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Mathieu functions, a visual approach J. C. Gutie ´rrez-Vegaa)and R. M. Rodrı ´guez-Dagnino Instituto Tecnolo ´gico y de Estudios Superiores de Monterrey, 64849 Monterrey NL, Me ´xico M. A. Meneses-Nava Centro de Investigaciones en Optica, 37150 Leo ´n, Guanajuato, Me ´xico S. Cha´vez-Cerda Instituto Nacional de Astrofı ´sica, O´ptica y Electro ´nica, 72000 Puebla, Me ´xico ~Received 25 June 2002; accepted 26 September 2002 ! The behavior of the Mathieu functions is illustrated by using a variety of plots with representative examples taken from mechanics. We show how Mathieu functions can be applied to describestanding, traveling, and rotating waves in physical systems. Some background is provided onnotation and analogies with other mathematical functions. Our goal is to increase the familiaritywith Mathieu functions in the scientific and academic community using visualization. For thispurpose we adopt a strategy based on visual recognition rather than only looking at equations andformulas. © 2003 American Association of Physics Teachers. @DOI: 10.1119/1.1522698 # I. INTRODUCTION Increasing attention is being given to scientific and engi- neering problems that lead to differential equations of theMathieu type. 1Their solutions, known as Mathieu functions, were first discussed by Mathieu in 1868 in the context of thefree oscillations of an elliptic membrane. 2These functions were further investigated by a number of researchers whofound a considerable amount of mathematical results thatwere collected more than 60 years ago by McLachlan. 3Dur- ingthelastdecadetherehavebeenseveralarticlesdevotedtonew analytical results, numerical techniques, andapplications. 4–9 Mathieu equations occur in two main categories of physi- cal problems. First, in applications involving elliptic geom-etries, for example in the analysis of the vibrating modes inelliptic membranes, the propagating modes in elliptic pipes,and the oscillations of water in a lake of elliptic shape.Mathieu equations arise after separating the wave equationusing elliptic coordinates. 10,11Second, Mathieu equations arise in problems involving periodic motion, such as the tra-jectory of an electron in a periodic array of atoms, the me-chanics of the quantum pendulum, and the oscillations offloating vessels. 12–17 Despite the existence of many applications, what attracts our attention is that Mathieu functions are barely mentionedin modern textbooks of mathematics for physics andengineering, 18others just present a limited discussion,19–23 and older texts that gave some account of Mathieu functionsare now out of print. 3,24–26Nevertheless, when facing a prob- lem that leads to the Mathieu differential equations, we canalways consult the invaluable handbooks byAbramowitz andStegun 27and by Gradshteyn and Ryzhik.28However, this ex- perience can be rather painful if one is trying to learn aboutMathieu functions for the first time. In most books we find ahigh density of equations which, from a didactic point ofview, can be scary for the unfamiliar reader. We believe that this lack of literature compared to that for other special functions is because the behavior of Mathieufunctions is relatively rich and consequently more difficult tounderstand. Moreover at least five different nomenclaturesare in use ~see Appendix A !, and the computation of the Mathieu functions and their eigenvalues still presents somenumerical difficulties. 8,9 The purpose of this article is to facilitate the understanding of some of the qualitative features of Mathieu functions andtheir applications. We believe that the visualization ofMathieu functions will be helpful in achieving a better com-prehension of their basic characteristics. This work is in-tended for students, teachers, and researchers who are unfa-miliar with Mathieu functions, and who are interested ingaining visual insight. We shall restrict ourselves to includ-ing the minimum formulas needed to explain the basic char-acteristics, the different notation, and the classification ofthese functions. The behavior of the Mathieu functions isillustrated by using a variety of plot types with representativeexamples taken from mechanics. We show how Mathieufunctions can be applied to describe standing, traveling, androtating waves in physical systems. II. VISUALIZING MATHIEU FUNCTIONS Over the years the visualization of Mathieu functions has been limited to simple curves for a few values of their pa-rameters. Jahnke and Emde 29were perhaps the first authors to include three-dimensional surface representations of theMathieu functions, whereas Abramowitz and Stegun 27pro- vide many useful two-dimensional plots. At present, moresophisticated computational tools are available, and hencefar more complicated behavior of Mathieu functions can beexplored in a variety of two- and three-dimensional plots. If the two-dimensional Helmholtz equation ]2U ]x21]2U ]y21k2U50, ~1! is transformed from rectangular coordinates ( x,y) to elliptic coordinates ~j,h!by the formulas x5fcoshjcosh,y5fsinhjsinh, ~2! and a solution of the form U5R(j)F(h) is sought, it is found that R(j) and F~h!must satisfy the equations 233 233 Am. J. Phys. 71~3!, March 2003 http://ojps.aip.org/ajp/ © 2003 American Association of Physics Teachers d2F dh21~a22qcos2h!F50, ~3! d2R dj22~a22qcosh2 j!R50, ~4! whereq5f2k2/4 andais the separation constant arising from the eparation of variables method. In the literature, Eqs. ~3!and~4!are known as the ordinary and the modified Mathieu equations, respectively.27However, in applications involving the Helmholtz equation in elliptic coordinates,Eqs. ~3!and~4!are better identified as the angular and the radial Mathieu equations. 24Their solutions are the angular Mathieu functions ~AMF !and the radial Mathieu functions ~RMF !, respectively. This nomenclature becomes obvious when we observe in Fig. 1 the similarity between the ellipticcoordinates ~ j,h!and the polar coordinates. The elliptic vari- ablehhas a domain 0 <h,2pand plays a similar role to a polar angle, whereas the variable j, with domain 0 <j,‘ behaves as a radial variable. The line joining the foci (6f,0) corresponds to j50. Notice that the polar coordi- nates could be considered to be a special case of the elliptic coordinates in the limit f!0 when the foci of the elliptic coordinates collapse to a point at the origin. In this limit, the angular and the radial Mathieu equations become the well-known harmonic equation and the Bessel equation,respectively. 3,27As a consequence, the angular Mathieu func- tions transform into the trigonometric functions cos hand sinhand the radial Mathieu functions become the Bessel functions. A. Angular Mathieu functions The angular Mathieu equation ~3!is a linear second-order differential equation that has two families of independentsolutions, namely the even and the odd angular Mathieufunctions, F m5Hcem~h;q!,m50,1,2,... , sem~h;q!,m51,2,3,... ,~5!wheremis the order. The notation ce and se comes from cosine-elliptic and sine-elliptic, and was first suggested by Whittaker.22Nowadays, it is a widely accepted notation for the AMF ~see Appendix A !. The behavior of Mathieu functions is fairly complicated, particularly because we need to understand both the variable hand the parameter qdependence of the functions. Physical considerations are usually such thatAMF are periodic30with period por 2p. The values of ain Eq. ~3!that satisfy this condition are the eigenvalues of the equation; for ce mthe eigenvalues are usually denoted as am(q), whereas for se m they are represented as bm(q). According to the Sturm– Liouville theory,18the eigenvalues form an infinite set of countable real values that have the property a0,b1,a1 ,b2fl. Each function ce mand se mis associated with an eigenvalue amorbmwhich in turn depends on q. In Fig. 2 we plot the functions ce m(h;q) and se m(h;q) for several values of mover the plane ( h,q).31Note that Eq. ~3! becomes the harmonic equation when q!0, when it is evi- dent that ce mand se mbecome equal to the trigonometric functions cos mhand sinmhasqvanishes. The range of the plots has been limited to @0,p#, because their entire behavior can be deduced from the parity and symmetry relations pro-vided in Table I. The parity, periodicity, and normalization of the AMF are exactly the same as their trigonometric counterparts. This is, ce mis even and se mis odd, and they have period pwhenm is even or period 2 pwhenmis odd. The AMF have mreal zeros in the open interval hP(0,p), but they cluster around p/2 asqincreases. The normalization for the AMF is E 02p cepceqdh5E 02p sepseqdh5Hpifp5q, 0i fpÞq.~6! The physical meaning of the parameter qdepends on each application. For instance, the classical problems of the vi- brating modes in an elliptic membrane10and the probability distributions in an elliptic quantum billiard32are mathemati- cally equivalent, that is, in both cases Mathieu equationsarise after separating the wave equation or Schro ¨dinger equa- tion in elliptic coordinates. However, as we will discuss be- low, in the first case qis related to the eigenfrequencies of the vibrating modes, whereas in the second case qis associ- ated with the characteristic energies of the eigenstates in the billiard. An interesting one-dimensional example where the AMF occur is the quantum pendulum. 14–16Consider a plane pen- dulum of length Land mass Moscillating under the action of gravity. The time-independent Schro ¨dinger equation cor- responding to this problem is 2\2 2ML2d2C du21@V~u!2E#C50, ~7! where uis the angular displacement from the vertical, V(u)52MgLcosuis the potential energy, and C~u!is the wave function associated with the energy E. The boundary condition to be imposed on Cis that it be single valued in u, that is, C~u!has period 2 p:C(u12p)5C(u). Equation ~7!can be rewritten in the form of an angular Mathieu equa- tion~3!, by defining u52h, ~8a! Fig. 1. Elliptic cylindrical coordinate system. The curves j5constant are confocal ellipses and the curves h5constant are orthogonal hyperbolas. 234 234 Am. J. Phys., Vol. 71, No. 3, March 2003 Gutie ´rrez-Vega et al. a54E ~\2/2ML2!, ~8b! q522MgL ~\2/2ML2!. ~8c! In view of Eq. ~8a!, the boundary condition is C@2(h 1p)#5C@2h#, that is, as a function of h, the wave func- tion has to be periodic with period p. From Table I, such solutions are AMF of even order: ce 2r(h;q) and se2r12(h;q) forr50,1,2,... . All otherAMF are excluded by the periodicity condition. We note further that characteristic values of the energy Emare defined by the eigenvalues amof the Mathieu equation; from Eq. ~8b!we find Em(\2/8ML2)am. Also, we can see from Eq. ~8c!that, for the quantum pen- dulum, the parameter qdepends only on the given physical constants of the problem, and that Eq. ~7!really corresponds to a Mathieu equation with negative q. To satisfy the usual definition of the Mathieu functions, we must perform the change of variable h!(p/22h) according to Table I. When visualizing mathematical functions, a suitable plot depends on the purpose. For example, the surfaces shown inFig. 2 were done following a mathematical point of view.These diagrams allow us to appreciate at a glance the evolu- tion of ce and se as functions of horq. In Fig. 3 we adopt a physical point of view to show the probability distributionsC2of the quantum pendulum. As stated, the wave functions are written in terms of AMF of even order, ce 2r(h;q) and se2r12(h;q). We might plot these solutions against the polar coordinate uin a rectangular system ( u,C2), but a more meaningful picture is usually a polar diagram. In Fig. 3, we can visualize how the probability distributions of a quantumpendulum vary with u. B. Radial Mathieu functions The solutions of Eq. ~4!whenqis positive are the even (e) and odd ( o) oscillatory radial Mathieu functions of the first and second kind, R5HJem~j;q!,J om~j;q!,firstkind, Nem~j;q!,N o m~j;q!,secondkind.~9! Forq,0 the solutions are known as the evanescent radial Mathieu functions R5HIem~j;q!,I om~j;q!,firstkind, Kem~j;q!,K o m~j;q!,secondkind.~10! In elliptic coordinates the radial Mathieu equation ~4! plays a similar role as the Bessel equation in circular cylin-drical coordinates. Because Bessel functions are betterknown than Mathieu functions, visualizing their analogies isa practical way to gain an insight into the qualitative behav- Fig. 2. Graphical visualization of angular Mathieu functions cem(h;q) and sem11(h;q) over the ( h,q) plane. The function ce0(h;q) is never negative, al- though oscillatory. Table I. Symmetry relations for AMF. Function r50,1,... PeriodParity about h50Parity about h5p/2AMF withq,0 ce2r(h;q) p Even Even ce2r(h;2q)5(21)rce2r(p/22h;q) ce2r11(h;q) 2p Even Odd ce2r11(h;2q)5(21)rse2r11(p/22h;q) se2r12(h;q) p Odd Even se2r12(h;2q)5(21)rse2r12(p/22h;q) se2r11(h;q) 2p Odd Odd se2r11(h;2q)5(21)rce2r11(p/22h;q) 235 235 Am. J. Phys., Vol. 71, No. 3, March 2003 Gutie ´rrez-Vega et al. ior of the RMF. It is known that Bessel equations have four families of independent solutions,18namely the ordinary Bessel functions JmandNm, and the modified Bessel func- tionsImandKm. Each Bessel family splits into two Mathieu families, for example, the Bessel function Jmsplits into the even Je mand the odd Jo mMathieu functions. Thus there are eight independent families of RMF. This abundance of ellip- tic solutions has provoked confused notations in the litera-ture which often complicates the recognition of the RMF andtheir relations. We refer the interested reader toAppendixA,where a comparative table of notation for the Mathieu func-tions is provided. In Fig. 4 we show the first-order RMF for different values ofq. Similar to the Bessel functions, the RMF have a de- creasing, oscillatory non-periodic behavior. Conversely tothe AMF, the radial solutions oscillate faster as qincreases. In Fig. 5 we plot the RMF Je 0forq55 in two different views. Adopting a mathematical point of view, we show in Fig. 5 ~a!Je0as a function of the argument j. The Bessel analogy of Je 0is indeed the lowest-order Bessel function J0. LikeJ0, the function Je 0is oscillatory, decreasing, and non- periodic. By comparing Je 0with respect to J0, we can ob- serve that the maximum at the origin of Je 0is not as domi- nant as in the case of J0, and that Je 0oscillates faster as j increases. Figure 5 ~b!shows again Je 0, but now physical insight is gained by plotting it as a function of sinh( j) instead of j. For instance, as we will see below, Je 0could represent the radial dependence of a vibrating mode in an elliptic membrane.10In this case the argument jis associated with the radial elliptic coordinate ~which is dimensionless !. However, to visualize the spatial behavior of the mode, it is required to plot it against a coordinate with a length dimension such as xory. Let us consider the yaxis. By setting h5p/2 in Eq. ~2!,i ti s clear that the yaxis is written as y5fsinhj. In this manner, the plot in Fig. 5 ~b!could show the behavior of the vibrating mode as a function of the normalized coordinate y/f. The plot in Fig. 5 ~b!reveals an important property of the radial Mathieu functions: they tend to be damped periodicfunctions when they are plotted against a spatial coordinate likex;cosh jory;sinhj. This characteristic is interesting because often the physical patterns are associated with the asymptotic behavior of the mathematical functions. III. VISUALIZING STANDING WAVES IN AN ELLIPTIC MEMBRANE As stated, the angular and the radial Mathieu equations can be expected to appear in any problem involving theHelmholtz equation expressed in elliptic coordinates. 10,32 Consider the free oscillations of an elliptic membrane with Fig. 3. Polar diagrams of the probability distributions of a quantum pendu- lum as a function of u. Fig. 4. Plots of radial Mathieu functions with q51 ~solid line !,q52~dashed line !, andq53~dotted line !. 236 236 Am. J. Phys., Vol. 71, No. 3, March 2003 Gutie ´rrez-Vega et al. semiaxes aandb, and focal distance f~see Fig. 1 !. The boundary of the membrane is expressed in elliptic coordi- nates as j5j05arctanh(b/a)5constant. IfZ(x,y,t) is the vertical displacement as a function of time of a point located at ( x.y), thenZsatisfies the two- dimensional wave equation ]2Z ]x21]2Z ]y251 v2]2Z ]t2, ~11! where v25F/s, with sthe surface mass density and Fthe uniform tension per unit length in each point of the mem- brane. By assuming a harmonic time dependence Z(x,y,t) 5U(x,y)cos(vt), the wave equation becomes the Helmholtz equation ~1!, wherek5v/v. After applying Eq. ~2!to trans- form the Helmholtz equation into elliptic coordinates, and taking the separable solution U(j,h)5R(j)F(h), we ob- tain the angular and the radial Mathieu equations ~3!and~4!. In this case, the parameter qis given by q5f2 4k25f2 4v2 v25f2v2s 4F. ~12! A vibrating mode can be considered as a standing wave, that is, each point on the surface vibrates harmonically with an amplitude U(x,y), but all points have the same fre- quency. The modes are given by appropriate products of ra- dial and angular Mathieu functions, namely, the even Ze m and odd Zo msolutions:33 Zem5Jem~j;q!cem~h;q!cos~vmt!, ~13! Zom5Jom~j;q!sem~h;q!cos~vmt!, ~14! wherem>0 for even modes, and m>1 for the odd ones. These wave solutions must satisfy the Dirichlet condition at the elliptic boundary, that is, Z(j0,h,t)50. This occurs only if the radial functions vanish at the boundary, namely, Jem~j0,q!5Jom~j0,q!50. ~15!As seen in Figs. 4 and 5, the RMF are decreasing- oscillatory nonperiodic functions. As a consequence, for a given order m, there are an infinite set of possible values of qthat satisfy Eq. ~15!. Letqm,nbe thenth zero ( n 51,2,...) of the radial functions Je mor Jom. According to Eq.~12!for eachqm,n, there exists a corresponding eigen- frequency given by vm,n5(4Fqm,n/sf2)1/2. In Fig. 6 we show the first even and odd vibrating modes in the elliptic membrane. Notice the symmetry and the anti- symmetry with respect to the xaxis of the even and odd modes, respectively. The radial nodal lines ~elliptic lines !are defined by the zeros of the radial functions emand Jo m, whereas the zeros of the angular functions ce mand se mde- fine the angular nodal lines ~straight or hyperbolic lines !.W e can see that Um,nhasmangular nodal lines and nradial Fig. 5. Two views of the Je0radial Mathieu function: ~a!plotted against j~mathematical view !,a n d ~b!plot- ted against sinh j~physical view !. Fig. 6. Plots of the first even and odd standing modes in an elliptic mem- brane. The shaded regions represent crests ~maxima !or valleys ~minima !on the membrane surface. For instance, the even mode Ze11presents one crest and one valley. 237 237 Am. J. Phys., Vol. 71, No. 3, March 2003 Gutie ´rrez-Vega et al. nodal lines including the boundary as a nodal line. The cross points between radial and angular nodal lines correspond to nodal points of the membrane. In particular, the mode Um,n hasm14m(n21) nodal points. The origin is a nodal point ofUm,nonly ifmis odd. To gain some intuition about the numerical values in- volved, we show in Table II the first natural frequencies for an elliptic membrane with a55 cm, and b53 cm. To make appropriate comparisons, the corresponding eigenfrequencies for a circular membrane with radius a55 cm are included as well.34For numerical purposes, without loss of generality, we have choose the tension and density of the membrane such that AF/s51. Note that for each circular mode, there are two corresponding elliptic modes, the even one and the odd one. FromTable II we can see that, for a given Um,n, the even and odd modes vibrate with different frequencies, in fact, the even modes have lower frequencies than the oddmodes. The reason is that the even modes vibrate along thelargest axis of the ellipse, whereas odd modes vibrate alongthe shortest axis. IV. VISUALIZING TRAVELING AND ROTATING WAVES IN AN ELLIPTIC LAKE Finding the oscillations of water in an elliptic lake is a classic problem. The oldest references to this problem maydate back to 1924 and 1927 by Jeffreys 35and Goldstein,36 respectively. To show the interesting phenomenon of rotatingwaves inside the lake, we extend their analysis by includinga confocal elliptical wall inside the lake. The geometry of the confocal annular elliptic lake is shown in Fig. 1. In terms of elliptic coordinates, the inner and outer walls correspond to curves j5j15constant and j5j05constant, respectively. If Z(j,h,t) is the vertical dis- placement of the water surface from its equilibrium position, thenZsatisfies the wave equation ~11!, but now v25gd, wheregis the acceleration due to gravity and dis the un- disturbed depth. By following the same procedure described above for the membrane, the oscillating modes in the lakeare given by Ze m5@Jem~j!1ANem~j!#cem~h!cos~vmt!, ~16a! Zom5@Jom~j!1BNom~j!#sem~h!cos~vmt!, ~16b! whereAandBare constants to be determined, m>0 for even modes, and m>1 for the odd modes. The wave solutions ~16!must satisfy the Neumann condi- tion at both elliptic boundaries. This condition states that the normal derivatives of Ze mand Zo mvanish at each point of the boundaries. For even modes we have Zem8(j0,h,t) 5Zem8(j1,h,t)50, where the prime denotes the derivativewith respect to the radial variable j. By evaluating the de- rivative of Eq. ~16a!at both elliptic boundaries, we see that the Neumann condition is fulfilled only if Jem8~j0;q!1ANem8~j0;q!50, ~17a! Jem8~j1;q!1ANem8~j1;q!50. ~17b! This is a set of two linear equations with a unknown co- efficientA. According to linear algebra, a nontrivial solution to this set exists only if the following characteristic equation for the unknown parameter qis satisfied: Jem8~j0!Nem8~j1!2Jem8~j1!Nem8~j0!50. ~18! This characteristic equation can only be solved numerically.31Let us denote qm,nas thenth zero of Eq. ~18!. Once the value of qm,nhas been determined, the correspond- ing value of Acan be calculated using any of the boundary condition equations ~17!, and thus A52Jem8(j0;q)/ Nem8(j0;q). Finally, recalling that v5kv, the frequency of the mode ( m,n) is given by vm,n5@4v2qm,n/f2#1/2 5@4gdqm,n/f2#1/2. For numerical computations, let us choose the values j1 50.5 and j251.5. The eigenvalues of frequency corre- sponding to the first oscillating modes are listed in Table III in units of A4gd/f2. We can see that the odd modes have lower frequencies than the even modes ( ovm,n,evm,n). Note that this behavior is contrary to the results obtained for the modes in the membrane ~see Table II !, where evm,n ,ovm,n. There is a simple physical explanation for this fact: for the membrane the even modes oscillate without obstruc- tion along the largest axis of the ellipse, whereas the oddmodes oscillate along the shorter axis. In the confocal annu-lar elliptic lake the even modes tend to vibrate along thelargest axis as well; however, now the internal wall is anobstacle that changes the relative distances. With regard to Fig. 1, let us denote as a 8the horizontal separation between the outer and the inner elliptic boundaries along the xaxis, andb8the vertical separation along the yaxis. We now apply Eq. ~2!to show that b8is always greater than a8.W e have a85a02a15fcoshj0cos(0) 2fcoshj1cos(0) 5f(cosh j02coshj1). Analogously for b8we obtain b8 5f(sinh j02sinhj1). The difference b82a8gives b82a85f@~sinhj02coshj0!2~sinhj12coshj1!#. ~19! By expressing the hyperbolic functions in terms of exponen- tial functions, we can write Eq. ~23!in the simpler form b82a85f 2@exp~2j1!2exp~2j0!#.0. ~20!Table II. Eigenfrequencies vm,nin rad/s for the circular and the elliptic membranes; a55cm ,b53cm . Order mCircular Even elliptic Odd elliptic n51n52n51n52n51n52 0 48.1 110.4 65.9 168.5 1 76.6 140.3 91.5 191.2 116.5 220.72 102.7 168.3 118.9 215.0 139.8 243.0Table III. Eigenfrequencies in A4gd/f2units. mevm,no vm,n n51n52n53n51n52n53 0 0 1.0849 2.0583 1 0.3508 1.2748 2.2629 0.3151 1.1017 2.05882 0.6458 1.4586 2.4499 0.6396 1.3667 2.26933 0.9222 1.6973 2.6099 0.9213 1.6665 2.4875 238 238 Am. J. Phys., Vol. 71, No. 3, March 2003 Gutie ´rrez-Vega et al. We now can appreciate that b8is greater than a8for any j0.j1.0. Hence, the inclusion of an internal elliptic wall leads to even modes to vibrate in shorter distances than odd modes, and consequently ovm,n,evm,n. In Fig. 7 we plot several patterns corresponding to the first standing modes. In particular, the mode e,oum,nhas 2m(n 21) nodal points. In Fig. 8 we show a three-dimensional plot of Ze 12(j,h). Observe that the Neumann condition is satisfied at both elliptic boundaries. A. Traveling elliptic waves in the confocal annular elliptic lake Similar to the propagating waves in a rope, the standing patterns shown in Fig. 7 may be regarded as the result oftraveling waves propagating in opposite directions. To beprecise, outgoing and incoming elliptic waves propagate ra-dially while reflecting at the elliptic walls. To understand this traveling behavior of the Mathieu solu- tions, it should be mentioned that analogous to the Hankel functions H m(1),(2)occurring in Bessel equations, the solu- tions of Eq. ~4!can be expressed in terms of the even and the odd Mathieu–Hankel functions of the first and second kindHem(1),(2)~j;q!5Jem~j;q!6iNem~j;q!, ~21a! Hom(1),(2)~j;q!5Jom~j;q!6iNom~j;q!, ~21b! where the super-index ~1!is associated with the positive sign and~2!with the negative one. Similar to the Hankel func- tions which are often used to represent outgoing and incom-ing cylindrical waves, 18the Mathieu–Hankel functions can represent elliptical waves propagating radially in the positiveand negative direction of j. By making use of these func- tions, the wave solutions in Eq. ~16!can be rewritten in a traveling wave format Zem5Hem(1),(2)~j!cem~h!exp~2ivmt!, ~22a! Zom5Hom(1),(2)~j!sem~h!exp~2ivmt!, ~22b! where the time dependence has been expressed in complex form. We can more easily visualize the traveling wave behavior by analyzing the asymptotic expansions of the Mathieu– Hankel functions as qincreases. For instance, for even func- tions He(1),(2)(j,q);v21/2exp(6iv), where v5exp(j). The substitution of this approximation into Eq. ~22a!yields Zem;cem~h! Avcos~6v2vt!. ~23! Equation ~23!can now be recognized as a wave traveling in the positive or negative direction of the coordinate v.W e may imagine that the outer boundary reflects the outgoing wave generating an incoming wave. Similarly, the incomingwave is reflected by the inner boundary producing again anoutgoing wave. In Fig. 9 we plot the Mathieu–Hankel func- tion He 0(1)5Je01iNe0in a three-dimensional complex space. This graphical representation of He0(1)is unusual in the lit- erature. Whereas we had to link up in our mind the separate and disconnected characteristics of the real and imaginaryparts of the complex function, we can now see the relation ata glance. Observe in the plot the tendency of the RMF to be periodic as the coordinate sinh jincreases. Fig. 7. Plots of the first even and odd standing waves in an confocal annular elliptic lake. Fig. 8. Surface plot of Ze12. A top view is shown in Fig. 7. Fig. 9. Evolution of the zero-order Mathieu–Hankel function Je01iNe0in the complex space. The separation of the three-dimensional curve with re-spect to the axial axis is the modulus of the function. 239 239 Am. J. Phys., Vol. 71, No. 3, March 2003 Gutie ´rrez-Vega et al. B. Superposition of modes: Rotating waves We now are interested in the superposition of an even and an odd mode, that is, Zm,n5Zem,n1Zom,n. Because their oscillating frequencies are different, the pattern produced by the superposition varies in time. In Fig. 10 we plot Z12at six increasing times; each constituent mode is plotted separately in Fig. 7. The resulting wave is a rotating perturbation thathas angular momentum different from zero. 37 The cross points between the nodal lines of Ze m,nand Zom,ncorrespond to nodal points of the resulting wave. If we stay on the water surface at a fixed point on a nodal line, we can observe that this point moves harmonically. Similarly,the motion of any point outside of a nodal line is the super-position of two harmonic motions with different frequenciesand amplitudes. V. CONCLUSIONS We have graphically presented some properties of the Mathieu functions. Our main goal is to motivate their useand familiarity. As with many other special functions, it isnot necessary to know them in great mathematical detail inorder to use them in applications. The approach presentedherecanbeusedwhetheronewishestobecomefamiliarwiththe functions or as a starting point for a deeper analysis. Wehave illustrated in a diversity of plot types those Mathieu functions that appear more frequently in physical applica-tions. In our experience, the best way to visualize the behavior of Mathieu functions when one is beginning their study is tothink in terms of their more familiar analogies: ~a!AMF are analogous to trigonometric functions and ~b!the RMF be- have like Bessel functions. The variety and peculiarities ofthese functions are so rich that it is not possible to showthem all within this article. Instead, we provide detailedmathematical information and more plots of Mathieu func-tions via Ref. 38. We believe that the Mathieu functions de-serve the attention of the authors of future textbooks onmathematics for scientists and engineers. ACKNOWLEDGMENTS The authors are grateful to the reviewers who suggested many improvements to the article. Financial support wasprovided by the Consejo Nacional de Ciencia y Tecnologia~CONACyT !,M e´xico. APPENDIX A: COMPARATIVE NOTATIONS OF MATHIEU FUNCTIONS As a quick reference for those beginning their study of Mathieu functions, we summarize in Table IV the most com-mon notations used in the literature. The two most commonnomenclature conventions for Mathieu functions are those ofMcLachlan 3and Morse.25McLachlan’s nomenclature has its origins in the notation used by the first researchers of theMathieu functions and is arbitrary. This arbitrariness in thenotation, in addition to the existence of different normaliza-tions, often leads to a confusion. Despite this fact,McLachlan’s terminology is the most commonly employednotation in the scientific literature. Morse’s nomenclaturewas created by thinking about the connection of eachMathieu function with its corresponding Bessel analogy.From this point of view, Morse’s notation is advantageous,because it facilitates the visualization and classification ofthe Mathieu functions. Fig. 10. Plot of the superposition Z125Ze121Zo12at six different times.The resulting is a anticlockwise rotating wave. The time values are given in Af2/4dgunits. Table IV. Comparative notations of the Mathieu functions. This paperMcLachlan ~Ref. 3 ! Gradshteyn ~Ref. 28 !Erdelyi ~Ref. 26 !Morse ~Ref. 25 !Abramowitz ~Ref. 27 !Stratton ~Ref. 24 !Refs. 19–23 ce ce ce Se ce Se ce se se se So se So seJe ce ce Je Mc (1)Re(1) Jo Se Se Jo Mc(1)Ro(1) Ne Fey Fey Ne Mc(2)Re(2) No Gey Gey No Ms(2)Ro(2) Ie Ce( 2q) Io Se( 2q) Ke FekKo GekHe (1),(2)Me(1),(2)Me(1),(2)He(1),(2)Mc(3),(4)Re(3),(4) Ho(1),(2)Ne(1),(2)Ne(1),(2)Ho(1),(2)Ms(3),(4)Ro(3),(4) 240 240 Am. J. Phys., Vol. 71, No. 3, March 2003 Gutie ´rrez-Vega et al. APPENDIX B: MATHIEU FUNCTIONS AND THEIR COEFFICIENTS Because the angular Mathieu functions ce mand se mare periodic, they can be expanded in terms of Fourier series. The corresponding expansions fall into four classes, accord- ing to their symmetry or antisymmetry, about h50 and h 5p/2, namely, ce2r~h;q!5( k50‘ A2k~q!cos@2kh#, ~B1a! ce2r11~h;q!5( k50‘ A2k11~q!cos@~2k11!h#, ~B1b! se2r12~h;q!5( k50‘ B2k12~q!sin@~2k12!h#, ~B1c! se2r11~h;q!5( k50‘ B2k11~q!sin@~2k11!h#, ~B1d! wherer50,1,2,... . The recurrence relations between the co- efficients can be derived by substituting these series in the Mathieu equation ~3!. For instance for the functions ce 2r,w e obtain aA05qA2, ~B2a! ~a24!A25q~2A01A4!, ~B2b! @a2~2k!2#A2k5q~A2k221A2k12!, ~B2c! wherek>2. There are two common approaches to finding the coeffi- cients of the Fourier series. The first one is based on trans-forming the recurrence relations in Eq. ~B2!into continued fractions and to apply further algebraic methods to find theroots; see the details in Refs. 3 and 27.The second method isbased on constructing an infinite matrix starting from therecurrence relations. The eigenvalues and eigenvectors of this matrix are the eigenvalues aof the Mathieu equation and the coefficients Aof the Fourier series; see the details in Refs. 8 and 9. The RME in Eq. ~4!is obtained from the AME in Eq. ~3! by setting the change of variable h5ij. Therefore, we can apply this change of variable to the AMF in @Eq.~B1!#to obtain the following expansions: Je2n~j;q!5( k50‘ A2k~q!cosh@2kj#, ~B3a! Je2n11~j;q!5( k50‘ A2k11~q!cosh@~2k11!j#, ~B3b! Jo2n12~j;q!5( k50‘ B2k12~q!sinh@~2k12!j#, ~B3c! Jo2n11~j;q!5( k50‘ B2k11~q!sinh@~2k11!j#. ~B3d! a!Electronic mail: [email protected] 1L. Ruby, ‘‘Applications of the Mathieu equation,’’Am. J. Phys. 64,3 9–4 4 ~1996!. 2E. Mathieu, ‘‘Le mouvement vibratoire d’une membrane de forme ellip- tique,’’ J. Math. Pures Appl. 13, 137–203 ~1868!.3N. W. McLachlan, Theory and Application of Mathieu Functions ~Oxford Press, London, 1951 !. 4J. C. Gutie ´rrez-Vega, M. D. Iturbe-Castillo, and S. Cha ´vez-Cerda, ‘‘Alter- native formulation for invariant optical fields: Mathieu beams,’’Opt. Lett.25, 1493–1495 ~2000!. 5J. C. Gutie ´rrez-Vega, M. D. Iturbe-Castillo, G.A. Ramı ´rez, E.Tepichı ´n, R. M. Rodrı´guez-Dagnino, S. Cha ´vez-Cerda, and G. H. C. 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Emde, Tables of Functions ~Dover, New York, 1945 !, Chap. 11. 30Because of the periodic behavior of the AMF, these functions are oftenreferred to as periodic Mathieu functions ~Ref. 19 !. 31Numerical libraries for computing the Mathieu functions ~in particular the radial ones !are not readily available; for this reason, we have developed our own numerical routines based on the theory of Mathieu functions.Roughly speaking, we applied matrix methods ~Refs. 7, 8, and 19 !to find the eigenvalues a mof the Mathieu equations. Afterward, the AMF are computed applying Fourier series, whereas Bessel-function-product-serieswere used to evaluate the RMF ~Refs. 3, 19, and 27 !. A very practical list of properties and formulas of expansion series of Mathieu functions issupplied by Gradshteyn ~Ref. 28 !. We compared our results with those from recent papers ~Refs. 8, 9 !, and an accuracy of more than 10 29was obtained. The roots of the characteristic equations ~18!were calculated 241 241 Am. J. Phys., Vol. 71, No. 3, March 2003 Gutie ´rrez-Vega et al. using a Newton–Raphson method.We have taken advantage of the graphi- cal facilities of Matlab to show the plots of Mathieu functions. 32H. Waalkens, J. Wiersig, and H. R. Dullin, ‘‘Elliptic quantum billiard,’’Ann. Phys. ~N.Y.!260, 50–90 ~1997!. 33Second radial functions Nemand Nomare excluded because the wave solutions and their derivatives must to be continuous at j50. See Fig. 4. 34The modes in circular membranes are well studied in textbooks ~Ref. 18 !. The natural frequencies are given by vm,n5(F/s)1/2rm,n/a, whererm,nisthenth root of the JmBessel function, ais the radius, and Fandswere defined in Sec. III. 35H. Jeffreys, ‘‘The free oscillations of water in an elliptical lake,’’ Proc.London Math. Soc. 23, 455–476 ~1924!. 36S. Goldstein, ‘‘The free oscillations of water in a canal of elliptic plan,’’ Proc. London Math. Soc. 28, 91–101 ~1927!. 37P. H. Cerpeley, ‘‘Rotating waves,’’Am. J. Phys. 60, 938–942 ~1992!. 38http://homepages.mty.itesm.mx/jgutierr/mathieu.htm Bladder Glass. Nineteenth century students saw many demonstrations involving the effects of atmospheric pressure. In the bladder glass, a piece of a nimal bladder is tied with twine over the top of a vessel open at both ends. The lower end is placed on the bottom plate of a vacuum pump, and only a few strokesof the pump are needed to produce a pressure differential of essentially one atmosphere. The bladder bursts inward with a bang loud enough to wake thesleepers in the back of the classroom. This demonstration was last done at Washington and Lee University about 1900, and then it was set aside until Idiscovered it eighty years later. ~Photograph and notes by Thomas B. Greenslade, Jr., Kenyon College ! 242 242 Am. J. Phys., Vol. 71, No. 3, March 2003 Gutie ´rrez-Vega et al.