteance between isothermal inside window surface and the bulk air temp ofthe room. .. ° . 2 singewind =BT=HOF (85°Fandside) R=. SL funbwdedk downed 1.0 Amunian Waste, _ Moe eK|. co rfa eeRMee e > ge x +2006 : -3- ‘The convection layer isperhaps only 1cmthick. Wehave assumed natural convection. Ifsomething actstomakethisnotso,eg,aheatventblowsdirectly onaglass e window, you have forced convection and resistance decreases, perhaps alot! One assumes that there isawindy exterior soyou cannot assume alayer onthe outside ofthe window. Nevertheless lets compute it: Convection layer oneach side ofwindow. Iftotal temp diff is40F, then only 20F drop ineach convection layer, sodT=12C for each layer. This gives R=.26 rather than .20 for each layer, sototal isR=.52 and R=2.5 inBritish units. You have more than doubled Rbyhaving aIpyer oneach side. Unfortunately you cannot really control the external layer due towind. Storm window. (wide) Lets assume that storm window isthick enough that youhave “bulk inbetween soyou can treat asseries stuff: Ee aefi) Iseiy=?© Now you have not one but three surface convection layers and abulk still air layer as well. Ifthe central area isonly afew inches, you may not have any “still air" in there but rather asingle cell asshown inred for convection. Inthis case you cannot really include acentral bulk air insulator. Solets assume that isthe case. Then roughly youget8Cacross each layer which means R=.3foreach layer soR=0.9 for total system which means R=5roughly for the combined storm window. Ifthere are large leaks where air can flow from the room into the air space, then you are really back tothe single window again, and the second layer has hoeffect, ie, both sides are atthe same temperature. Thus Ithink itisimportant toseal storm layers. Curtains: really the same asthe storm window analysis except you have tomake sure nogaps attop and bottom and air cannot flow through them! Otherwise much less effect. (We have not yet done radiative effects). e ‘ -h- ®AirGapwindows: letsassumegapissosmallthatconvection cannotoccurinside the window. Assume a0.25" gap ofair. Air insulates about 40times better than glass (seeKvalues), sowewould getR=1.3from thegap. (British). Addtothis R=1.3forroomside convection layer sothat R=2.6British forwuch awindow, ascompared to1.0for single window. Iamnot sure what commercial double glass airgaps are. Ifitisonly 1/8th inch gap, then yougetR=2.0fordouble glass. This agrees with Utah book and Andrews program, more orless. Heat Loss thru front wall glass at457 Club .Assume 9panes at1meter each so area =9m. Assume 40F difference soR=1 British. This means R=.2SI. Temp diff is25C. Then Qdot =BX9x25/.2=1125 watts. =3800 BTU/hour. This is 92,000 BTU/day orabout .9therm perdaywhich means 27therms/month =$13.50/month. Sunfe went ahod| HeatLossthrough cetling/ Assume 1000squarefeetofceiling forupstairs. AssumeRell mainly from batting.’ Then Qdot =1000x40/14 3600-BTU/hour .This isi 3333 -another .86therms/day. drside.cmvectinba 180amottrn 4 @ odene3sey” Heat Loss thru walls. Ilooked through back room speaker. Walls aré brick +1.5" batting erushced inthere +sheetrock s0,R=t+4.0+.3=eTsay.Assume1500 ftofwall, soQdot =1500 x40/be?=32,700BTU/houroutthewalls.(I assumedwindows~wall).Thisse.gegtherns/say.!2. Heat Loss thru floor: call itsame asceiling because batting isthe same. Soabout 1.0therm/day through upstairs fllor. Summary: counting only conductive heat loss, wehave this summary: (upstairs only) 25F outside ceiling loss 1.0therm/day 65Finside floor loss 1.0therm/day wall loss 3.0 themm/day (windows all curtained) front glass 1.0therm/day kitchen glass 0.6therm/day Kitchen windows: big ones are 50" x60" each =21square feet =2meters each. Other is60'x34!=14feet =1.3m*sototal isabout 5.3meters, soabout 0.6therns leaks out there ifnot covered. How good isthe covering??? Ball Park? Gas use isabout 10therms/day very worst case, both furnaces, somaybe 7therms/day upstairs, and this agrees nicely with above sum. ~5- eConclusions: asquaremeterofwindowassuming Rellets3therms/month leakouty=Whichis$1.50month, assuming 25°Foutsdie, ie,acoldmonth. Ifyoucoverthat window perfectly, you might reduce the loss byfactor of5.Acrude cover like loose curtains probably buys you afactor of2. What improvements can bemade? 1.Byshutting offthat back room, probably saving 10-15% ofthat $100.00 month upstairs bill because that much less ceiling walls etctoheat. Save $15.00/month. 22 2.Bykeeping all curtains closed over all 3%meters ofglass inhouse, save perhaps effective Reckumexiiit ciHoccicxSixcfexdhmkms Lamm MREOOEE AXEL EMER LEA Check this: backbed has 1.3x2 =2.6 meters. Bed has 1.7 meters. Kitchen has 5.3 meters. Front is9meters. Slide door is3.3 meters. Total upstairs is 22meters. With R-l loss through total upstairs glass would be24therms/day. With curtains raising Rto 2,say,wesave1.1therms/day or33therms/month or$16.00/month. ak, nfsfayWy mathsey 2")puracsitersSSSianinkshy8°17gor. eRadiative LossthroughWindow:Itragicreadsomewherethatthenightskyhaseffectiveradiation temperature ofabout -/5C/ even though this isnot the external air temp. Since OCis‘273 K,wecan compute what 1meter ofglass does inthis ragard: é aw. \ 1QeS9x10- |(ansH8e) —(273-yse)) ]=250wallemtaag*|LP ED1.2410 2.910" This isabout double the conduction loss, bythe way, all the more reason tokeep curtains closed atnight. During the day things are different, especially for southern windows. : Infiltration Losses. Assume aleak that isvery small, perhaps only 1ft?/minute or 60t3/nour. Assume LOFtemp difference. Each cubic foot ofairrequires .02BTU toraise 1F(Utah book page 25). Thus 60ft/nour 40degrees means about 50BTU/hr or 1200 BTU/day or.O1therm/day or1/2¢/day notvery significant. (15¢/month). Ofcourse 100such small leaks could addupto$15.00/month. Based onthis, Iseenoreason to gocrazy plugging little leaks everywhere. 1-24-84 Fuselacesaarouergey)sob: ©damp, Oeewnt LokayeontoanchordaadeJpage—9 fy a ne ¢ ae . 2 \Die He Sis Typrenk wm-windy)aniftnd ? iaste p=a rr oe 5Pe [:OE%10oy/aey. Ap=mkoT Whebons!acum findandsabaVeJOem/ac, ~Yin’fate, : 3 ay. ay. Drmplhahn wet x [Lit/ae =7208Jinn tpSpe i aQincomet LeGoethe dngcomemaytad. : ~ he Co ©opendown: VoaXdeaPr%100K, Somakes 30XRonyrSoca,FrontswredimenbinbteheatooR,Cue© Penta Oeantatweg) seyantag So Ihave been thinking about glass fireplace doors asa"fix" for damper leakage. lets try tomake quantitatice cost analysis ofdoing this. Weknow that our cold months therms useisabout 7therms/day. Roughly atherm costs $20/month. Insummer, youuseabout 1therm/month forhotwater andgasdryer use, that is why summer bill isabout $20/month. Inwinéer this rises to$140/month inthe woxst few moths. Lets think only about these months. The entire house leaks 7therms/day onone ofthese months. IfIwere toleave thedamper open allthe time, andifdowndraft isabout 10cm/sec (move your finger toseethat this isreasonable) or4inch/sec, andifdamper is 100square inches area, then you would lose about 1/3therm/day inthis way. That isfully one third the complete loss through the ceiling, orfloor. That represents anadded $7/month onwinter bill. Even inthis case, you would have togoalot ofyears topay back a$1000 glass screen. With the damper closed, yes there isleakage. Ihave eyeballed itnow. Itis roughly the area shown onright edge ofthis page, its intwo pieces along one edge ofthe damper. Icall it1/4" by12"=3square inches. This is100times enller than when damper i6open! Thus, loss ismaybe 7¢/month. Truly this issomething weneed not worry about. Yes, there will beaslight draft. Candle shows how draft is mich more when damper is open. General conments about heat loss of457 house 1.24.87 @ First,thebigpicture. Backgroua ‘gas‘tse forhotwateranddryerseems tobeabout'l therm) day.Thisis100,000 BTU/day. Thisgasusescosteabout.$20 per month. Nearly all ofthis ishot water use, mainly for showers. Dryer is 25000 BTUgason,soonehour/week at15000 BTUmeans 15,000 BTU/week or1/50 therm day, completely negligeable. Hot water heater holds 40gallons and recovers at33.6gecntons/hour ‘for100Fandfiresat40000BTU,soyoumightsayittakes50000 BTU toheat up40gallons. Iwould guess than myshowers almost use the whole tank, somaybe 1shower =0.4 therm. The rest iedishes and cleaning and other hot water use, laundry, dishwasher, etc. All inall, Iprobably use about 1therm/month hotwater +gasdrier andthis gives abill of$20/month or $21,0/year. Now, total gasbill isabout $700/year. Youmayconclude that theheating bill istherefor about $460/year. Roughly the heating season isNov thru April or6months. Average monthly heating bill istherefore $77. Consider that $460/year heating bill. Wethink weknow how that heat "leeks" outofthehouse. House isabox. Top=15%, floor =15%, walls =35%including draped windows except IRandK,15%thru front LRglass, and 10%thru kitchen glass. Crude estimate: ifyou were toreplace entire IRglass, you would raise Rfrom 1to2,andheat loss would becutfrom 15%to7%,saving about $35/year. However, cutthis inhalf because Ihave plastic on2/3ofthis glass andcurtainuse,80maybesaveabout$i5/year.probablyeachwindowis$200,ortotalwall ewould be$2000 including sliding glass door. Pay back time isthen about 133 years! Room would bealittle warmer, but Ijust can't see the economy ofdoing this! Much ofthe cost isofcourse labor toinstall those new windows. Its like factory airversts installed airinacar. Itmight beworth doing inanew house, but not asaretrofit. Ifyou want itwarner inthere, just upthe heat. Mytopthree plastic sheets probably save 5%oftotal geating bill or$23/year, 0itdoes pay for itself. Ithink this 100 year payback time istypical for awindow:analysis. Goodman said $200 forabig window installéd, and Ithink you only save $5/year atmost, socall it40year payback, not worth it. The fireplace doesnt even register! With damper closed, Ifigured 7¢loss in worst winter month. Definitely glass $1000 deal there not worth itfrom that point ofview. While fire isactually burning, there can beadifference, but not really much. SoOKhereistheconclusion. Youhaveinherited anenergy non-efficient house. Walls are not very good, nor isceiling, nor are windows. The cost ofreally upgrading the whole thing isprobably onthe. order of$5000-$10000. You might save $230 per year, for pay back of20-40 years. Iwould rather some other owner worried about this and not me. Jwill just pass iton. Probably cheapest thing you could doisdouble attic insulation. This would cut maybe 7%or$35 orr annual bill and would cost maybe $300, so7year payback. Even that ismarginal. Morover, heating e costsarenowdropping. ~“ ?fasticundally)<2| ’ t. hey Dovid Swain, Atlanta . You(andyourcomputer) could become pretty popular save usinheating costs over aseason isnotquite ES when word getsoutthatyoucananalyze thebenefits ofaseasy todetermine. Onewaywould betokeep .homeimprovements onfuelbills.Thisprogramisin_recordsofourheatingbillsforoneseason,make .Microsoft (Apple,PET,OSI,etc.)andAtariBASIC. __theimprovement, andthenkeeprecordsofour : heating bills forthenext heating season. There : Lately there hasbeen agreat deal ofinterest in aretwodrawbacks tothismethod. : saving energy inthehome. Nobody needs tobe First, theseverity oftheweather willvary %reminded thatfuelcostsarerising.Weallwantto.fromoneyeartothenext.Ifthefirstyearissevere|°reduce ourenergy bills. Thewaytodothisis andthesécond ismild, ourheating billswould be simple: reduce household energy consumption. _lesseven ifwemade noimprovements. This : There areanumber ofways thiscanbedone. problem canbecorrected byadjusting theheating : Thecheapest wayistochange habits. An __costs using weather data forthetwoyears. : example would besetting thethermostat back to Thesecond andbiggest drawback tothis alower temperature andwearing heavier clothes. Method isthatyoucan’t find outifanimprove-Ifyou'renottookeenonthat,thenextalternativementiscosteffectiveuntilafteryouhaveinstalled Cy @iscimprove theabilityofthehousetoprotect_it.Ifitturnsoutnottobecosteffective, itistooyoufrom theelements. Insulation could beadded latetodecide nottoimplement it! es tothewalls, floors, attic, andheat ducts. Weather-___ What weneed isaway ofpredicting savings. stripping could beapplied towindows anddoors. Ifweknow theweather andtheheat losscharac- Stormwindows anddoorscouldbeadded. teristicsofthehouse,wecanestimatetheheating *Improvements such asthese reduce the cost. Bycalculating theheating costs based on " amount ofheat thatthehouse willlosetothe __,_heat losscharacteristics ofthehouse both before outside. Butwhich oftheabove items would save andafter theimprovements, wecanobtain theusthemostmoney?Whichonewouldcostthe._estimated savingsduetotheimprovements. Thisleast toimplement? Or,better yet,which will". _,is-what theprogram heredoes. give thegreatest savings fortheleast amount of”” Togather thedata needed bytheprogram, cost? It’sthislastquestion wereally want to __*youwillneed tomake some measurements andanswer, . +.dbserveinsulation levelsinyourhouse.ThefirstThebestmeasure ofthecosteffectiveness,of* thing theprogram calculates istheheatlossoftheanenergysavingimprovement isthepayback *house.Heatlossofahousedepends onthreeperiod. That issimply theamount oftime (in things: thethermal resistance, known astheR- years) ittakes forthesavings inenergy costs to‘, "value, ofthestructure; thetotal areaofthestruc- adduptothetotal costofinstalling theimprove- *tireexposed totheelements; andthetemperature ment. Obviously, theitem with theshortest «difference between theinside andoutside ofthe payback period isthebest candidate forim- house. Sowesimply need thearea, R-value, and plementation. Todetermine thepayback period, __thedifference intemperature. .wemust know twothings: how much itwillcost Theonly problem isthatdifferent parts oftomaketheimprovement, andhowmuchitwill__thehousehavedifferent R-values. Windows willsaveusonutility billsforayear(aheating season). have alower R-value thanwalls, forexample. In Obtainingtheimprovement costrequirescon.general,youcandividetheexternalareaofthe e@sultingacontractor or,ifweplantodoitourselves, houseintofivecategories: windows, doors,walls, abuilding supply store. ceiling, andfloor. Theprogram requests informa- . " tion oneach ofthese fivecategories inturn.Predicting Effectiveness Forwindows itrequests height, width,Finding outhow much thetmprovement will number ofwindows (itcalculates total window 84COMPUTE: Jonuary.1983, . areafromtheseitems), andtypeofframeand theprogram. Theprogram willstillgiveyouvalid} °number oflayers ofglass.Thenumber oftypes _results forsavings andpayback. However, usingand/orsizesofwindows isrequested first.Most_thecorrectoutsidedesigntemperature givesyouhouses willhaveseveral sizesofwindows, and _theadvantage ofseeing whatthefurnace sizetheremaybestorm windows onsome andnotonwould beforyourhouse withandwithout theothers.Theprogramallowsforuptotendifferent_improvements. Infact,heatingengineers usethe |typesand/orsizesofwindows. Ifyouneedmore,_samebasicmethodasthisprogram doestosizechange thedimension ofSinstatement 180. furnaces forhouses. Onlyonesizeandtypeofdoorisallowed. If Whentheprogram finishescalculating the youhavesliding glassdoors, youshould consider heatlossofthehouse afterimprovements, itisthemanother typeofwindow. Youneedtoget__ ready todothecostanalysis. Firstyouareaskedtheheight, width, andnumber ofdoors. Re- forthetypeofheating fuelyouuse:electricity,member: theseareexteriordoorsonly. =Information neededforthewallsconsistsofTable4WinterDesignTemperatures type ofconstruction and R-value oftheinsulation 5 ~ inthewall. Ifyouenteranegativenumberforthe cry TEMPERATURE » R-valueofthewalllinsulation, theprogram will *MONTGOMERYAL, 6 giveyoualistoftypicalR-values forwallinsula- JUNEAUAK nei tion.Togettheareaofthewall, theprogram asks PHOENIXAZ«ne oor fortheceiling height, totalperimeter ofthehouse, SACHAMEARA © B *andthenumberofstoriesinthehouse.Thepro- DENVERGO Po Bowegramwillcalculate thegrosswallareafromthis »HARTFORDCONN 5 “dataandsubtractthetotalwindowanddoorarea DOVERTDELEFL :15- toobtaintheproperwallarea. TALLAHASSE 3 . OneHand Calculation HONOLULU HI ~4 BOISEID 10 Theonlytimeyouhavetocalculate areayourself SPANCHIELDIL q isforceilingandfloor.Fortheceiling,youwillbe INDIANAPOLIS IN 4askedforthenumberofinchesofinsulation in DESMOINESIA 3theatticandthetypeofinsulating material, For TOPEKAKS 6 thefloor,thetypeoffoundation isrequested. BATONRonentA 10Inaddition totheheatlossesmentioned so AUGUSTA ME +® . far,therearetwoothers.Thefirstoftheseis BALTIMOREMD. 20 ad infiltration ofoutside airthrough cracks inwin- BOSTON MA 10dowsanddoors.Theprogram asksifthewindows LANSING MI 6anddoors areweather-stripped. Ituses thisinfor- ST.PAULMN a0 JACKSON MS, 24 mationandthetotallength ofthecracks around JeercRSON erryMo 2windows anddoors tocalculate infiltration. The HELENA MT asotherheatlossisintheheatductsfromthefurnace LINCOLNNE ototheheatregisters. Theprogram asks ifyour CARSON CITYNV 7 heatductsareinsulated andwheretheyarelo- SRR NH 7cated.Thisconcludes theinputneeded forcal- SANTATERE iculatingthetotalheatlossofthehouse.Atthis ALBANY NY 3point theheat losses aredisplayed, andyouare RALEIGHNC 20 asked ifyouwish tomake improvements tothe BISMARCKND 219house, COLUMBUS OH 7Iftheanswer is“Y”,youwillbeaskedifyou SXTAHOMA CITYOK 3wishtoimprove eachitem. Youcanmake im- .HARRISBURGPA B .provements tooneitemortoanynumberof PROVIDENCERT. ore~,items.Asyouprobably noticed, thefirstquestion ‘COLUMBIA SC BTyouareaskediswhattheoutsidedesigntemper- NASRVADETN oxi.atureis.Theoutside design temperature formy |NASHVILU eeearea(Atlanta, Georgia) is23degrees. Theoutside SALTLAKECITYUT «3designtemperatures forotherareasaretabulated BURLINGTON VT 7inTable1.Foramorecompletelist,consultone RICHMOND VA Sw t) ofthereferences listedaftheendofthisarticle. f.OLYMPIAWA gBe oFActually, youdonotneedtoputanyspecific MADISONWS. |iciee1Shee. #temperature inhereaslongasitislessthan75 +,CHEYENNE Wy"©“SGES®Beya.# degrees, theinside design temperature used by ERA: Cae SE 8 commer Jerson, — ictidtiniabhdnale [a 2 “+fueloil,ornaturalgas.Nextyoumustinputthe__listcanbefoundinanyofthereferences. Thelast costperfuelunitoftheheatingfuel. thingyoumustinputisthetotalcostoftheim- ©esNote thetthisunitcosts indollars, soif provements youmade. From thisdatathepro- natural gasinyourareais35cents pertherm, you gram calculates thepayback period jnyears.should input .35dollars pertherm. 1gotpretty popular inmyneighborhood Usingthisdataandthe‘heating degreedays,_whenwordgotoutthatmyhomecomputer theprogram calculates thetotalenergy needed tocould calculate howcosteffective itwould beto heatthehouse fortheentiré heating season. The _addinsulation. Ihave alsolearned agreat deal degree daysandname ofthecityareonline7010. about myownhome fromrunning thisprogram. Youshould change thislinetoreflect your own’ Much ofwhatIconcluded waswhatIexpected, location. Some sample degree days fordifferent butsome conclusions surprised me.Theprogram cities arelisted inTable 2,andamore complete _candefinitely help home owners inassessing —____ home energy improvements; itcanalsoenable a Table 2:Yearly Heating Degree Days home owner tospot dishonest “energy-saving” I PEED, Ee a] Schemes pretty quickly.ekORGS«PeeDEGREEDAYS!, 3 >,MONTGOMERY. shoes: 22985Bie’ References Sf)JUNEAUAICHESasha sorsgzate< y,,|1,ASHRAEHandbook1981Fundamentals. Atlanta, RigsPHOENIXAZ 2.53"Sigg“176505.ice Georgia:AmericanSocietyofHeating,Refrigerating SeTLEROCASsotalgage,nats£|andAir-conditioning Engineers,Incorporated, 1981. AsSACRAMENTO CNECTHES.sotgevsis©.to|2.OtherHomesandGarbage,jimLeckie,GilMasters, ESHARTFORDSee Gz3s¢iF"332"|HarryWhitehouse,andLillyYoung.SanFrancisco, EeWILMINGTONDEL <>900 California: SierraClubBooks,1975. - [>TALLAHASSEE FL80411485.93“~~|3.Refrigeration andAir-Conditioning, Air-Conditioning 4 +IREATIANTAGASES GAsDatsS_+|andRefrigeration Institue,EnglewoodCliff,New AigBoIsetone, pene BESaosS Jersey:Prentice-Hall, 1979. fa —SPRINGRELDIE ACP. 5an0> ,INDIANAPOLISING Jf"|5699, +> Program 4:Microsoft BASICQi. DEMONS FE, 8TOPEKAKS@*-3) 05,pt5182 100PRINT"{CLEAR}{2 DOWN}HOMEEN |}.LEXINGTONKY “953°, 4683¢ ERGY PROGRAM [” BATONROUGELA ”:.'* 1560: 116 PRINT: PRINT*PORTLANDME ,~*~‘7511; 126PRINT* BYDAVID SWAIM.AMOREMD afSeat ..|130PRINT" P.0.BOX720126otEANSINGML. Lerte’ 6908 146PRINT ATLANTA, GEORGIA 303 “o»MINNEAI SS. age. . ‘ST.LOUIS MO: >» 4484 16REMCOPYRIGHT 1981DAVID C.SWA wilHELENAMT, ' 8129, IMIr drLINCOLN NE: 5868 179REM otRENONV = |6332*+|189DIMA(6)-Q(6),R(6)/RW(4,3),D(4) for SONCORDNHBe rIW(2,3) ,S(19) incALBUQUERQUENM .4348 390omBray OUENS(5)eC(8)1D '*AeA. te See 200REMWINDOW RVALUES .BISMARCKND~. 3851. 210DATA1.01,2.22,1.815,3.155 * COLUMBUSOF>)y-=*5211 [email protected],1.667,1.437,2.137 * OKLAHOMACITYOK. ..3725 . 230 DATA .909,2,1.724,2.564 |. SALEMOR : |ATS 248 REM DOOR RVALUES *HARRISBURGPA 5251 250DATA .41,.75,-95,l-1 *PROVIDENCERI * 5954. 268REMFLOOR RVALUES ANDTEMP CORCOLUMBIASC’’ “4 2s R trRAPIDCITYSD: 7345 : 278DATA 3.2,0,3-2,30,1.23,0 * NASHVILLETN j.. *A' 3578 wiAUSTINTX 2.Se 7 289REMCEILING INSULATION RPERIN “i SALTLAKECITYUT.+.6052. cH - leBURLINGTON VT+ >8269 298DATA 3.5,3,2-5,4.5,5.5@iwctmonnye sss "5865 300NS(1)="WINDOWS* =NS(2)="DOORS":N OLYMPIAWA™? 3,7. 536-0 $(3)="WALLS"CHARLESTONWV'....- 4476, . 310 NS(4)="CEILING":NS(5)="FLOOR *" |. ,MADISONWS: "iri 7863,7 im 320REMDUCT MULTIPLIERS Plex,CHEYENNEWYS 7" o.,7380 336 DATA .2,.15,.1,-15,.1,.05,-1,.8 aOe FOR ee DATA 0covertrs.69 340 DATA .2,.15,+1,e1,+1,-05,-05,.0 830 INPUT" {CLEAR} {02 DOWN}DO YOU Wr se 51.05 SH TO IMPROVE FLOOR";A$ 350REMAIRCHANGES PERFOOTOFCRA 6.40TPLEPTS(AS,1)="¥" THENGOSUB 5 cK a0 360 DATA 39,74,52,24,32,33 858 INPUT" {CLEAR} {G2 DOWN}DO YOU WI 370 REM READ WINDOW RVALUES SH TO IMPROVE DUCTS";AS ; 38@ FOR F=1 TO 3 860 IF LEFTS(AS,1)="Y" THEN GOSUB 5 396 FOR G=1 TO 4 206 400 READ RW(G,F) 878 GOSUB 6600:REM REPORT RESULTS 410 NEXT G,F 888 Q2=TO/DT 420 REM READ DOOR RVALUES 890 PRINT:PRINT"HIT RETURN TO GET S. 430 FOR I=1 TO 4:READ D(I):NEXT I AVINGS" 44@ REM READ FLOOR RVAL AND TEMP C 900 GET AS:IF AS="" THEN 900 ORR 919GOSUB7600:REM CALCULATE AYEAR . 450 FOR I=1 TO 3:READ RF(I),TC(I):N OF SAVINGS EXT I 999 END 4 460 REM READ INSULATION RPER INCH 198@ REM WINDOW SUBROUTINE g 470 FOR I=1 TO5:READ IC(I):NEXT I 1010 I=1:1F PK>1 THEN 1040 Fes 486REMREADDUCTMULTIPLIERS 1026PRINT" {CLEAR} {DOWN}HOWMANYDIF Ff490 FOR KD=1 TO2 FERENT TYPES OFWINDOWS"; £ 500 FOR K=1 TO3 103@ INPUT NX 3-8510FORJ=1TO3 10401X=1:CW=02A(I) =0:Q(1) <0 ad520READDM(KD,J,K) 1059PRINT" {DOWN} AREWINDOWS WEATHE aes530 NEXT J,K,KD RSTRIPPED"; bed * 540REMREADAIRCHANGES FORINFILT 1668INPUT WWS a;RATION : 1076 IPLEFTS(WWS,1)="¥" THEN IX=2 ee :556FORI=1TO2 1080 FORJ=1TONX xa566 FOR J=1 TO 3 190 PRINT"SIZE";J:IF PK>1 THEN 1166 3 570 READ IW(I,J) 1109 PRINT"NUMBER OFWINDOWS"; e 580 NEXT J,t 1110 INPUT NW 590 REM INSIDE DESIGN TEMPERATURE 1120 PRINT"SIZE OF WINDOWS (H,W) FT" 600 IT=75:PK=1 A 605 GETA$:[FAS=""THENGO5 1130 INPUT H,W 610 PRINT"{CLEAR}{DOWN)WINTER OUTSI 1140 S(J)=H*W*NW 4 DEDESIGNTEMPERATURE"; 115@CW=CW+(H+W)*NW ¥ 620 INPUT OT 1166 A(I)=A(I)+5(3) $ 630DT=1T-oT 1178 PRINT"TYPE OFWINDOWS" $s649 GOSUB 1006:REM WINDOWS 1180 PRINT’ 1, SINGLE GLASS" be 658 GOSUB 2000:REM DOORS 1190 PRINT" 2. SINGLE +STORM" bs 660 GOSUB 3000:REM WALLS 1200 PRINT" 3, DOUBLE PANE" i 670 GOSUB 4000:REM CEILING 1210 PRINT" 4, TRIPLE (DOUBLE +ST i 680 GOSUB 5600:REM FLOOR ORM)" L 696 GOSUB 5260:REM DUCTS 1220 INPUT G 700GOSUB 6000:REM REPORT RESULTS =1239PRINT"TYPE OFWINDOW FRAME"716 Ql=T9/DT : 1240 PRINT" 1,WooD"726PRINT"{DOWN}DO YOUWISHTOMAKE ©1259PRINT" 2,METALORJALOUSE"IMPROVEMENTS?" 1260 PRINT" 3. FIXED" 736 GET AS:1P AS="" THEN 733 1270 INPUT F 740 PK=2:1F AS="N" THEN 999 1280 RM=RW(G,F) 756 INPUT" {CLEAR} (02 DOWN}DO YOU WI 1296 Q(I)=Q(I)+S(3) *DT/RM SH TO IMPROVE WINDOWS";AS 1360R(T)=RM 760IFLEPTS(A$,1)="¥" THENGOSUS 11316PRINT" {CLEAR} {DOWN}"7990 1326 NEXT J 770 INPUT"{CLEAR}{@2 DOWN}DO YOU WI 1336. IN(I) =0.018*DT*IW(IX,F)*CW SH TO IMPROVE DOORS";AS 1349RETURN 780IFLEFTS(AS,1)="¥" THENGOSUB 2280REMDOORS SUBROUTINE ’909 2010 I=2:1F PK>1 THEN 2680 790INPUT" {CLEAR} {62DOWN}DO YOUWI=-202@PRINT" {CLEAR} {DOWN]NUMBER OFDOSH TO IMPROVE WALLS";A$ ORS"; 800IFLEPTS(A$,1)="Y" THENGOSUB 32030INPUTN e“690 2040 PRINT"SIZE OF DOORS (H,W) FT"; 810 INPUT" {CLEAR}{62 DOWN}DO YOU WI 20506 INPUT HW SH TO IMPROVE CEILING";A$ 2060A(T)=HFWEN 820 IFLEFTS(A$,1)="¥" THEN GOSUB 4999 20708 CD=(H+W) *N2COMMsoruon1983 : a ae — *2688 PRINT*{DOWN}TYPE OFDOORS” 3358 RETURN2099 PRINT" 1.WOOD" 3500 REMLIST OFINSULATION RVALUES2108PRINT" 2,WOOD+STORM" 3510PRINT" {CLEAR} {DOWN}LIST OFINSU@2ns PRINT" 3,METALURETHANE CORE LATION RVALUES, WALLS". 3526PRINT" {DOWN} NOINSULATIPRINT 4.M1 LYS'NEC ON(AIR) =.94" ne ETALPOLYSTYRENE 3530PRINT" BATTINSULATION INWA 2130 INPUT T LL=11"2148 R(Z)=D(T) 3540 PRINT" HALF INCH ASPHALT BOA 2158 Q(1)=A(I) *DT/R(I) RD=2.42160DW=138 3550PRINT? 1/2INGYPSUN ORPLAST2170PRINTY(DOWNJARE DOORSWEATHERST 3560parwn’|1/4INWooDFIBERBOA 21SoIPLeers(ows,1)="¥" Tuepues 3570PRINTS, FIRORPINESHEATHE=*DT*DWH =1, 2200IN(T)50.016*DE¥DWFCD 3500print® |3/4INPLYWOOD PANE - D= 1. 3020PRINT(CLEAR)(DOWN)T¥PEOFWALLsoaBRENT:PRINT. . RETURN 3636PRIME’(DOW)|1,JORICKVENEER 4009REMCEILING ROUTINE : Ee Sat 4626HI=.61:H0=.61:1F PK>1THEN4060 : 3070 PRINT "5, MASONRY BLOCK" 4038 PRINT"({CLEAR}{DOWN}WHAT ISTOTA £ ‘|3080 PRINT *6.Loc" LCEILING AREA" 53096 PRINT “7. OTHER:" 4040 PRINT"OF THE HOUSE"; 3100 PRINT * ENTER CALCULATED R 4850 INPUT A(I) F VALUE DIRECTLY" 4060 PRINT"HOW MANY INCHES OFINSULA t 3110 PRINT * WHEN ASKED FOR INS TION INCEILING"; z ULATION RVALUE" 4070 INPUT CI Z 3120 INPUT TY 4080 PRINT"TYPE OFINSULATING MATERI 313@ ONTYGOTO 314¢,3159,3160,3170, AL" 3186,3190,3208 4099 PRINT"{DOWN} 1,FIBERGLASS"3140 RM=.2*3.5:GOTO 3210:REM BRICK 4169 PRINT "2,MINERAL WOOL" ia 315@ RM=.08*5: GOTO 3216:REM STONE 4110 PRINT "3,VERMICULITE ORPERL ‘ 3168 RM=.87: GOTO 3216:REM WOOD ITs" bs 317@ RM=.2*2: GOTO 3210:REM STUCCO 4120 PRINT "4,CELLULOSE FIBER" : 3188 RM=2: GOTO 3210:REM MASONR 4136 PRINT "5.U-F FOAM(DOWN}" 3 y 4140 INPUT T 33190 RM=1.25*8:GOTO 3210:REM LOG 4150 RM=CI*IC(T) 7% 3200 RM=O:REM OTHER 4160 R(I) =HO+RM+HT ee 3219 PRINT" FOR LIST OFRVALUES F 4176 Q(1)=A(I)*DT/R(I) caeORINSULATION" » 4186 RETURN got3220 PRINT" ENTER -1FOR INSULATIO 5000 REMFLOOR ROUTINE fitNRVALUE" 5010 I=S:IF PK>1 THEN 5040 pot3230PRINT"INSULATION RVALUE"; 5020PRINT"{CLEAR} {DOWN}WHAT ISTOTA3240 INPUT RI LFLOOR AREA"; ot 3250 IFRI<@ THEN GOSUB 3500:GOTO 32 5030 INPUT A(T) d30 5640 PRINT"HOW MANY INSOFINSULATIO itd 3260 R(I)=HO+RM+RIFHISIF PK>1 THEN 3 NIN FLOOR"; Tg 349 5050 INPUT FI:IF PK>] THEN 51103270 PRINT"HOW MANY STORIES INHOUSE 5060 PRINT"TYPE OFFOUNDATION" 2% "i 5870 PRINT’ 1.OPEN CRAWLSPACE" P 3280 INPUT NT 5080 PRINT" 2.ENCLOSED CRAWLSPACE 5 3298 PRINT"WHAT ISTHE CEILING HEIGH OR BASEMENT" T(PT)"; 5090PRINT”3.CONCRETESLAB" xq @3306 INPUT CH 5100 INPUT TF 54 3310 PRINT"WHAT IS TOTAL PERIMETER ( 5110 R(I)=HO+FI*3,1+RE(TF)+HI ag Pry"; 5128 Q(I) =A(I) *(DT-TC(TF))/R(T) *d 3320 INPUT P - 5130 RETURN d 3338 A(I) =NT*CH*P=A(1)-A(2) 5200 REM DUCTS : 3349 Q(I)=A(Z) *DT/R(I) 5218 DI=.1 4commaJano.88 | * 5220 IFTP=3 THEN KD=3:RETURN ELUSED" 5230 PRINT"{DOWN}IS YOUR DUCTWORK IN 7049 PRINT" 1,ELECTRICITY" SULATED"; 7650PRINT" 2.NATURAL GAS" 5246INPUT D$:IF PK>1 THEN 5318 7068 PRINT" 3.FUEL OIL" i 5250 PRINT" {DOWN}LOCATION OFHEATDU 7076 INPUT FT:PC=.55| cTs:" 7088 ONPTGOTO 7169,7260,7398 in 5260 PRINT" 1.ATTIC ORCRAWLSPAC 7896 GOTO 7638H EB" 7108 REMELECTRICITY| 5278 PRINT" 2.UNCONDITIONED BASE 7119 PRINT"IS HEATING UNIT AHEAT PUMENT" MP"; | 5288 PRINT" 3.INSLAB FLOOR" 7128 INPUT HP$:ER=3413 1 5298 PRINT" 4.INSIDE CONDITIONED 7136 IFLEFT$(HPS$,1)<>"Y¥" THEN 7150| SPACE" 714@ INPUT"ENTER EEROFHEAT PUMP";E i5300 INPUT KD RrER=ER* 1000 15318RETURN 7158INPUT"AVERAGE $COSTPERKWH";C | 6008 REM WRITE AREPORT O:FUS="KWH" : 6019 PRINT" (CLEAR}","HEAT LOSS EVALU 7169 E1=INT(E1/ER+.5)| ATION" 7165 Ml=E1*CO1 662@ PRINT:PRINT:TQ=0 7178 E2=INT(E2/ER+.5)1 6030 PRINT" ITEM"," AREA"," R~VALUE” 7175 M2=E2*cO +"HEAT LOSS" 7188MS=M1-mM2 |6040 PRINT ,"SQ.FT.",," BTU/HR":PRI 7198GOTO 7460 1NT 7206 REM NATURAL GAS ‘ 6058 FOR I=1 TO5 7210 INPUT"AVERAGE $COST PER THERM ~6060 A(I)=INT(A(I)*160+.5)/100 OFNATURAL GAS";CO 6876R(L)=INT(R(I) *100+.5) /100 7229 EL=INT(E1/(163000*PC)+.5) i6086Q(I)=INT(Q(I)+.5) 7225M1=E1*CO | 6090 PRINT NS(I) ,A(I) -R(I) ,Q(I) 7230 E2=INT(E2/(103000*PC) +.5) H 6100 TA=TA+A(L):TQ=TO+Q(I) 7235M2=E2*CO | 6110 NEXT I 7249 MS=M1-m2 i 6126 REM PRINT INFILTRATION LOSS 7258 FUS="THERMS":GOTO 7400i 6130 PRINT" INFILTRATION",,INT((IN(1) 7386REMFUELOIL i+IN(2))/2+.5) 7310 INPUT"AVERAGE $COST PERGALLON | 6140TQ=TQ+(IN(1)+IN(2))/2 OFFUELOLL";CO 1 6150REMCALCULATE DUCTLOSS 7320E1=INT(E1/(138800*PC)+.5) |6160 X=T0/(A(5) *CH*NT) :3=3:K=3 ~7325 mi=e1*co J6170 IFX<45 THEN K=2 7330 E2=INT(E2/(138000*PC) +.5) 6180IFX<35THENK=1 7335M2=22*cO t6190DI=.15+.85%* (3-K) 7346MS=M1-M2:FUS="GALLONS"6200IFLEFTS$(D$,1)="N" ANDKD<2THE749gREMGIVERESULTS 1 N6240 7416 M1=INT(M1*166)/109| 6205 IFKD>2 THEN DI=0:GOTO 6240 7420 M2=INT(M2*10) /108 |6210 TFOT<15 THEN J=2 7436 MS=INT(MS*100)/10@ 6220IFOT<@THENJ=1 7446INPUT"{DOWN}TOTAL $COSTOFYOU |6236 DI=0M(KD,J,K). R_IMPROVEMENTS";CI ; 6240 PRINT™DUCT LOSS",,,INT(DI*TQ+.5) 745g pa=INT(CI/MS*1000)/1000 i 6256 TQ=TQ+TQ*DIee 7 7460REMREPORT SAVINGS ANDPAYBACK i! 6260 PRINT ,"eee@eeee", ,"@eeeeece: 7478 PRINT" {CLEAR}","ANALYSIS OFIMP7 6270 PRINT" TOTAL", INT(TA),,INT(TQ) ROVEMENTS” 6288 PRINT: PRINT 7480 PRINT:PRINT 6299 PRINT™DESIGN CONDITIONS:" 7498 PRINT,,"ENERGY NEEDED" 6308 PRINT" OUTSIDE DESIGN TEMP";07508PRINT"ORIGINAL HOUSE",E1;PUS |T 7510 PRINT"IMPROVED HOUSE" ,E2;PUS 6318 PRINT" INSIDE DESIGN TEMP";I7520PRINT, ,“eeeeeeaee" :T 7538PRINT, "SAVINGS" ,E1-E2; FUS 6320 PRINT"TEMPERATURE DIFFERENCE";D7548PRINT T7556 PRINT, ,"OPER. COSTS" 6330RETURN 7566 PRINT"ORIGINAL HOUSE","$";M] 7088REM FIND SAVINGS USING DEGREE-D 7576 PRINT“IMPROVED HOUSE","$";M2 .aAYS 7589PRINT, ,"@e@@eeeeee™7810 DD=2961:DD$="ATLANTA GA" 7598 PRINT, "SAVINGS" ,*$";MS .7612 El=INT(Q1*DD*24) 7600 PRINT: PRINT, "PAYBACK" ,PB;"YEARS 7014 E2=INT(Q2*DD*24) . 7636 PRINT"{CLEAR}TYPE OF HEATING FU 7616 PRINT:PRINT 98 owPEM Jono ny, |’ 7626 PRINT"ABOVE ISBASED ONONE YEA 230 DATA .909,2,1.724,2.564 : ROFOPERATION" 240 REM DOOR RVALUES 7630 PRINT"IN. "7DDS 250 DATA .41,.75,.95, 1-1 ; 260REMFLOORRVALUESANDTEMPCORR Oi RETURN 270DATA3.2,0,3.2,30,1.23,0 8900 REM DRAW HOUSE 280 REM CEILING INSULATION RPER INCH 8616 PRINTCHRS(142):PRINT:PRINT:PRIN 290DATA3.5,3,2.5,4.5,5.5 T 300 NS(1)="WINDOWS":NS(11)="DOORS":NS ! 8020PRINTSPC(8);* Til (21)="waLLs™ . 8030 PRINTSPC(8) ;"{REV}) {| 310 NS(31)="CETLING":NS(41)="FLOOR #* OFF)" S15NUC1)972NL (2)#52NL(3)=eNL(49020 8040PRINTSPC(8) j"Tyisieisiei¥ 320REMDUCTMm - ULTIPLIERS 8050 PRINTSPC (8);"ROOOOOF(REV} 330DATA.2,.15,-1,.15,.1,-05,-1,.05,_{OFF}" -05 8060 PRINTSPC(8);"Trisieivisi¥e"""Z¥ 340 DATA .2,.15,.1,.1,-1,-05,.05, 05, .05 8078PRINTSPC(8) ;"T<><>{REV}!{OFF}1< 350REMAIRCHANGES PERFOOTOFCRACK2Ooy{REV}! (OPFJi¥" 360DATA39,74,52,24,52,33 8080PRINTSPC(7);"S8HHHHESHEEHHEHORE ©520FoRpertoe ESaoe” 390FORG=1TO48698 RETURN 400 READ TEMP:RW(G,F)=TEMP 410 NEXT GrNEXT F . 420RENREAD DOOR RVALUES Program2: 430FORT=1TO4:READTEMPsD(1)=TENP: Makethesechanges inProgram4fortheAppleIl, NEXTI100HOME 4VTAB2sPRINT "HOME 440REMREAD FLOOR ®VALAND_TEMP COR ENERGY PROGRAM"en 450FORI=1TO3:READ TEMP:RF(1)=TEMP S510HOME sPRINT "LISTOFINSU, TREAD TEMP: TC(I)“TEMPENERT. ZATIONRVALUES, LS! 460REMREADINSULATION RPERINCH 3520ubRiNt PRINT* NOINS 470FORI=1TO5:READ TEMP:IC(I)=TEMP. NEXT T 40350HOME ¢PRINT “WHAT ISTOTAL 480 REM READ DUCT MULTIPLIERS CEILINGAREA 490FORKD=1TO2 eo”PRINT :PRINT “1.FIBERGL 500 FOR K=1 TO 3 Sesstadl 510FORJ=1TO3 4130PRINT :PRINT "5.U-F FOA 520 READ TEMP: DM(KD, J+K#4)=TEMP M'sPRINT 530NEXT J:NEXT K:NEXT KD S020HOME :PRINT “WHAT ISTOTAL 540 REM READ AIR CHANGES FOR INFILTRA FLOOR AREA?"; TION 5230 PRINT +PRINT “IS YOUR DUCT S50 FOR I=1 TO 2 WORK INSULATED?" 360 FOR Jel TO S 5250 PRINT “LOCATION OFHEAT DUC 570 READ TEMP: IW(1,3)=TEMP 18: 580 NEXT J:NEXT T 6010 HOME +PRINT “HEAT Loss Eva 590 REM INSIDE DESIGN TEMPERATURE LuaTION* 600 IT=75:PK=1 7aa0 PRINT +INPUT "TOTAL &COST 601 7:73? "Press tobegins"; OF YOUR IMPROVEMENTS";CI cosweywea 7470 WOME : PRINT “ANALY: t — ; Jea0 PRINT [onooooooooe 620INPUT OT 8000 RETURN’” 630 DT=1T-0T Boro BOSO"DELETE™ 640 GOSUB 1000:REM WINDOWS 650 GOSUB 2000:REM DOORS 660GOSUB 3000:REM WALLS Program 3:AtariVersion 470GOSUB 4000:REM CEILING680 GOSUB SO00:REN FLOOR 100 POKE 82, 0:PRINT “CCLEARD(2 DOWN? 690 GOSUB S200:REM DUCTS HOME ENERGY PROGRAM” 700 GOSUB 6000:REM REPORT RESULTS 4110 PRINT 3PRINT 710 @1=Ta/DT 150 GOSUB 8000 720 PRINT "DO YOU WISH TO MAKE IMPROV 170 OPEN #1,4,0, "Ks" EMENTS?:"3 180 DIM AC6),G(S),R(),RW(4,5),D(4),1 730 GET #1, A:AS=CHRS(AD W(2,3),3110) 740 K=2:IF AS="N" THEN 999 190 DIN’ RF (3), TC(3) ,NS(5#10),1C(5S),DM 750 PRINT "CCLEAR)(2DOWNDO YOUWISH (2,15), IN(2) ,AS (1) NL (5) TO IMPROVE WINDOWS";;INPUTAS 191DIM WHS (1), DUI), D1), DDS(20),H 760 IFAs="Y" THEN GOSUB 1000 i Ps(1),FUS(10) 770 PRINT "{CLEAR}(2 DOWN3DO YOU WISH 200 REM WINDOW RVALUES TO IMPROVE DOORS";:INPUT As : 210 DATA 1,01,2.22, 1-615, 3.155 780 IF As="Y" THEN GOSUB 2000 i 220 DATA .909, 1.667, 1.43752. 137 790 PRINT "{CLEAR>{2 DOWNDDO YOU WISH 1 seover89comwure 37 | i | + TOIMPROVE WALLS";:INPUT AS 2100PRINT "C3SPACES?2. WOOD+storn /i { 800 IF AS="Y" THEN GOSUB 3000 ” * ! 810 PRINT *(CLEAR?(2DOWN?DO YOUWISH2110PRINT“C3SPACES?S. METALURETHA TOIMPROVE CETLING"3: INPUT As NECORE" 820IFAs="Y" THEN GOSUB 4000 2120 PRINT "(3_SPACES?4. METAL POLYST 850PRINT "CCLEAR}¢2 DOWNDDG You WISH YRENE CORE™ TOIMPROVE FLOOR"; :INPUT As 2130 INPUT T . 840IF AS="Y" THEN GOSUB 5000 2140 RET) =DITD 850 PRINT "(CLEAR}(2 DOWN>DO YOU WISH 2150 Q(1)=A(1)EDT/RET) TOIMPROVE DUCTS";:INPUT AS 2160 Du=138 860 IF AS="Y" THEN GOSUB 5200 2170 PRINT “CDOWN>ARE DOORS WEATHERST 870 GOSUB 6000:REM REPORT RESULTS RIPPED"; 890 @2=Ta/DT 2180 INPUT Dus 8970 PRINT :PRINT “HIT RETURN TO GET S$ 2190 IF DWs="Y" THEN DW=69 AVINGS™ 2200 IN(1)=0.01esDTEDWECD 900 GET #1,A 2210 RETURN 910 GOSUB 7000:REM CALCULATE AYEAR 0 3000 REM WALLS SUBROUTINE FSAVINGS 3010 1=32HO=0.172H1~0.68 999 END 3020 PRINT “CLEAR? (DOWN? TYPE OF WALL 1000 REN WINDOW SUBROUTINE. CONSTRUCTION" 1010 I=1:1F PK>1 THEN 1040 3030 PRINT “CDOWN?<S SPACES?1. BRICK 1020 PRINT "(CLEAR? CDOWN>HOW MANY DIF VENEER" FERENT TYPES OF WINDOWS"; 3040 PRINT "CS SPACES?2. STONE™ 1030 INPUT NX 3050 PRINT “(3 SPACES?S. WOOD SHINGLE 1040 1X=1:CW=02A¢1)=02@¢1)=0 s* 1050 PRINT "(DOWN) ARE WINDOWS WEATHE 3060 PRINT "(3 SPACES>4. STUCCO” RSTRIPPED"; 3070 PRINT “(3 SPACES?S. MASONRY BLOC1060INPUTwus Ke | :1070 IF WHS="Y" THEN Ix=2 3080 PRINT “C3 SPACES)S. LOG” 1080 FOR J=1 TO NX 3090 PRINT "(3 SPACES)7. OTHER:”. 1090 PRINT "SIZE "jJ:IF PK>1 THEN 116 3100 PRINT "(& SPACESIENTER CALCULATE ° DR VALUE DIRECTLY™ 1100 PRINT “NUMBER OF WINDOWS“; 3110 PRINT "Co SPACESDWHEN ASKED FOR 1110 INPUT NW INSULATION R VALUE” 1120 PRINT “SIZE OF WINDOWS (H,W) FT" 3120 INPUT TY ; 3130 ON TY GOTO 3140,3150,3160,3170,3 1130 INPUT H,w 180, 3190, 3200 ' 1140 S(3)=HauENW $140 RM=O.283.5:G0TO 3210:REM BRICK ' 1150 CW=Cu+(H+M) EN 3150RM=0.0685:G0TO 3210:REM STONE | 1160ACT)=ACT)+563) 3160RM=0187:G0TO 3210:REM WOOD 1170 PRINT "TYPE OF WINDOWS" 3170 RN=0.282:G0TO 3210:REM STUCCO |! 1180 PRINT "(3 SPACES}1. SINGLE GLASS 3180 RM=2:GOTO 3210:REM MASONRY . 3190 RN=1.2548:G0TO 3210:REM LOG 1190 PRINT “C3 SPACES?2. SINGLE +STO 3200 RN=O:REN OTHER } eet 5210 PRINT "(3 SPACESIFOR LIST OF RV 1200 PRINT "(3 SPACES}3. DOUBLE PANE“ ALUES FOR INSULATION" 1210 PRINT "(3 SPACES>4. TRIPLE (DOUB 3220 PRINT "(3 SPACES}ENTER -1 FOR IN Les STORM)= SULATION RVALUE” 1220 INPUT 6 3250 PRINT “INSULATION RVALUE"; 1230 PRINT "TYPE OF WINDOW FRAME” 3240 INPUT RI 1240 PRINT “<3 SPACES?1.Woon” 3250IFRICOTHENGOSUB3500:GoTO 323 1250 PRINT “(3 SPACES)2. METAL OR JAL o buse" 3260 ROD SHO*RM+RIGHISIF PKL THEN 33 1260 PRINT "(3 SPACES}S. FIXED” 40 1290 INPUT & 3270 PRINT “HOW MANY STORIES IN HOUSE 1280 RM=RW(G,F) “3 1290 Q(I)=Q(1)+59)#DT/RM 3280INPUT NT 1300 RUDD=RM 3290 PRINT “WHAT IS THE CEILING HETGH 1310 PRINT “{CLEAR} (DOWN "5 TOCETO5 41320 NEXT 3 3300 INPUT CH 1330 INCI) 20. o1esDrerw(1x,F) ecw 3310PRINT“WHATISTOTALPERIMETER ( 1340 RETURN Lata 2000 REM DOORS SUBROUTINE 3320 INPUT P solo Tenth PROT HEN 2050 3330 ACI) aNTECHEP-A(1)—A(2) 2020PRINT "C{CLEAR> (DOWNDNUMBER OFDO3340 Q(T)=AC1) SDT/R(IDors"; 3550 RETURN 2050 INPUT N 3500 REM LIST OF INSULATION RVALUES 2040 PRINT “SIZE OF DOORS (H,W) FT"; 3510 PRINT "(CLEAR>(DOWN>LIST OF INSU 2050 INPUT H.W LATION RVALUES, WALLS" 2060. A(T) =HaWEN 35320 PRINT (DOWN?(8’SPACES?NO INSULA 2070 CD= (Hew) Nn TION (AIR) =294" 2080 PRINT "(DOWN? TYPE OF DOORS“ 3530 PRINT "C4 SPACES)BATT INSULATION 2090 PRINT "C3 SPACES)1. wooD™ IN-WALL =41" 68communeseen83 “|. 3540 PRINT “(4 SPACES)HALF INCH ASPHA 5310 RETURN LT BOARD = 2.4" 6000 REM WRITE A REPORT 3550 PRINT "(3 SPACES)1/2 IN GYPSUM 04010 PRINT "(CLEAR)", "HEAT LOSS EVALU R PLASTER = 1.39" ATION" 3560 PRINT "(4 SPAGES?1/4 IN WOOD FIB 6020 PRINT :PRINT :TO=0 ’ ER BOARD =1.12" 6030 PRINT "ITEN"," AREA"," R-VALUE” 3570 PRINT “(6 SPACES>FIR OR PINE SHE ,"HEAT LOSS™ATHING =1.92" 6040 PRINT ,"SQ.FT.",," BTU/HR*:PRIN \ 3580 PRINT "(6 SPACES)S/4 IN PLYWOOD T 4 PANELS = 1.88" 6050 FOR I=1 To 5 4 3590 PRINT "(13 SPACESI1/2 IN PLYWOOD 6060 ACI) =INT(A(T) £10040.5)/100 = 1.57" 6070 R(T) =INT(R (I) #100+0.5)/100 3600 PRINT :PRINT 6080 Q(T) =INT(Q(I) +0.5) 3610 RETURN 6090 PRINT NS(1810-9, (I-1)B104NL(I)), 4000 REM CEILING ROUTINE ACD RCD),a0 4010 I=4 6100 TA=TA+A(I)2TQ=TO+O (1) : 4020 HINO. 61:HO=0.61:1F PK>1 THEN 406 6110 NEXT I ° 6120 REN PRINT INFILTRATION LOSS 4030PRINT “CCLEAR?(DOWNIWHAT IS TOTA 6130 PRINT "INFILTRATION", yINTCCIN(L) LCEILING AREA” $IN(2))/2+0.5) 4040 PRINT "OF THE HOUSE™; 6140 TO=TQ+ (INCL) #IN(2))/2 4050 INPUT TEMP: A(I) =TEMP 6150 REM CALCULATE DUCT Loss 4060 PRINT “HOW HANY INCHES OF INSULA 6160 X=TQ/(A(S)ECHENT) 23=3zK=S TION IN CEILING"; 6170 IF x<45 THEN K=2 4070 INPUT CI 6180 IF X<35 THEN Kai 4080 PRINT "TYPE OF INSULATING MATERI 6190 DI=0.15+0.058(3-K) aL” 6200 IF Ds=*N* AND KD<2 THEN 6240 4090 PRINT “(DOWN? 1. FIBERGLASS" 6205 IF KD>2 THEN DI=0:GOTO 6240 4100 PRINT " 2. MINERAL WOOL” 6210 IF OT<15 THEN Juz 4150 PRINT." 3. VERMICULITE OR PERLT 6220 IF OT<O THEN J=1 Tes 6230DImDM(KD, J+Ke4) 4120 PRINT" g.CELLULOSE FipeR* 6240 PRINT “DUCT LOSS",,, INT(DIsTO+o. q - 5) 3150 auccrsiccTs o2e0 peme roee gis0 Ruecrarcer) 6260 PRINT ,"C8 RI", ,"¢B RI"17@:% RADHOFRNSHT 4270print*toral "!,rur(ra), ,incre 4180RETURN 4280 PRINT 3000REM FLOOR ROUTINE 6290 PRINT “DESIGN CONDITIONS:” S010I=5:IF PK>1 THEN soso 6500 PRINT “(3SPACES)OUTSIDE DESIGN 5020PRINT "(CLEAR? (DOWN?WHAT 18TOTA TEMPE" ;0TLFLOOR AREA": 7he.soso NUTR AREAS ren 6310 PRINT "(4 SPACES?INSIDE DESIGN T TON IN FLOOR"; maS050 INPUT FISIF PK>1 THEN 5110 pr 5060 PRINT “TYPE OF FOUNDATION" 550 RETURN 5070 PRINT “C3SPACESD1. OPEN CRAMLSP 7000 REMFIND SAVINGS USING DEGREE-DAAce” 5080 PRINT “(3 SPACES)2. ENCLOSED cra 7010 DD=2961:DDS= "ATLANTA GA” 5090 PRINT = =70BEINT "CSSPACES)S. CONCRETE SLA7955 pRINT “(CLEARITYPE OFHEATING FU* 5100 INPUT TF et USED" S110 RCT) HOFF IES, 14RF (TED HI 7o40 PRINT " 1. ELECTRICITY" B1z0 GiIdoacd)SDTTECH))OREDD 7050PRINT*2.NATURALGAS” 288 Rennucts 7070 INPUT FT:PC=0.55 . 2208 Bere 7080 ON FT GOTO 7100,7200, 7500 5220 IF TR=S THEN KD=3:RETURN 7090 Gara 7030 5250 PRINT “CDOWNDIS YOUR DUCTWORK IN 7100 REM ELECTRICITY SuLATED"; 7110PRINT "ISHEATING UNITAHEATPU 5240 INPUT DSiIF PK>1 THEN S310 i4 > 7120 INPUT HPs:eR=3415 5250 PRINT "CDOWNDLOCATION OFHEAT DU5139 IpupacSeys ThieW 7150 5260 PRINT "C4 SPACES?1, ATTIC OR CRA 7140 PRINT “ENTER EER OF HEAT PUMP;: WLSPACE* INPUT ER:ER=ER#1000 5270 PRINT "(4 SPACES?2. UNCONDITIONE 7150 PRINT “AVERAGE $COST PER KWH";: 5280 PRINT "(4 SPACES)S. IN SLAB FLOOD 7160 E1=INT(EL/ER*0.5) A Ro 7165 Mimei sco 5290 PRINT “C4 SPACES>4, INSIDE CONDI 7170 EZ=INT(E2/ER+0.5) \ TIONED SPACE™ 7175 M2=E2%C0 5300 INPUT KD 7180 MS=ni-n2 | Jonsny903 COMME 67 : fe 7190 GOTO 7400 2 7200 REM NATURAL GAS OFNATURAL GAS";3INPUT CO . i 7220 EL=INT(E1/(1030008PC) +0.5) |BBMIS FACTORY PRICING Hl7230 E2=INT(EZ/ (1030008PC) +0.5) ‘| 7235 H2=£28C0 i 7240MS=mi-n2 INSTOCK! IMMEDIATE DELIVERY! ! 7250 FUs="THERMS":GOTO 7400 i 7300 REN FUEL OIL | 7310 PRINT "AVERAGE $COST PER GALLON| OFFUELOIL"3: INPUTCO ]| 7320E1=INT(E1/ (1380008PC) +0.5) NOLOS| 7325Nise18co xen _| 7330 EQ=INT(E2/(1380008PC) +0.5) wos RANS| 7335H2=E28C0 AL 600ARI |“| 7340MS=M1-m2:FUS="GALLONS” ws> |L| 7400REMGIVERESULTS | 7410 ML=INT(H1#100)/100 | a 7420 M2=INT(M2#100)/100 . Hi i 7430 MS=INT(MS8100)/1003IFMS=0THEN Plus Hl :MS=1. 0-05 | H7440PRINT“(DOWND TOTAL$COSTOFYoU (9MPS6550RAMforPET | | R_IMPROVEMENTS" 5:INPUTCI ah’ gine&MPS6530-002,003for KIM: i7450 PB=INT(CI/MS#1000)/1000 SSoueus @MANUALS ' |7460 REM REPORT SAVINGS ANDPAYBACK CROSS Skvn3BkSTATICRAM H i7470PRINT"CCLEAR)", "ANALYSIS OFIMP MABE eeSa i |ROVEMENTS* B i 7480PRINT:PRINT @KIM-4MOTHERBOARD i 7490 PRINT ,,"ENERGY NEEDED“ ©KIM PROMMER 1 7500 PRINT “ORIGINAL HOUSE ",E1;"% “3F KIM-1 &4CompatibleH us vs. EpromProgrammer\ 7510PRINT"IMPROVED HOUSE“,£25" "5F S08 ovMaTH i. us re F| 7520PRINT ,,"¢9RD":? cron? ChipswithtstingA 7530PRINT;*savines",e1-c2;* *;rus|NAM kiMiextEXPANSION BOARD { 7840 PRINT KiM-1 Plugatble PROM, Ram 7 7550 PRINT ,,"OPER. COSTS" ondVOBoard 7560PRINT “ORIGINAL HOUSE",“s"5m1 ©RS-232ADAPTER hi 7570 PRINT "IMPROVED HOUSE")<3"ytd Forkia i7580 PRINT 44°C? R>"2? @POWER SUPPLIES '7590PRINT >"SAVINGS", "$";MS ©KIMREPLACEMENT KEYPAD. ‘ 7800 PRINT PRINT ,"PAYBACK ",PB;" YE ! ars” . i 7610 PRINT :PRINT f eeOROpeEayions PASE ONONEYEA STANDARD MICROSYSTEMS i 7630 PRINT “IN ";DD8 i 2480 RETURN ae AUARTS FLOPPYDSCCONTROLLERS !8010 PRINT :PRINT :PRINT :PRINT ! 8020 POKE 95,0:? 3"(5 SPACES) <I>co?" BAUDRATEGENERATORS CRTCONTROLLERS }8050POKE85,0:? “CH?¢10SORTED CJ)"8040 POKE 85,8:7 “CV} <I> CO>¢1(OICIP (09 C13 CO¥ C19 60> CBD" 8050 POKE 85,027 "CY) CK? CL? CK? CL? «K> CL CK? €L2 CK? CL? (BCS METAR>" FALK-BAKER 8060POKE85,87 “CV?(1)CO)(12CO>(I>.(03 (19 C09 (1) (09 (BP (19 C4 UBD" 8070POKE85,822 "CV>(KI(LI(KD(L?CED ASSOCIATES CY)CK?(LDCKDCL?(BDCDC4ETREca" 9080 POKE 85,777 “C21 Ho" 8090 RETURN e 382FRANKLINAVE.@NUTLEY,NEWJERSEY07110 Sr(201)661-2430 Toreceive adiitional inforrnation from . aavertisersinthisissue,usethehonayreader a service cards intheback ofthemagazine. WAITEORCALFORCATALOG nN i"00COMMITJonson.189 f THENITINOL HEATENGINE [ Discussion and Demonstration a a forthetransformation isdetermined bytheexact x proportionsofnickelandtitaniuminthealloys yi3variations inthecomposition willcause-the trans- aehe, formation tooccurattemperatures whichmaybe 1&3ad belowthefreezingpointofwater,orabovethe et's,— boilingpoint. LO,Ne Se Duringtransformation thealloyundergoes abrupt 7 / a\ changes initsphysical, mechanical andelectronic fe™ a ae “y properties. Itisprimarily thechange inelasticoe RE properties thatmadethematerial interesting forLoge SNNeA i] application toheatengines.Abovethecriticaltem-co ed perature (inthecaseofthematerials usedinthe a. uae If protype enginethisisintheneighborhood ofhotaoess Za tap-water) thematerialissimilartospringsteel, qte AAR>Ly) butbelowthethresholditmaybeaseasilyde- a formed itwillrapidly andforecfully return, onre-@ Sca iud heating,totheshapeithadbeforeityasdeformed.ee “ — Thisistheshape-memory effect onwhich the design oftheoriginal prototype wasbased. The TheBerkeleyPrototype engineusesacranksystemtodeformNitinolwires oncooling, andtotake power offasforce isex- Certain metallic alloys exhibit ashape memory erted onheating. Afteraperiodofoperationinthe effectwhen heated andcooled across aspecific engine however, thewire elements developed an temperature threshold. Oneoftheprojects under increasingly pronounced automatic shape changetheSolatEnergyProgramoftheLawrence Berkeley oncooling.Theunanticipated appearance ofthis Laboratory’sEnergyandEnvironment Division is “double memory” hascontributed toanoverall thedevelopment oflowtemperature heatengines improvement intheperformance oftheprototype based onthisprinciple. Suchengines maymake since itwasfirstdemonstrated inAugust 1973. practical theconversion ofsolar and other forms The machine hasnow made over 21million revo- oflowtemperature thermalenergytousefulmech- lutionswithnodeterioration oftheoriginal Nitinol anicalwork, suchaspowering anelectric generator. working elements. ‘Theshape memory alloyusedintheLBLprototype TheLBLHeatEngine Development Project involves engine isanickel-titanium intermetallic compound. thework ofaninterdisciplinary teamofengineers, named 55Nitinol, whose dynamic properties were inventors andscientists whose backgrounds empha- firstobserved attheNaval Ordnance Laboratory, sizetheareas ofmaterials research, thermody- Silver Spring, Maryland, inthelate1950's. The namics andphysical research. Theproject issup- name Nitinol isanabbreviation oftheelements’ ported bytheU.S.Energy Research andDevelop-symbols NiandTi,andtheinitials ofthelabora- ment Administration. tory.Thealloy,composedofnearlyequalnumbers by @& ofnickelandtitanium atoms, undergoes asolid Ridgway Banks,Techical Asocate, Inventorstatephasetransformation (change incrystal struc- Lawrence Berkeley Laboratoryture)onheating andcooling. Thethermal threshold University ofCalifornia, Berkeley Air Conditioning. @1.Thedensityofdryairat20°Cand1atmosis1.2gm/liter=1.2kg/m?. 2.Thespecific heatatconstant pressure ofairisc,=(7/2) R. 3.R=6.3 joules/mole/deg where deg means Kelvin orCentigrade. 4.Forair, 1mole =26grams (nitrogen) 1BTU =1054 joules Thus,R=.337BTU/m/deg and GC,=12Bru/a?/deg 1.2 °So,oneBTU's cooling power canlower thetemperature of1mofairby1°C. Example: Consider myhomeoffice. Volune isxix}=50m?roughly. Repeat theabove: ©,=0.7BrU/n?-°F whereIconverttoF. Suppose inonehours timeIwanttocoolthisroom10°F. Thiswould require 350BTU/hr. Example:Supposeworldwereat90°Fandmyofficewerecooledto70°F.Ifcobler @ sere turned off, Ibetroom would rise to75°F in5-10minutes. Suppose 5minutes. ‘Then ineffect heat in-flow rate is5°in1/12 hour or60°F/hour. Toexhaust this heat flow would therefore require 2100 BTU/hor. Cost estimate: tocool myoffice might take 2000 BTU/hor typically, sothis is 2500watt machine running half time which is4kwheach hour. At10hps/day that is2.5kwh/day =75kwh/month =$2.25 added onto electric bill each summer month. conversion: 1BTU/hor =.3watts 80 300watts =1000 BTU/hour Efficiency :Let F=heat exhausted/work applied. Ideally, this factor isTyoSTT) Soforanoffice, aT=15°K andyoucould get300/15 =20=F. Butthat isonly theideal. Zemquotes F=§asperhaps typical. Ofcourse F=(cooling watts/ elec watts) soiftheso-called EER=BTUH/elecwatts, then F=(1/3) EER. Soacheap machine with EER-6 has F=2which isnot very impressive. The best machines have EER-9 soFa3. QM Uys” vf ao 6 Messiah andthemeaning ofbrasandkets.1.More than once Ihave been ledback toQMandthequestion: what isaket, - what isabra, what isamatrix element. This time Iwas trying toprove the Wigner Eckart Theorem and came tothe conclusion that Idid not know what Iwas doing. Ithink Messiah understands this stuff. 2.According toMessiah, page 246, aket isone ofthose abstract vectors which span the Hilbert space. The space defined bythese kets isalinear vector space over C,the complex numbers. This space has additional hot- shot niceness properties which make itaHilbert Space. Alinear operator takes any ket into some other ket. Asyet, these aspace has noscalar product, nometric, nonorm, nonothin. Abra isdefined tobeadifferent kind ofvector. Abra isafunction whose argument isone ofthekets. Itissupposed tobealinear function. Bach linear function you can think ofdefines abra. Any linear combination ofsuch functions isalso alinear function, hence, abre. Thus, the brasspanalinearvectorspaceoffunctions calledthedualspace.Note -@_that wecan speak ofafunction asdistinct from its value atsome point (ie, onsome ket.) These bras are supposed tomap the kets into C. Next, weare tosuppose that toeach ket there corresponds abra in thedualspace suchthatthisbraofthatketisapositive realnumber. This number is called the norm of the ket. This correspondence between bra and ket issupposed tobeantilinear: COeewaagale oA.abakSe eamreprnding" bra. _ (= aly+ bl => Qa Sdil+ Vel Note that ifyou defined aset ofkets asanorthonormal basis (in the abstract), the above supposition would lead you tothe vector interp. for "ss the“ket, and‘the transpose star forthe bra. Also, the above guarantees that . the norm will bereal and positive. Finally, itguarartes that. the scalar product will have the star reversal property. <alyp = oye Fe apt . r) 3.Theeffectofanoperator Aonaketisalreadyclear.Theoperation ofAonabra issupposed togive you anew linear function, anew bra. Indeed ef . wy -2- 6 itdoeswiththefollowing definition: . : wuw del=CHA Qn\uy =(lA) ep=Gel(AlW>) You couldd prove rigorously that this new bra sodefined isanew linear function, Thus,(no parentheses areneeded)in thiscase, Later involune two page 638 Messiah discusses antilinear operators, inwhich case the parenthesis _does make adifference, but not here. 4. Inpassing, webhiould have itclear that complex conjugation isonly intro- duced when you write the corresponding bra toaket. You donot get ithere: GA =_2(AW) KL(eA) =&(<x\A) 5.What isthehermitiean conjugate ofanoperator A.?Consider thebra (a/A. Since this isafter all abra, itmust have aconjugate ket. Wecan hopefully obtain this conjugate ketbyoperating on/a)with some linear operator, since conjugation ofket tobra orbra toket isantilinear. This linear e operatorwhichgenerates thatconjugate ketiscalledthehermitean conjugate WD othe - - - yi .+ (<ajA =<ul => itp=Al) detines A . 6.Bytheway,inpassing hereisaproofthatconjugation isanantilinear operation: * 3 * * CALOulderely)=MleMe=AGila]aMG(I) Here Iuseaspecial notation: (x/[/a) ]means "the brathat istheconjugate oftheket/a)".Youcanseetheanjjtlinearity. Theconjugation operation is linear except that, constants get starred. --- _1. Now, once wehave defined thehermitean conjugate inthe above way, it at once follows thatt _ *\ ALA) =[CMer)eny KatA)eng) =<1(A'S) xI KALA LY=CulAtley® eo Qneobservesthat(b/Atmustbetheconjugate ofA/b)also.But(b/herewas. arbitary. Itreally isAtthat isaconjugate operator toA,because Atcan -generate all the conjugate kets that Acan-make, -——-- --- - of ’ -3- 6 8.Messiah goesontodiscuss lucidly allthebasicsoftheQMMath,although I now recall that when Itried toread this once Igot lost inhis discussion ofprojectors. Ishould have just skipped those sections then. Anyway, after @complete and dynamite presentation ofthe ABC's ofmatrices, Messiah starts inonmatrix representations ofbras, kets, and operators. 9.Isure wish that when Iwas studying this QMin221 that someone said: Goand read that chapter 7ofMessiah. IfIever teach the course, that chapter will a _beassigned with projection partially deleted. e _. e@ - — -ee 2 aia Operatorsve‘Wovealumction Quomtum. WMeckamiod Guoss) ©Whaabostiak Walaa sporeQM.woudd sory e Pile=ele> CHO= poe) wre Fieodiieontiol oprate, Noodo wepdate GdoPat @noose Sear stale momolieatems : ep =SG-%) <p’lp> =SC p) Ghamathomdiraiomey aaguanwes drat <mip> =eae =$0 Vine compule <KlPix’oaSlows GAPIN> =Sdpdp!<x\p><pl Ple'><p'e> =Sapap’oPpS(prp’) SPE Can)! =Sap-p Expt(p&-x)) Cry! @ =Sap: [2&]oO) pow =-ia[8e-%y] ®Vos,letVE)=<a")besomemormolitenble wantfumctum, soHwitNH2oOok+0and—-oWw waetamSan? SEIPIED =Sax’GelPiercxdar> =Sax!2dSRI] YO) =Sd! Ley Seow] =poate -Sax! SG VRHE) si £46) =O4®) ©Orquwsrak, wemaydajime MhaCsorigk) womsSametin operat on! <lAly =AVG) areser poemoaeeTaleBERDaeeeomeaaaatinptionwatakin oe ARI =HRI) =GH and Die +cBaseieal mecbomied yields TeSdlmoddgn squatiad. é ~2-’ .lookok rowernane Yon conch2K A. ,©naleook.gagescnet ananand sravnhe ros“rowuw "Wasyire e 3spiteOseeorapeins eens cote SSOnepond ALLdeey=BCR)|Lem> W\ded=ton2ia> WaldmS =am)Qn WLDm>=Com(Qweid <rony =SylSra! A= EZlawirdomal byDefine(Serigt) worrSunclion operatnn ae:Kodth|Qmy=L*K<odided be :o)faantilyranesenigtcpmatn wheomgysanrca B= 2xP @--<¥ 4)WaosolveMesquatine jG)tofindDek <og|Qm> =YouCo,4) <esleod =S(22) 2)VoleUapassing) drokGreatHumetinnd onasoluctionts do » MrSood. ha Www polenkiah isgh: -3take ann.ofreprwacanreonn Sioa gee e £)seOranexghora convemienk laowiasJumetins do ©HoningoidokMur,wawsaale: @DWowaretheondfyrndated QDok we CR bal XD 7 @®Fuak, Sokeomer @).Wewank A= Sok lAd<K1 wher 12> =ASHP =IAS©les. Seats Shag=5@-B) bamad, —<alad =vt8Gra) ; Ons aya =CAND<4thaSedCn>2USTraeShBESMlewey . =Eseee SCN-2) =-Ldg CARD @Draw wecomsony: RVMale =Sake SRVWalROSH 6 =Saw Leia] 21>HRD =-hd¢ VCR) ‘ 1 a. Clearing upthe problem ofactive and passive transformations. sConsider thefollowings _ a a ve) oO) Here,thereisnomentionofx".Youcould aeShia . ~add&@dePinition thatx'==x.Notice * * that innoway isM®aunit operator, : —— —— -~—The-system stays fixed: and the function o changes. Thisisthe"active" wayto A)=TYR) =“¢(-x) do.transformations. Thisistheway -Ithink Tinckham does it(te "countours . move, coord system stays fixed.) Wow consider? Keon Here wehave the passive method of , > doing transformations. ThefunctionJe x)= v(x) stays right where itie(and inthatYO)=WNC) ac)‘ senseMPhereisaunitoperator) butan Pre) (a) thecoorsystem moves, f'(x8) isaAX fs ——functionasseeninthenewsystem. \% Pa 4 x Asafunction ofitsargument, f' - Po _ isadifferent function from f._ 1 ~-¥@)= ¥C8) ++6). — Uycontradiction developedasfollows: startwithf(x)anddoapassive_ transformation.Intheoriginal system nothing hashappened soyouget e Mt(x)=f(x)asshown above, Nowstart withf(x)anddoanactive transformationa6-that youhaveNof(x) =f(-x). ThedIequated thetwotransformations andconcluded that f(x} =f(-x) which isalie. Comment: Once you have chosen atransformation “method” you really have to stick with itoryou will screw up. When an-author uses Pfor parity, he can mean two very different things. Bjorken and Drell seem toconsistently use the passive point-of-view: for example onpage 71: — HEEL Pott ee HO. xex a QER=PAG=Gt) [hiaaroanBeclawGrrfata\ - (2). =P R@= RE) veo 22-2 - This last line shows how BDwould say alegendre function transforms. However, lote-ef books would-tell-you:— - -- -- - - — sk PRS =ACL =cyke) -_—— -—_ That isthe active method agaain. Enough said. ‘Themeaningofatransformation, —QWiashatdunes anon Wak © CostomeBWs. Somhow I-havs developedwuentalblookagaiust ‘understanding something that- Iknow isquite trivial. The context isthe PCT wavefunction and knowing whatitmeansinB+D,but-thetrouble hasnothing todowithdiracia, -Consider the following one-dim mirror "transformation": ‘ MVO=VO) S xlemx 7 a. _- , Wweve) ~ a v v - Fa*« 1-9)=46) - - -- v . x a _— 7 oct"8 ieMTMostintuitively, IthinkofM,assomethingwhichreflectstheKees BHg (x)"inthemirror" togivethenexfunction 9(x)asshown) This isthe active transformation where yon hold tight toyour reference systen and let the "thing" dothe moving. = a ~--- --x. NG 6) eo *» eo |gaye ere) =n _ D . x? Se a(t Thepassiveinterpretation isdifferent. ——_. Pia) Inthiscase,youleavethe"thing" fixedwhere itisbutyouhopinto ng adifferent "system". Tyepictureyouthendrawdependsinwhoseframeatyoudrawit,Drawn inthesameframe asthestart frame youget: €, - ;,--— -4 =i Cie) VO) j oe) - =. we ——, —— Ok ah Sa _ i 4 Onthe right Ihave drawn itasitwould look from the new system (sort ofBeayaxe5)sIfwe8tosorieaiuiiy“Variable 8Wehave: -- 4- H0)-— fp¥O- —- oe -- -- —[=?s _ are -= nyClearly, X(s) andY'(s) arenot thesame function’ ¥(1) =3but ¥'(1) =-.3. Wehave shown that inthi particular case thefollowing istrue: — was— YER ;: wo! od ok e ME)=PE=4G art)=NE1)[=ate,=1G4G)=HOY) fhequestion thatremainsisthis:WHYDOTHEBOOKSPUT|THATPRIMEON¥777Ttheprimedoesn'tseemtomakeanydifference. XA 4, - 4 Butconsider theexample onthenextsheet: embabttio —_ - —Camecd oryose,yeKe ra an ardtypebewefen, Illustration ofthe statement: "Btransforms asapseudovector " Start with acurrent loop and-a-RHCS denoted (x,y,z). Next-toit,draw theparity transformed setup: ¥ an sree - -—~abt_.i's Bese92 Sa|. -ee. Sen sy : a aee a P_ge! x! 6 - -Af). = i Boyan) CH= 6%) May | a Inthe transformed frame wedrew the loop first, then the eurrent, All electron velocities changed ‘sign, asdid all coordinatess Then inthe new-frame I-used the RHrule tofind the direction ofBatthose corresponding points. (This RHrule iscoordinate system independent). . -ne After drawing inthose Bdirections, wenotice that the direction ofBattheparity-related pointisthesame,whichsuggests thatBisapseudo-vector. Notice that the new system (x',y',z') igaleft-handed one, Now we cam practice with the transformation statement, The statement -r)thatBisapséudovector underparityiswritten:~ nd ~ Be es .aN _Bt =2RaX) =+BAO eae Keoh.He+t Tocheckthisoutfortheonepointwehavepicked, out: _ Vat S : 44%Bee)=4C%heaeatsuaBee - +8) =YQ) on _. a Here, B'(x',t') isthe maganticYield observed intheframe S'atS'location x's We-must realize that-B' and Bare-two completely-different functions-of two variables (x,y) here. For example, it —..BS?) =lo-3s4 2st. eee ‘ —Te BEA =wa3s +%st__ _sie _BGS =BC5;¢) Interms ofthe above pictures, let (x,y) =(2,1). Then = >7 - = . ‘~-&@)=42Crorea.ning 4BeyaBey)1ua : > 7 @. Bea=42(macy 4Body)aBie) wee. W aly Syurmasue|), BRYN=nue _- _-an L Noticethat the parity statement does NOT say: —y ||. - ae_—_ —_ ——-St)= 0BaH=BCR SSE eaniteNOguetta UNEN= ROY SO Tua weeee —— a. —-S sks. S+4S8=9 i= s®=Ae +6S. ig Se Anent. SBA ecoy _ReaisBipchinwened:- - -—---- - feats vaeSeelenwm-E AliR-0._ _.© OM.(detbh+SeCeO-EAST =o| — -Aneta apewre—- — -- Voor thes o,Cos: — ae -HEAGHtBE10>sR Roe — wQekeMepodinl aquadions Jnogedaan tania polenta —2NG@\SaSuredann - -- Meodeageater? Tay Rawle. Geen _6 — Be aw@yeigea ds Mi =Ake aaF easy _ eee . _ - _ —S_Betae 4Ce-v-)aso — “Ss amaeaad de2Sa-F aXGuMye 2. TALL, bateens ee, coReled odpak Tg,eS ondgett, NaioeRosaiEyed CoB oe ee 3 yet , ———A22D.9 sion (@seRe ve veoe ee DS GEER ILSed NO)=EVO ~| Sa ee Ee Soins oteenaenaeddal €=2uf- Am _ — \Sats Cendtade} yasof _ oov —fe Qoleri.Sunaseiviiensdnoee ee se(eidcn Sey =oOo waste Woe) x© ja fnBatons)-_—-- les 2mEL w= wh _ ©RaSaagcues. \Ua nee oye eTwoeSa seMee ey Cael shet —-Sdion aquedcn RaBkST= = ~—heCadky _YS =Goren olo’= GEDWsalah =aGck\ho _ . @icceS aSW ME SWgh APS TRCake GeecNyTae KPA Bala KeSEPed’ads) lo —.Qoswominds~----- _wee -———_— _freak ty §yeaut a _ ak OO —‘ — v en —.. |=aGseNev of LL — I aSo oe-2 [Ma Shalev oo . a fae =C=2ach ator Q\eYf ne Geom chem O02 Oo Ga tacSaakg) +ECLA) 4COLLENE SO SSR UR 8|fee $ePERE = MA GaelNa(ente Seyheo * Se Aca sous)Yi—Pw OE_— 2 WT[Sen aahfeeteKeSTS Oo| Nb sqvading solvedoySh=2°(Be) @Mone Ranes-ce Boyee -=— idieras |nert on MW - a,Sk ofa Meagh =ae MN =ora tsalanNoeWwe :~ oe , Rtoy -\ en eee aThy ste-—gaplee =piny)=Vey She pl gray =WENT SaNEOO (OC SN SS0Oe @Ses aseg Cyety OS SeotkbaWUxyake Colayauy = alearcenesra42.alsBayu“akcaravisVlvert) — ste fate 4(aot adsNU afoeyy [vs okhvt XRov| BN_ rae = <2 Wsa@ythy .leced vf RY @ —-_{R= heCoated ¥ ee a-aPRSeeCaS +Leerigtt 4o@-9b a ee esmnSSCO Wanselues 2 BSS2 Qaat-\ te (a-\\oo — ys aeS =GS-8) Ce os ONSgarGay a Lo Banhagas RAhesVYGyet)~ao)RE WE seen asah-aeas] oo EE = wee wth . — oe ee a(axd) 2USSRLESSRGeaaes] OO : woehe ~oe _|_ - Uo=(efof)ey. : . oe[ESGeely |oe (eee) +ZAghaer=¥ (ivy)+owCue)=o ote a ®yttG@atyyh tab=2=ORLY) +Kati Ona) +dasCat=o7 Ss,LS—xQark-DE colon) ® (Gt) Gda8)+Byrd =04 wR eSCate Kg)+ECEolecayox)=07 =>|s+Learet —“Eeey+[oe Worn) x7Jrg=O worded VoEGE=KX(ax) @ Ooner cmeoe P+ef +ami =o Xe R= Gate —BRK | = x aedgO)=abankale+UBCorwV % feta: FG)=xXEa Car’) wise. Lstk4B=GR= YOu . e us ASsafe aOmelasCN~GSMgLegaedewe Sey iy dea PekSe d= 1ACasnt) +aG@-2) oe Mea akom2sDeseo 9 le TLgah Dn TT SS eer 7 Sun 84GATS RW 0 oe eae TS vlonddim dow Sako leme fiaie, topalea _Neugeli MaeiahNeGyy EY STSRE AGEMG@e) TOT | Be _ gun sonishsabusbiwo 2-0N00 - Ss SoY=aOn! ReZoran =Aaone LeAdare =ZFaatnndin) |eanGnayGenteC200) OnanOO Ger Gains a GanGood Ora rts} ry nat Boar Cree) 7 - DS date rn) PM ne La BnGL (eho ae Cee) on™ ~Lom mete A ne Sor GearQuer} 2°]4an(ewe e-Pa> & nares Khea,Temteel=0. | Qua Borat \~< a a \nakaos QueSuumcabres condtui: LaGes= @—2how)-wks=LEBehe) A Qn. = Qon>€%6*(aie = (Lo®O/ 2axc\ 0, oG2 Grr” - =(WEE V%3~C)/2-€ neWoaarEe}a dor Fo(wGeesN\ a. SorWoomeRony naluadra anh _s Te S,TatReways mam eS@Q=Zhawt ZTE) EDERUER Ga) = “Too oak. Bas YoigBO — @ ee See Taan, OST __a nr, Sa ee KT 7 oo Tee eee__ SpeDeSoBeBMG) ____ ts BS Raabe so Ty Salen OgiegSE a aa SS et | eS neee-- Sa ares aed 4SEGhor§) -sale Sse 8ee Os ——Grosse Docesighscanengga — oo—Do wBus p=DK=Idx5 ---—___--—. ---—ee a(x)st.€=Be=2a\e oe Ss $20,265Aas| gcaswks ods DORI =EBA ~7Weama \ea2d neLZSate SsSegoe a alana) peek Cosy-—_. =Ses~K- — "OM ok2¢4as (5eek 22.Geo (ante) =~) aa-rel)_s.a(-an) =~ata+) >ont (yeeeE TO Ne smGemriey- »YoSeunmwasgt 2—--- ee -erECAR%D)BAoak a Peaee ok nd Ce Que, sabudin ie ee woe an ee aeoe RSE Te a not¥Os Seey ec ee fo [osm meaGSaz oth Ookalg HewSheLOG) LeG@') ee eee xeWeey 8 Aw. =Zann --— .« —~ gee pes, - ~mate:RS Ee Ae SxO™ ae OGawd VoGt)= t oe Quantum Mechanics. @., quantum mechanics, certain quantities arequantized: 1.a} Inthe analysis ofa"particle inabox", (infinitely high potential walls), what causes the energy tobequantized? 2.{is} Consider the more complicated case ofthe hydrogen atom,erenyteentrei-_pobentiel— problems Asyouknow, thewavefunction maybefactorized into theform ‘Y*(r,0,f) =fey(r)Yy(218) where Yisaspherical harmonic, andfistheraddal wavefunction. Theangular momentum Qisquantized sothat A=0,1,2.... (in units offi),andtheenergy isaksoquantized ,theeigenvelues beinggiven byt 2 2=-2 wheren=1,2,3... anda,= ne,OFee 2a,n @)tmthisproblem, whatisitwhich causes thegnaukxx angular momentum andenergy tobequantized? (The following two parts ofthis problem are hints for this part.) (\p)Theradial equation forthehydrogen atomcontains theenergyE.Foranyvalue (thi ofBthisdifferential equation certainly hassolutions. ffEisnotaneigenfalue, ewhatcould yousayabout theredial wavefunction ap2 : (Q)Whenm=0,¥gq(018)=Pg(cos®),aLegendre polynomial. Itispossible toextend theméaning ofthefunction Pg(2))ftoallrealvalues (infactcomplex values) of theparameter 4. IfJisareal number butisnotaninteger, whet would you guess would betrueabout thefollowing integral: ¢anduh.) \ IosSCPp(2)Pa (3) ~ pouch}xConsider theearth andsunasa,qua ntum mechanical system like thehydrogen atom. Isthe energy ofthe earth quantized? Isthe angular momentum ofthe eearth quantized? fajxRampute the"Bobr redius" fortheearth-sun problem (Roywh Please make the following computations only towithin 10orders ofmagnitude: (tmaahax (a)Compute the"Bohr radius" oftheearth-sun problem. sees) (b) What isthe energy ofthe ground state ofthe earth-sun system (the “Rhydberg") (c) What infact isthe energy ofthe earth (ofcourse ignoring all other e celestial objectsandsettingE-Oforinfinite separation)." (4) What isthe prinipal quantum number mixthexemkh nofthe earth? (e) Comment o) 7theroleofgnincelestial mechanies. . , Relativity a. @2)= =NapOkdxt =>Aw.dk== Ca: ds AS (teePtemdat =medxt (one —_____ Woutna IndBaoind: P=PAomdgiy— _ ;AG_ ce a Tom aekao: mat Sm tsJeTom (2) unebud:A=omafEaux N=ple Basa dkGy aaoma©.ExonENE (ex\,d8—9. _Frond Xirl>\amd1). Gua,amtWoldsofON oePO ——@ WeuawkYouscodundtdobeoauas dks. So,neplase(0)opmuaatlig cnn — +o —o= FB sato > Ofde0 odae “a = =A(Ede) Edat swede= ®OstrmaMy, Gro Aadprnabeinn eh=Edy DMG — ié) ee _ I. ; —_usham\emnemedas ocoaderabing igzeaecnntrbent a ae 7 AS anVORvOrNETINTOne)%-“eamlngManeiaHT7 papa tig OW albaesrChor,unpunsanh—oealyerd AS_ bth necomee dS agt=Egshe,0,0,Ggsh6,0,0, goo) a = 6 fs=3) AS okdy \ a, —0Bae a a ode =shor_- ’ —@)+0)sayahd FESghe _ _ Sn ee _--—oseel ronanitUnSakdintS20\ytoegh . he Qedne A @duglasadume 7 7Te) I@ake=GOSk oSBE 0) 6@)_= dor=ch3)= Nv =oe @dk=cheAS=chGS)ds _ —@_dxaW@3d= Gags)dR Repl 5paaBitesoeZUG IAEAa é sof a. 5 WB WeaonTroe 008)casnck(5)Whack ontVE] ound ml)? w= VG)=On(30)=MWTMGY= g ; gud¥-dv ¢- =q—\ cea I[siWsCc r =of he squockdont unt¢one: squoskiacs conf,©ea — (Lax > .. ‘ Y= 3.15 %I0 ae — a Yu=a:bom oa,ax0* b_.105 @ Luss hbaac b=3.15xso7 & (on qeARem|e [eee=AEDyyt\ oo aa Lame]braye| °4 [ej ™~ .+ @bon7Soaissfa . % . y=oh(S\=\ (a)Taleo : 3 x x . \ BA) 15 _ 3.6 3.8 : 3 0.0 0. cs : 6} @ Vowanask gens gS>>|bogahGeaaOorhauslit ©epdogalocks conduct mag?Cuosletee=(5,0Le :~fe? . _ aQn2xmeDon30,000x.03h.pA) 6 Sa_quarkun. eg,=10.3year=((Momubile ,b=ck(S) x22 og dg LeISOb yore . , s ™ eu (onnsd.rocleck) 3. _ dleatk YOyeaa cavniojaan Kawa. (heoor(h urmdch ska\ ch 2.6%, Noonssk Slaaah4.3.0 . Wdaanedo ookawi$=oe(oti)xQm[2x WO), sirppoas SfCat=waYar) SokXoo=axDy a Rett:Sw=Onlaict= ma+23m So, Finamdemoda a= 2.n=6 =SW= b4$13.8=Sy Aosunas 4y2>108year —_ @shalhipdepanUeda 8aDy: TEE TES _.Syy=obs =_ok = ao=2.5yeas dy=Sh(5,)=sh(ja)=3.26 £=[3.0year. monk, rude do molepel., mn phradex “ al. ee aePSa? ok dp Sas 0 nak. Mm ° =. oa ;Feof= =FPS Poxsoaafal ilkoadYui pS=F =E¥8 Qiao, weemmcduncke: omdp?=ef:p=ekeefee — 6 em AE eeBp ee ag eg a DS SiiceKEWOD, p=PAO) amdpazpA),amd en=dkay=Vag aC<-> Co es a 9B seetev xb (em imopooabad) a 2egEOap 2[EsaB[ —-}, sounaetaerahonscam lorchenteedjonDarseemedariate Wig aAld candi id._causAobed AVI Wiweare, ous ust nosroaiherr (flak: “a ee Sey: ox eRe pws ed®. a - : = <a ce arefae _ tenn, =eB _ — a es | a a —@ Example: atPEP,R=20m=22W0gs.ac=SHageror fn - —R= WPEs) oprxoEC)GEE DRG) Reg) yaher - Qya)=lbw? Ee st 4 _arnajaDumalatien otondaae— E= [56V =@=5,00qaue .¢ E=200d) =)8=23,000 gauee <—mdSuprcenduclera . —) ©Fist,0golitean AracussionVesaatadnouda.canoe tJABSABaer |Pamdhoard. olosenasrs ——— -f£ Wiis ore — ingfe EXNO. Omsock, wesholosomols onLimea7Aycpordinabe quisQhaasspade+outoeSg -CAD Re|ptsSO | _Ganuwa Dek!fesseVi,sowhewweboopontobedain,adie onsen agen): a a -, _ThagoutfedecreeTE aid)do Nappa 2 a - SS oe OfomusseQuacassrralosske,ts aod inoobeien |LL ee ee | — -aa _Wabremak stl4spaaliust., Qhsooa'scpa(ohAsstonfieoor__. a_awustaCunatigha.“baobathsboatasmbydhehoin,___]____cosenvay, ‘Tem\DaaneaDaChegalled grees—lmests_ __=—Bonmmuke|soflrsagsaathULustalonask”(uitla resteaAoketem) 2 Re ReEL RS -——-Le boost2. . So ByBax5“BilFae=|atx[=BB] aa —MeSvccétquassatnas of|h2.G.G.. comme ond ole - eo@Povscae: vadmaleswattog, i egy sae —. __adieded) ADIOIae + + 7PaKy=100[ostmaalbaa.fame“Ke,Somansolaaenata)Ba ae chapel _ ©Apodeer desea in.DedeKOaeDatevecknN71, WhackTeNyt Omemstdod—boosholsomesok.___a AR geeee ee -te, WeRRGMM Unodpwieqi00Qrsaesuemth? MMesos,soa ©HAGEQR) Bay25hNeYonBlansoakonions sane |. cnn _ | ee 8 1Nr=Bs6)Bal) =aV|fStansonedeuist,alooQrsasorgee ~@NeateleSkVShst La¥ eee a ee eee a 7 Yauikake" “6 So . she Gey 9|fo fw Ge ee L--- a iStoejoeee ae - @.©TREO OTP SRTSYP| f=[RS 8|]smohxe |RACv)ER(rs)fee N°}]9phtet2]. wpe—2wpe ego 'ibeepety oolal | deethe emsiwe xeOy)~G——sh sieht emoig ga glagis en Oe ene) OsTh Lye.|Oe_- es 8||A °o4 ---.! - — Lio @..:10. PL) -oeott -yfaoH =o — ee +Lbes] {-oarne*] ee - I lan basedworthayes _LIBEloss.Rte Gd senGuaaspadewachoxRaeangabeCompan] Se deat_.toaeeenaleCg AhaBatata askgad]MaccMakqeondsaan AGuonapasesondWy1HJase —O DaahendShgueUe&Guasieanualapankoee 4 ow OM . pwr. Rock AtoDurotan.’ Geaadiiit, bspraviovedaaswecem Vwiebok ---— -omolamgaUKTaste a - —____- ee —A0= Arjde ___SUNSeeensenategatiysenttaaih —S 7ihety“What oon!BanoodaTLednewands panenssrind akOy,(an. —@BxramaT,—hnasnnen aaaawinguue eddinedo ——4|aa penton Sucolung olfdoDarDepST WRG ghalmealapgtaeete —@Aa8sctles daa aieqnunntc Zhloadcg aa ee awee 12Wp) .©.bnwiortiadfoeT,De!Spy,prarsaied dasotopaaamuse of _an BRE Ook a ©WosBochon LonQala arclakdingSpoely=Norbu --=.Olaradriny bovateolCRiulim QR,— ayet -, asee —ee ee ee Se Cee Rikhee,p=Orde| —Thauraa_parsaacon|a ~~@®Soin(ieadgcban wotpastacai,aqsacch,pedaunbehs -@ AS eBBexSOne odSee, eerRL, 3eeoe .afr[esa5]--849 aen en a SO@Wasa Deachowes achivemiagutke. dandd oOAREA Reshetacwadaenatajestusoagealie fall <BR2 ’ y|AWa areesee me yhewe Fal aq ~ aoeyoe —— a—Ra,oaun)® =,Ove : So l_——- eepg O cg ageTrentCARE a ap- Venti adaCosas ANTa2 ae"spinmtit ernjimaKee dnSO nl a aa x + 1. ae, ee EySauliong—wan NeSafe OOgay oFCQuman pasiuin) BO ~ @®Nnocrutd eatomagauas Duty? grokkOpoxide co EO BY=EDSTHIDSSine a a ee RN). tee AUD 7 Sxgstinpak vsmddmde,WOayoQaakMeeTamas Jacke a Gassa—| ———1 {-toe ._-@Wuonetin|ilMonaaWs(gag)Seca, IEA|0"p86k(i962)ee neBatenet guean Teka Cataa) dadanantalltQuDeroMau:. —gt mt tpMarian =PhyBest=PraBot a ay ia Paeseanenenn ees——— —eee --4Weian)BedL 1. -os ee enee RE _Were~fige WLS pate, omd[pg ="7WY se wn.SOTTO Etat]a eeeeon)AS0ThFew SY—— htSak =a=i - @Fanen, stake|L,3>)hespltbaiy veent.VPosdien- Rack|_| —a4=ZL(SSAnte — —eswom =Vind rasan=YiandessVbar|6aJdaSeRHSach FFAigchsaans poadon): gauoa Sodus .“TelawASPROBLEM, ~*7Weckomes. Naw): OX _dasuny i ©< Dasaninobel“Be=wel.| NRE GE Ro,afousautt poadidand TAF Baalcesta withaoerand |z Spin Siakaoatyobvimal) 4 aAnsaersiig, do.Golshu, WhoFaoiayslasnaensates. —,—___oe RS =OA —_- 1 a ' ee kek Wigpaceshae _ Se a a . Martin /Macaiivan Soden) Os eres naeeenSee MWewnaOritKGL YdadeqeSxSunde ded_doatiaeup1awet ia,anddondsy furvnschany Lo.--dd,=pising: %~phe___.__ | @©Wrnpa Oepiou,Mkosama OaSey | Fetcilly, oshave Wapeb=gTS.T] Tine cetg_pakalS, amedocbodenorQueQi deestsGotBefcanene okOeS _slat t<S,aneash’8 ~~. a .en ch re _(ussays(hakGeSuny pussadea anacont. _. ips Reydegoorsundiausa PodeinS=spesei © Dy-“dex Gein=ASSpBakdak—— eS\seme|)yy=$8=fpRDaaOR:tate SEEa| We Tt . a “2.Connon Pasnooind splnmileonas.dapeda. a BDYraomshigPEycdanaga ©Te a ES 7ala! A anaesagile FaansSechou, puboga A=Ateei=SWSwe_ones oeASMoersnaMak BS Shean oR _ Ce -—_—-.—[mat eie0 v Nae WM,Yt|ee tec aaee[\ ee _—- Isgace 7 4 12, 2008_quy seytSSAswondaad bys gh eS XBSRS RSET TV Coit) =ESE RY ae a aeda -7~eea —— TTT rn as — ayarticle e070 Visngdob,| 2 CMMonMtrmam SUB) MH oe Me Wt woe Wrasse duscoony IN eethe-XBox syeen. .-.BSTCMwann-faia -- faut nner 195%———CoEMu__Leelfony.a ASS GoDWon=Meabiypin Ngeea[EN WF820 --————WCesas.[158ClMaun[Figromane.- MorTaactieay [950 _ee - ( fl —.. -Geidtugu.\Waliw Footaste, _pS47equation Crab). __ ABWardtoSieutyheaxpasssion buO26)vaJatonoguasiade of Sip. Fistweona_nacoquive Qaotarose dondiagale os. erin motedion Sa“stele UsuallywanfuiAastakeeuag ——seaitsing lea,Bonibost” SeiwaCite)ittacksRuler —Ranekein, alt,ikJasonsnckOncZana)pentains Desenabcnuan— aDepabiasofDarsiede”Cit).DbiaaDosoasenceQeek —Cepia aSake”Gtsowaweluusken”, o_alistale olosed auto eteoad. Badusuma,OeZogidn) rmeama.ttMogowYoul§9). —enmachin JamassadeZog|be =CogLE[a>=omdo)bs Soy gpgbeckde0%)awh(leh),peoui eptiaskl de ~rotentnke Top=BegsSeTySy)mnOssdedeCie),weewkd Taste SyaeachingguiaUspyudanan Seponl ow Pa NinwenGavel, Wbden dosQans C120),atslam tin(2a)2 Rivage,SeeUse=SUdeneSeVin=oeYoon Ga Uigysteokescarssanclessd [159]heen BaeVionSiedeeaad tise <eplu> Awa,woeuaeCrackanadenal semanofSop, wat Dowandinale cog.eeiatofSe oewer, atdanwaackHotenecokonsedtodeciigHarn enplick, Rawaihe CWee: 2 Wow,ssebeseDroTaeSsoausdinQuiasoay Cepuval3. Byortle) oobeondeonMaspooinwaockade LY.Oiaiabesos Semdt! dematappeartnteeJustpoten. WowubdassotQaot TTinomsiguesteled 5became, Grnwanedonsnanans Cast, “ay,Eq.2.40.935.4Wilde), - - ZLn!syQ<daisyomy =|T's'w> _ wnatWataansangECR)Ug=Wals'y>.Abasosceail, NUMA)wey=CRs), acOaae)stele,so: Z.cham Xtuv'lwS =Zana =<alsnd. feQue.sandaseoATp. oe / ~ en a ae f r te ota6PACSGTein6 OD ower wedim) Bot oe —©sees NGSGmeOsQosagdea :MsGino -<alyicotmia> —— £o\9i'60WiCO> _ _@&Make ThePCaGeneralization: Boe oo <oLApCle> =Be<b13.Zoom. pspacpa.oo ARColo" Goled=-ErCblfcdle> =+E CPO _ OA HyRGESHY f= =ig,MEY Tus, @Becomes oo gakCLP gular =4ig.Coley YS]ay, AnaAroncewecerMeort[gem ep] a Akiak. TiaBaseyacayeEe=80Malexpan Oo | Le OAR wean SS <aBi.Co!Mp =FeLol Riona> =ahEnptZoKoga) - —oe - a —a woo - P<Alay ==AptCulie—= a eget SEG IND BeebgeWsShue ee _Le a esEy(teuWires aNp==BeggAuer ep gph OMAIA VIN==Eg,pt.Some © NN iyte ; eee ->) ___fsconsheuttr=%tevTeaOatgSdapat 2555 - —— ‘Vandal dimeiou,umsk koa?Suir.ISG) a&@l> 4 Ohok innCoe)=€* @)=2%Q noilookAYSsa=ipftcrackaoefy,Omnia (26)Bi= tak dain (ead)=£%saage.Conackiny: Lo Ginionddum(Geoodun (AM\= E*(rom, tWS=Eteesceenousape, So. 4 udfagl=80) aksduncliesl= E™oaim®.r) PwGALLsalele— 3)Tua, dim(Fj)=Eaodomrod. —_ Sk osQian beoilJondad dad Quia.aathoallSeudasdca)4 @ pi gan ~\Toandade AYNboheSermon wuds eOnp280Goa,SheAndoderOreocdkedshadeoY a+ Fo elas] appar oesrgurstodis Atts. Yer, [4,4,0, Mort, wnsLeoleotChaaniokursole eunraik octetamdgust, omcrwnctin dodhaedeutlosenQorr Manned: . (Auge) =(6°)ei(EY Ore emmmecthm weShou: e TApe stCRY4EY)=ent de Waydoweatroose DeprideLeeoar .WhamadeworovoteGas"oniadheulomic Stole curt” wilt someofr peemrdo selon wwem! OlurmudaSto, Y=. Ondaadnanohioiet aOe22dbTink Ormem teOateeTE|louse dotwiepian. QAOM2%)DnutdhaaAweAt.Soubat?onst Wechard indaroyesen, plindasom!! Tuaoktecree:but SwanBaryon‘enousMeetare2inedcmicadvwNNOnMasvac coun, AE=1,WAKO, SoWwWadwixde, doThingpteceBHM, Reeo=(Rey aeMating eee, e wy Stactin, WG. stedemart ieMure: @ a beTae<¥e|A*@loy=Cg".Eas <qAeris=<a)SMe ory =6ealroaty, Bsaltorp= SMeagy:Sy Fea<a)afl =e’. f. A253SNABAO|S> =Sep e VesusrCRAG Draw » aA =bem bee Fao<a)anttonley =(a<q]bEPHOIsp =Waolage):bwer.Sores abs) o> =bt Uruwps, Dawuceopable egratusidr aSue . bent=Sar @ OwdPCAComauyWar.Wwacdhon : . «.FockokMrasmmiaaetnAwpre: “Y_getin, =EVscid Rgeeo" =Ba S&TES Slgow . tM 4sage~geirt)asovstea) eogto€ niadoeME ; ml$e10> , monn Borman ve: Ms.=+eh) Ae<m\de(ole> *FearN26Seen e=FTemba G01[ple Une.POACdoqehathahGif4doematude (hat: <n\ Meal =GPE)Sw ; 7Gale (aay YaweSantohes My:inareniadMsn,begk Ms.=Iq,UA)¥sa(—) Crue, uaeeae Sok layWta\ey =tirlig Boren GQ)SROs ae e“feolidelewateArreathoppronahtoting ny — Da,ALpt ©<n\SpA"(d|p>=leKe>|sacersts+qn)85]4) iOni,Onestqamnrrod Sonn?! moqodarn, Daumvsod|ondWodamawd SokaldtomahoneirrameG-poiandarokG07(umstanddassoumeuss), Bi sinenCiaVika, mearea huliens. yyCeesghiry traabeor Som?Wallwad.onan’ay0% hse, SpPQsvBnavorcol <x\AXe\p>=GainTCs)[an@nt'te+ Gis+Lytle),FA YudeyourAuamponne =*OOdoak: °SNBHO=Raerc)[3a2M%5+the+0\acy) C) =FR(302+ gh)Royssue) NawSoma.Tanpeinqfwatbegit,eougVeewo equoduia CSowD(2: teQ(T)_ aytyee (3)“ag—qe=RAMyJaq)+gna(*) Funally, Aockofq=o.Mare:g=aut)#40). +Sede(9)_2a3pJa=32)Gama?) @hea” : : aeg(t")=,fron: re *Aroaliy eadensertion ASa! 3 eaR G=2HGersxA My=138 =356MS We=135 Ja=Wa 3=Varsi46 Sapanvinsnh gywieSetent Comat: PRAReovnrclt soma urtele(RY)wail,oneHadag showy(44).Gre,OuGTR,aateber Cashageaster 42 evo ahdekh esercotumee ... Qddondum :iYPCACwane,CAR,dhomthepiaeautdwot waokderen se,BMS Oo=>fo, Also,ueadd Conclude Tho,lookingyof(3),am essbnekA(t)oaSu@)=al90’) opaleokq=st =<8ple*9 Fo3pQt).(ie,BeadsdoNombou's GIRdarivatind. , e oete ~ ePOAC: Goldberger Trieman Relation This business isfairly complicated and there are leads ofcenventiens tethrew you off what you're doing, The basic idea isthis: 1)weknew the width efthe weak pion decay. Wealso knew that the matrix element ofthe hadrenic axial weak current between the single pion state and the vacuum has avery simple form involving aconstant, the pion decay constant. 2)ifwemake theassumption that dyAY =¢pi, then itiseasytoshow that the constant ¢iethe pion decay constant. Netice that this constant isdefined differently beCommins and Adler. 3)Now consider the NNPI vertex for anoff shell pion. From the usual reductien formulas wecan write the Smabrix element interms ofthe pion field sand- wiched between the nand pstates. But wealso know this same matrix element from lowest order field theory interms ofthe pion coupling constant gp. Again, amost general form argument isused here. :4) 4)NowusePCAC inthis matrix element andusethemest general form again to bring inthe form factors gAand gP. Then weget afancy relatien relating ry BA,aP,&rsfpiyandsomemasses. . 5)finally, you leok atq=0and get almost the GTR. The last step istoassume asmoothness inthebehavier ofg,near small q®.This gives therelation. @ _©“epdoenondOU34a,CRA)Hanshawsiden.OsoCee a! nt L! eo fj (rr ee—— -~[Ssfiel=deupuae —__1@afel= nidypAg o- : ee a _ os !_iWadowQe,Anqoastcpain: [@,¥]=-2 38a atUda Ba,feqoekan. UsXsplat2LGsehYe Daan haan . opwa, vadaterimtndor,ask:QVl=teTe _ ey_ |* C&,¥are5© Manosk.on MY.ioon rr a — 3OndoledABeaniedaar:_ 1a aIs.rPels«ERC Oo ee eee Lo.2fests Ave(LARRY, TA)_=okieJopassanhe. -- Wakes vinhidDekLQ,R= heataeete dew auualLiondAn,Daogplwoke.To, annagBa$002000quad gah: oe oo ,GRFQ) SUG), OSB_!ae [email protected]&) aWN@b NOh Le wee | es _- --re ee eeLe Sitssaisnanan ———Co whoo ee aan mee iaen eee --| Sop nee | -—} . ne -| OO ye a . + LL _ _ nt ee Qagtcxqnwortty - ee WoSedktan adh,acadeienpcdentCaTpdayeumte wee te — ee =Spe teenyeet a (Qua, IS EEROO GeTOL) ay Saguan omandou Samoneone,son_ _. aa naRasa, aeopenOice)seespinug: YaoCasedQaENEF=cei WengenBete,CNEABettiyeoeonphdabanafalasat,: Decay omenVee2Qere = epShame eg a naneShows tebrubyse|——- —___-- Den, canoniaDolan gakabeg a12 Nat, VikaEELMoca(AORAOROIDAOH)— —Oe ane CheECoe es PO) 8 Le Se MWK SLLSGeet AAICO OL ©QOL,Oe,VeesoarAcaiBalily, qbanophleadonbindsea OoRai wh |Wesdy =SIG ee . , ©Sa,rosouatesooaseJandaA=ALTE aremge vokosfa\oole ~OMaacouicenuen dion———-——y A — (YreaslaecomRantaigsmaagiongfanOs t/a.Juot Oo ole ntLis(SENsolae Maa, pakacedone€=,saaadeaC\—e088) wn a Nenepee tachSededoeeebeal=ee Aaustenbabsitroeoan_we ee ©Garseag, SoncayadeSaabpalate emerMaack —@Gawd ee Odie 4a) tack es —- ---=nore herNawaloudol,Sereng OU NAS eteaeesabyaiid) som,Samesb|SeanSapa igenaga, Sg PLR Brome W(ie\=+1.OchAramdhomespin wit(spatial boOatAahctreaSacbeon oerYokoeKASEAS ---(adden conniecllanmuedMeg—-x)2NE Nek wold BrovnpubatA.grata.Qopidedynoneoicullalee —alllpreareadogerthan, .“Tad. emu pola.QuaoseMatee:PS - , / April25,2008 e@ DRAFT Cosmological Hilbert Space Geoffrey F.Chew Theoretical Physics Group Physics Division Lawrence Berkeley National Laboratory Berkeley, California 94720, U.S.A. Abstract AFockspaceof‘cosmological preons’—quantum-theoretic universe constituents—associates toa‘Milne spacetime’ based ontheLorentz group andto Gelfand-Naimark unitarygrouprepresentations. Lorentzinvariance ofMilne-universe e ‘age’accommodates ‘totalrelativity’. Two‘extra’dimensions ofthe 6-dimensional spaceoccupied bypreonsatcertainexceptional agesofthe universe define self-adjoint single- preon operators thatinclude acanonically-conjugate pairrepresenting preon energy and local time. Global spacetime divides into‘slices’ offixed macroscopic width inage,with ‘cosmological rays’ defined onslice boundaries. Self-adjoint-operator expectations at such aboundary prescribe throughout thesubsequent slice anon-fluctuating ‘mundane reality’—current densities ofconserved electric charge andenergy-momentum together with electromagnetic andgravitational potentials, Therayatthelower boundary ofaslice ispropagated totheupper boundary bycosmological branched Feynman paths across the slice thatcarry (divergence-free) potential-depending action. Amacroscopically-stable positive-energy single-preon wave function identifies either with aStandard-Model elementary particle orwith agraviton. (Unstable negative-energy preon wave functions remain tobeinterpreted.) Special relativity--Poincaré invariance~although inexact, is accurate forlow-density regions oftheuniverse atspacetime scales which arefarbelow that ofHubble and farabove that ofPlanck. “le ; * Introduction|r) Nonexistence ofunitary finite-dimensional Lorentz-group representations hasongbeensupposed topreclude, fordynamically-changing numbers ofparticles, aDirac-typequantum theory thatrepresents individual-particle properties such aslocation, momentum andspinbyself-adjoint operators onariggedHilbertspace.‘TheStandard Model, which employs asfoundation notparticles butquantum fields associated tofinite- dimensional Lorentz-group representations andwhichrepresents actionbyill-defined | localfield-product operators withperturbative renormalization procedures tomanage consequent divergences, hasfailed toaccommodate gravity. Problematic, furthermore, is theStandard-Model description ofbound states (‘condensed matter’), perturbation theory being unsuited tomacroscopically-stationary composite wave functions. Thediscrete quantum cosmology (DQC) ofReference (2)andthepresent paper introduces a‘cosmological-preon’ Fockspaceviaunitary (infinite-dimensional) 4representations ofthecomplex Lorentz-group. Anycosmological preon (henceforth oe throughout thispapersimply called‘preon’) is‘lightlike’—in thesenseofhaving se velocity candapolarization transverse tovelocity direction—and carries momentum, angular momentum andenergy. However, only very-special preon wave functions exhibit,throughexpectations ofself-adjoint operators, therelation between energy, momentum, spinandspacetime location thatcharacterizes ‘ordinary matter’. ADQCFock-space ray,representing theentireuniverse, comprises sumsofproducts ofpreonWavefunctions whosedisertequantum numbers allow certain macroscopically-stable positive-energy preon states tobeinterpreted aslepton, quark, weak boson, photon or graviton.(Acosmological meaningisgivenbelowfortheadjective‘macroscopic’.) e Reference (2)specifies ‘creation-annihilation’ Feynman paths whose action includes gravity aswellaselectromagnetism andweak-strong interaction.TheDQCFockspaceisbuiltfromPauli-symmetrized superpositions ofproducts ofinvariantly-normed single-preon functions. Displayed inthispaper isasingle-preon basis thatis‘6-labeled” bypreon energy, ‘momentum magnitude’, direction of momentum (2angles) andapair ofinvariant helicities. One ofthelatter—here called ‘velocity helicity’--is angular momentum with respect tovelocity direction while the other isthe(usual) angular momentum inthedirection ofmomentum. Preon energy isthe component ofitsmomentum inthedirectionofitsvelocity.Theterm‘momentum. ie magnitude’ hasacontinuous significance, through Lorentz-group Casimirs, thatparallels thesignificance innonrelativistic quantum theory ofaparticle’s discrete ‘angular- momentum magnitude’. Commutability ofthecomplete setof6corresponding self-adjoint operatorsderivesfromcommutability ofright Lorentz, transformations with lefttransformations. Ourusageoftheadjectives ‘right’ and‘left’ willbeexplained. DQC Hilbert space unitarily represents a/2-parameter group—the product ofright andleftLorentz groups. Action isright-Lorentz invariant, DQC right transformations being those employed by Milne todefine aspacetime )andwhich wecall‘Milne transformations’ toavoid confuusion with theEinstein-Poincaré meaning foraLorentz transformation. The 6self- adjoint-operator generators ofMilne transformations, which donotcommute with each other, represent preon momentum andangular momentum. DQC path action conserves emomentum andangular momentum butnotenergy—which associates tooneoftheJef? -2- Lorentzgenerators.Velocityhelicityassociatestoanotherleftgenerator,onewhich e@commutes with preon energy aswell aswith momentum andangular momentum. Tothebestofourknowledge DQC lefttransformations have no6-parameter- group precedent innatural philosophy butal-parameter leftsubgroup associates to/ocal- timetranslation (nottranslation ofglobal time--which wecall‘age’). Thegenerator of thisleftsubgroup represents preon (total) energy. Another U(/) leftsubgroup, generated byvelocity helicity, comprises rotations about thevelocity direction. IntheDQC algebra ofself-adjoint operators (which, because DQCaccords noapriorimeaning to ‘measurement’, weavoidcalling‘observables’)thetwoleftLorentz-groupCasimirsare ytequal tothetworight Casimirs (commuting with all12group generators).Reference (2)addresses theDQCactionofbranched Feynman paths. Thepresent paper iscomplementary-—-ignoring path action while addressing various 6-labeled bases forsingle-preon Hilbert space. Each basis corresponds toacomplete setof6commuting self-adjoint operators (a6-csco). Unitary Hilbert-space regular representation ofthe product ofright andleftLorentz groups provides aDQC path-contactable basis that A parallels theFeynman-path-contacting coordinate basis forDirac’s nonrelativistic +y quantum theory.“AnanalogofDirac’smomentum basisassociates totheunitary ‘ irreducible SL(2,c) Gelfand-Naimark (G-N) representation (‘unirrep’). Thealgebra ofpreon selfadjoint operators represents inDirac sense preon spatial location, velocity, polarization, energy, momentim andangular momentum, aswell as velocity-helicity andmomentum-helicity. Ofcourse notallthese operators commute with each other. Each oftheDQC bases discussed here associates toadifferent 6-csco. Inthe‘pathbasis’eachpreonis‘classicallyspecified’by(aproductof)the6 e@continuous coordinates ofamanifoldtraversedbyFeynmanpathscomprisingstraight lightlike ‘ares’ which may becreated orannihilated asapath progresses. (The local time along anyarchasunitderivative with respect toglobal age.) Apath propagates some “ray’--a fixed-age cosmological wave function whose norm lacks significance through probability orotherwise-to thesubsequent ray.Each preon ofa‘starting’raycontacts exactly onestarting arcofaFeynman path.Beforereaching theageofthe subsequent ray,anypatharcmaybeannihilated atacubic vertex—an ‘event’—where newarcsare created. Oneoftwo‘extra’velocity-associated manifolddimensions,bydefiningaself- adjoint operator associated toindividual-preon local time, allows reality inthe‘near-future’ofaraytobeprescribed byself-adjoint-operator expectations overthatray.The meaning of‘near future’ attaches toaDQC ‘macroscopic slicing” ofMilne spacetime that will bediscussed below. Cosmological rays aredefined only onslice boundaries. DQC Fock space comprises sums ofPauli-symmetrized products ofnormed single-preon functions. Attherisk ofobscuring total relativity, thepresent paper chooses toemphasize theunfamiliar labels onwhich depends afunction belonging toan individual preon. Special such functions thatrepresent elementary particles (quarks, leptons andweak bosons, together with photons andgravitons) willbeexposed. What wecall‘total relativity’ recognizes time arrow andabsoluteness ofmotion while respecting Mach’s principle inthesense discussed byWilczek. ©Binstein- Poincaré special relativity ignores time arrow andmotion absoluteness; special relativity furtherdisregardsMach.Application ofDQCtophysicsrequiresscale-based @approximation; DQC addresses anexpanding universe thatlacks meaning for -3- reproduciblemeasurement. Thegeneralmeaningof‘physics’andofspecialrelativityin e@particular isconfined tospacetime scales tinycompared tothatofHubble while huge compared tothatofPlanck. Reference (6)presents aEuclidean-group-based physics- scale gravity-less approximation toDQC thatmay bedescribed asa(Higgsless) ‘sliced- spacetime Standard Model’ (ssSM). TheDQC (global) age, invariant under both right andlefttransformations, plays a discrete role paralleling that ofcontinuous time innonrelativistic quantum theory. DQC Feynman paths connect successive macroscopically-spaced exceptional ages atcach of which isdefined acosmological Fock-space ray—inasense recalling S-matrix theory.DQCspacetime dividesinto‘slices’ofmacroscopic widthwhoseboundaries locateattheexceptional ages. Path branching—path-arc creation orannihilation--is forbidden atslice boundaries where arayisdefined—occurring atages interior toaslice where rays are notdefined. DQC dynamics prescribes quantum propagation inthediscrete S-matrix senseofan“instate”leadingtoasubsequent “outstate”withoutanywavefunction being defined between inandout. yAlthough notdiscussed intthepresent paper, theaggregation ofDQCHilbert space, path rules andinitial condition “spontaneously” breaks C,PandCPsymmetries.BecausetheDQCHilbert spacerepresents agroupisomorphic tothecomplexLorentzup(asdoesanalyticS-matrixtheory),itislawiblethatsonnecoomologicalthat some cosmological counterpart to‘CPT symmetry”willeventuallybecomerecognized. The here-examined infinite-dimensional single-preon Hilbert space comprises normed functions ofthecontinuous coordinates (path-basis labels) that ‘locate’ an individual preon within a6-dimensional manifold which isatonce aright andaleft e group manifold (common Haar measure). Thesingle-preon Hilbert space hasasafactor a ‘finite-dimensional Hilbert subspace ofdiscrete labels, invariant under both right andleft continuous transformations, labels thatarelargely ignored bythepresent paper. Discrete labels carried byboth path andraydistinguish different preon ‘sectors’ (¢.g., electron, up- quark, photon, graviton) byspecifying electric charge, color, generation, etc, Weshall here attend toaninvariant 2-valued parity-related “handedness” carried both bypath arcs(between path-branching points) andbypreons--in contrast topreon helicities that aremeaningless forapath arc.DQC Hilbert space correlates handedness to signofhelicity inassigning toeach preon sector aunique velocity helicity thatcoincides \_withmomentum helicity. vi G-Ndiscussed twodifferent basesforaHilbert spacethatrepresents unitarily theg groupSL(2,c), withoutattempting foreitheranatural-philosophical interpretation. The ? vectors ofonebasis—analog tothecoordinate basis ofnonrelativistic Dirac theory--are normed functions over the6-dimensional (continuous, left-right) manifold. Wecallthis thepathbasis because DQC Feynman paths traverse this6-space. G-N’s second basis-- thatwehere call‘G-N unirrep’--parallels theDirac-Fourier-Wigner momentum-spin basis ofnonrelativistic quantum theory that unitarily andirreducibly represents the Euclidean group. ®(The6-parameter compound Euclidean group isacontraction ofthe 6-parameter semisimple Lorentz group.) Although thetransformation connecting thetwo G-N bases isnotentirely ofFourier-Wigner form, wave-function norm ispreserved; the transformation isunitary. WemodifytheG-Nunirrepbasisby(unitary)Fouriertransformations ofwave- r) function dependence onapairofcomplex directional labels,soastodiagonalize -4- simultaneously preonenergy—the component ofitsmomentum initsvelocitydirection @ andcomponents ofmomentum andangular momentum insome arbitrarily-specified direction. Acontinuous Casimir label, carried over undisturbed from theG-N unirrep basis, wecall‘magnitude ofmomentum’. Two discrete labels arehelicity interpretable— components ofangular momentum invelocity andmomentum directions. The altered basis, which facilitates meaning for‘preon parity reflection’, wecallthe‘energy-unirrep’ basis. ‘Theuniverse spacetime identified byMilne inthenineteen thirties ©—an open forward-lightcone interior whose boundary allows ‘big-bang’ interpretation— endows ‘Lorentz transformation’ with acosmological time-arrowed meaning different from the Einstein-Poincaré special-relativistic physics meaning (ignoring time arrow) that augments Lorentz invariance byspacetime-displacement invariance. (Asufficiently large spacelike ornegative-timelike displacement may move apoint within Milne spacetime outside thatspacetime—ic., outside theuniverse.) Jncontrast toanEinstein boost between different ‘rest frames’ that each assigns adifferent setofvelocities tomassive entities within some ‘laboratory’ spacetime-localized region, Milne boosts relate toeach other different ‘local frames’ thateach associates toadifferent spatial location, Milne boosts--rightDQCtransformations--are spatialdisplacements atfixeduniverseagein& curved (hyperbolic) 3-space. Although Milne-spacetime flatness~manifested inDQC bystraight lightlike arcs within Feynman paths--might seem incompatible with general relativity’s association of gravitytospacetimecurvature,DQCgravitationalactionatadistancepluscreationand r) annihilation ofsoft-gravitonic arcsenables discretized curvature via‘gentle’ branchingsofstationary-action classical paths. ®AnyDQC path,asprescribed inReference (2),isan‘eventgraph’—a setofspacetime-located cubicverticesconnected byarcsofpositive- lightlike 4-velocity that carry energy aswell asdiscrete attributes. Bydisregarding Planck’s constant, general relativity ignores gravitons andapproximates byaspacetime- curving trajectory (e.g., anelectron trajectory) anarc-sector-maintaining stationary-action sequence ofDQC straight ‘hard’ arcsthatareseparated atgentle events by‘soft? gravitonic-arc absorption oremission. ‘The adjectives ‘hard’ and‘soft’ refer totheenergy scale setbyPlanck’s constant timestheinverseoftheagewidthofaspacetime slice—the timeintervalthatdefines cosmologically theadjective ‘macroscopic’. Both theStandard Model andtheReference(6)ssSMrevisionthereofattendtothePlanckconstant, whilemakingaG>0approximation andachieving flat3-spacethroughdisregard oftheHubbleconstant— suppressing redshift byregarding universe ageasinfinite. ThessSM physics approximation toDQC—differs from theStandard Model byitsrecognition of macroscopic spacetime slicing. Therevision accommodates softphotons andthereby maintains capacity (even while ignoring gravity andredshift) toprovide an electromagnetic theory ofphysical measurement. Milne SpacetimeTheopeninteriorofaforwardlightcone—what wecall‘Milnespacetime’--is the product ofalower-boundedone-dimensional‘agespace’withanunbounded3- rdimensional ‘boost space’. Thespacetime displacement from theforward-lightcone vertex (whose spacetime location ismeaningless) toanyspacetime point isapositive- <5 timelike4-vector(t,x).Definingthe“age”rofaspacetimepointtobeitsMinkowski @ distancefromlightcone vertex—ic., theLorentz-invariant modulus (P—x°c)"ofits spacetime-location 4-vector—the setofpoints sharing some common ageoccupies a3- dimensional (global) hyperboloid. Any point within such ahyperboloid may bereached from anyother bya3-vector boost. Once anorigin within boost space isdesignated, an arbitrary spacetime point isspecified by(z,f),wherefisthe3-vectorboost-space displacement from theselected origin tothepoint. Writing =in,where nisaunit 3- vector andfispositive, 1=tcosh, x=ctnsinh fp. oy The spatial-location label #will inthefollowing section andinAppendix Abe identified within thepath basis forDQC Hilbert space. Compatibility ofage discretization with Milne’s meaning forLorentz invariance allows DQC’s spacetime to betemporally discrete foritsFeynman-path quantum dynamics eventhough spatially continuous. Discretization occurs attwodifferent fundamental scales: (1)Any DQC Feynman path traverses anage-discretized ‘macroscopic slice’ ofMilne spacetime—a slice bounded above inageaswell asbelow. (2)Within each slice any(straight and lightlike) arcproceeds inPlanck-scale agesteps, whose precise value isestablished in Reference (2)from action quantization. Although consistency requires slice width tobeanintegral multiple ofarcstep, thehuge-integer ratio willnotbeaddressed bythepresent paper—which ignores arc steps.Beforeattendingatalltopatharcsthispaperchoosestoaddressthesingle-preon @ Hilbert-space pathbasis. Nevertheless thetermination ofpath arcsatexceptional (ray- age) hyperboloids might betaken asdefining thepath basis ofpreon Fock space. Toeach point ofboost space associates a“local” Lorentz frame inwhich f=0— ie.,aframe defined uptoarotation bythepoint’s location 4-vector being purely timelike inthatframe. Age change andtime change areequal inlocal frame. Inlocal frame an infinitesimal spatial displacement dxatage7relates toaninfinitesimal boost-space displacement dfbydx=crdf. (Aphenomenological meaning for‘local frame’ resides in theapproximate isotropy ofcosmic background radiation observed inthatframe. This meaning parallels thatofstandard cosmology’s ‘co-moving coordinates’. )Therotationalambiguity oflocal-frame meaning isreducedbelowthroughanoriginofa6-dimensional ‘space that(arbitrarily) designates notonly theorigin’s boost-space location butalsoan attached orthogonal andhanded (1,2,3)setof3referenceaxeswhichmaybeparallel transported along aboost-space geodesic from theorigin toanyother boost-space location. ‘The(3-parameter) global orientation ambiguity isaccommodated bytotal relativity—a DQC Fock-space restriction thatrequires rays tobeglobally rotationally invariant—unchanged when acommon Milne rotation isapplied toallpreons. Ray expectations ofMilne-boost generators arealsoglobally invariant. Total relativity might besaid tomean that both thetotal momentum andthetotal angular momentum ofthe universe arezero (6conditions). TheDQC universe isnotonly rotationally invariant but, associating right-boost generators with infinitesimal spatial displacements atfixed age, theuniverse isalso ‘classically-invariant’ under (non-abelian) boost-space displacements. ~~ -6-