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0 287 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 288 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 289 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 290 1 {CSTYLE "" -1 -1 "" 0 1 0 0 17 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 291 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 292 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 293 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 294 1 {CSTYLE "" -1 -1 "" 0 1 0 0 6 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 295 1 {CSTYLE "" -1 -1 " " 0 1 4 1 170 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 296 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 297 1 {CSTYLE "" -1 -1 " " 0 1 25 1 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 298 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 299 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "The file sound.txt contain s the procedures we need in this worksheet." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "restart; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "lib name:=libname,`c:/portugal/sound`:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read `c:/portugal/sound/sound.txt`; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT 256 48 "PROPAGATION OF SOUND WAVES IN SPHERICAL CAVITIES" }} {PARA 19 "" 0 "" {TEXT -1 47 "S. P. Lipshitz, R. Portugal and J. Van derkooy" }}{PARA 297 "" 0 "" {TEXT 263 60 "University of Waterloo, Wat erloo, Ontario, N2L 3G1 - Canada." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 299 "" 0 "" {TEXT -1 0 "" }{TEXT 262 8 "Abstract" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 252 "The propagation of \+ sound waves in spherical cavities is revisited from the viewpoint of a computer algebra system. This allows the easy generation and visualiz ation of solutions with different initial conditions for this conceptu ally difficult problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 320 "We analyze a classical problem in mathematical physics from the p erspective of a computer algebra system. The problem consists in the a nalysis of the propagation of sound waves in spherical cavities using \+ the Fourier method to solve the wave equation. This paper has an asso ciated Maple worksheet and Maple program file" }{TEXT -1 152 " which a llow the reader to re-perform all calculations and display the animati ons of the physical quantities using the Maple symbolic computation sy stem" }{TEXT -1 468 ". The physical parameters like the cavity radius, sound velocity and the initial conditions can be changed. The calcula tion is performed analytically as far as possible, and with success fo r many types of initial conditions. If it is not possible to solve the problem exactly, the solution is performed numerically. When the Four ier coefficients can be found exactly, the time spent to perform the a nimation is much smaller than when numerical integration is required. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "The Wave Equation" }}{PARA 0 "" 0 "" {TEXT -1 61 "The equations that govern the propagation of sound w aves are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "rho*Diff(v,t)+grad(p)=0" "6#/,&*&%$rhoG\"\" \"-%%DiffG6$%\"vG%\"tGF'F'-%%gradG6#%\"pGF'\"\"!" }{TEXT -1 22 " (Eul er's equation), " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(rho,t)+div(rho*v)=0" "6#/,&-%%Diff G6$%$rhoG%\"tG\"\"\"-%$divG6#*&F(F*%\"vGF*F*\"\"!" }{TEXT -1 24 " (co ntinuity equation)," }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " p/p[0]=(rho/rho[0])^gamma" "6#/*&%\"pG\"\"\"&F%6#\"\"!!\"\")*&%$rhoGF& &F-6#F)F*%&gammaG" }{TEXT -1 32 " (adiabatic equation of state)," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 34 " is the particle velocity ve ctor, " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 23 " the absolute de nsity, " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 43 " the absolute pres sure, and the quantities " }{XPPEDIT 18 0 "rho[0]" "6#&%$rhoG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p[0]" "6#&%\"pG6#\"\"!" }{TEXT -1 83 " are the equilibrium density and pressure of the medium respective ly. The constant " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 29 " \+ is defined by the equation: " }}{PARA 271 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "gamma=C[p]/C[v]" "6#/%&gammaG*&&%\"CG6#%\"pG\"\"\"&F'6# %\"vG!\"\"" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 7 "where " } {XPPEDIT 18 0 "C[p]" "6#&%\"CG6#%\"pG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "C[v]" "6#&%\"CG6#%\"vG" }{TEXT -1 90 " are the specific heats ( at constant pressure and constant volume respectively). For air " } {XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 17 " is around 1.4. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 " For sound waves of moderate intensity, these equations can be simplified \+ to the following linearized system of differential equations in terms \+ of the " }{TEXT 265 8 "acoustic" }{TEXT -1 65 "quantities, which we sh all henceforth denote by the same symbols " }{TEXT 266 1 "p" }{TEXT -1 4 "and " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 20 "[note These \+ are the " }{TEXT 267 10 "deviations" }{TEXT -1 49 " from equilibrium; \+ the acoustic pressure is thus " }{XPPEDIT 18 0 "p-p[0]" "6#,&%\"pG\"\" \"&F$6#\"\"!!\"\"" }{TEXT -1 26 " and the acoustic density " } {XPPEDIT 18 0 "rho-rho[0]" "6#,&%$rhoG\"\"\"&F$6#\"\"!!\"\"" }{TEXT -1 3 ".]:" }}{PARA 272 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p=rho[0] *Diff(Phi,t)" "6#/%\"pG*&&%$rhoG6#\"\"!\"\"\"-%%DiffG6$%$PhiG%\"tGF*" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(p,t)-rho[0]*c^2*Delta^2*Phi=0" "6# /,&-%%DiffG6$%\"pG%\"tG\"\"\"**&%$rhoG6#\"\"!F**$%\"cG\"\"#F*%&DeltaG \"\"#%$PhiGF*!\"\"F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "a nd" }}{PARA 273 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p=c^2*rho" "6#/ %\"pG*&%\"cG\"\"#%$rhoG\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Ph i" "6#%$PhiG" }{TEXT -1 28 " is the velocity potential (" }{XPPEDIT 18 0 "(v=-grad(Phi)" "6#/%\"vG,$-%%gradG6#%$PhiG!\"\"" }{TEXT -1 7 ") \+ and " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 48 " is the speed of sou nd in the medium, given by:" }}{PARA 294 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "c^2=gamma*p[0]/rho[0]" "6#/*$%\"cG\"\"#*(%&gammaG\"\"\" &%\"pG6#\"\"!F)&%$rhoG6#F-!\"\"" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 13 "Eliminating " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 69 " from the differe ntial equations above, we get the wave equation for " }{XPPEDIT 18 0 " Phi" "6#%$PhiG" }{TEXT -1 1 ":" }}{PARA 264 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Delta^2*Phi -Diff(Phi,t,t)/c^2=0" "6#/,&*&%&DeltaG\"\"# %$PhiG\"\"\"F)*&-%%DiffG6%F(%\"tGF.F)*$%\"cG\"\"#!\"\"F2\"\"!" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "It is easy to verify that the density " }{XPPEDIT 18 0 "rho" "6#%$ rhoG" }{TEXT -1 17 " and the pressure" }{TEXT 268 0 "" }{TEXT -1 1 " \+ " }{TEXT 269 1 "p" }{TEXT -1 86 "obey the same partial differential eq uation, and so do the components of the velocity " }{XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 130 " if the velocity field is irrotational, which would be the case for inviscid fluids (and, to a close approximation, also for air)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 " In spherical cav ities it is advisable to use spherical coordinates, so that " } {XPPEDIT 18 0 "Phi=Phi(r,theta,phi,t)" "6#/%$PhiG-F$6&%\"rG%&thetaG%$p hiG%\"tG" }{TEXT -1 7 ". Let " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 63 " be the radius of the cavity. The rigid boundary condition is: \+ " }{XPPEDIT 18 0 "Diff(Phi,r)=0" "6#/-%%DiffG6$%$PhiG%\"rG\"\"!" } {TEXT -1 5 " for " }{XPPEDIT 18 0 "r=a" "6#/%\"rG%\"aG" }{TEXT -1 48 " . For sinusoidal solutions of angular frequency " }{XPPEDIT 18 0 "omeg a" "6#%&omegaG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Phi(r,theta,phi,t) = \+ Psi(r,theta,phi)*exp(-i*omega*t);" "6#/-%$PhiG6&%\"rG%&thetaG%$phiG%\" tG*&-%$PsiG6%F'F(F)\"\"\"-%$expG6#,$*(%\"iGF/%&omegaGF/F*F/!\"\"F/" } {TEXT -1 12 ", we have:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Delta^2*Psi +k^2*Psi=0" "6 #/,&*&%&DeltaG\"\"#%$PsiG\"\"\"F)*&%\"kG\"\"#F(F)F)\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "wher e " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 23 " is the wavenumber and \+ " }{XPPEDIT 18 0 "omega=k*c" "6#/%&omegaG*&%\"kG\"\"\"%\"cGF'" }{TEXT -1 254 ". This last equation is called Helmholtz's equation; its solu tions in spherical coordinates involve the well-known spherical harmon ics. The series solution of the wave equation can thus be obtained fro m the solution of the Helmholtz equation, and it is:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$PhiG6&%\"rG%&thetaG%$phiG%\"tG,&*&,&&%\"AG6#\"\" !\"\"\"*&&%\"BGF0F2F*F2F2F2-%\"FG6%F'F(F)F2F2-%$SumG6$-F:6$-F:6$,&*(-& %\"RG6#%\"lG6#F'F2,&*&&%#A1G6%FF%\"mG%\"nGF2-&%#Y1G6$FFFM6$F(F)F2F2*&& %#A2GFLF2-&%#Y2GFRFSF2F2F2-%$cosG6#*&&%&omegaG6$FFFNF2F*F2F2F2*(FBF2,& *&&%#B1GFLF2FOF2F2*&&%#B2GFLF2FWF2F2F2-%$sinGFfnF2F2/FN;F2%)infinityG/ FM;F1FF/FF;F1FgoF2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "R[l](r)" "6#-&%\"RG6#%\"lG6#%\"rG " }{TEXT -1 37 " are the spherical Bessel functions, " }{XPPEDIT 18 0 "Y1" "6#%#Y1G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y2" "6#%#Y2G" } {TEXT -1 35 " are the spherical harmonics, and " }{XPPEDIT 18 0 "A1" "6#%#A1G" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "A2" "6#%#A2G" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "B1" "6#%#B1G" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "B2" "6#%#B2G" }{TEXT -1 68 " are constants which are determined once \+ the initial conditions for " }{XPPEDIT 18 0 "Phi" "6#%$PhiG" }{TEXT -1 25 " are specified. The term " }{XPPEDIT 18 0 "(A[0]+B[0]*t)*F(r,th eta,phi)" "6#*&,&&%\"AG6#\"\"!\"\"\"*&&%\"BG6#F(F)%\"tGF)F)F)-%\"FG6%% \"rG%&thetaG%$phiGF)" }{TEXT -1 87 " is introduced in order that we h ave the most general solution of the wave equation. " }{XPPEDIT 18 0 "F(r,theta,phi)" "6#-%\"FG6%%\"rG%&thetaG%$phiG" }{TEXT -1 25 " is a h armonic function: " }{XPPEDIT 18 0 "Delta^2*F=0" "6#/*&%&DeltaG\"\"#% \"FG\"\"\"\"\"!" }{TEXT -1 70 ". For the boundary conditions we are go ing to analyze, we must impose " }{XPPEDIT 18 0 "F(r,theta,phi)=1" "6# /-%\"FG6%%\"rG%&thetaG%$phiG\"\"\"" }{TEXT -1 22 ". The remaining part (" }{XPPEDIT 18 0 "A[0]+B[0]*t;" "6#,&&%\"AG6#\"\"!\"\"\"*&&%\"BG6#F' F(%\"tGF(F(" }{TEXT -1 70 ") is the zero-frequency term in the series \+ expansion. The coefficient " }{XPPEDIT 18 0 "A[0]" "6#&%\"AG6#\"\"!" } {TEXT -1 135 " has no physical significance, since the physical quanti ties are obtained from derivatives of the velocity potential. The coe fficient " }{XPPEDIT 18 0 "B[0]" "6#&%\"BG6#\"\"!" }{TEXT -1 105 " is \+ related to the total mass inside the cavity, and must be zero for the \+ equilibrium mass density to be " }{XPPEDIT 18 0 "rho[0]" "6#&%$rhoG6# \"\"!" }{TEXT -1 70 ". These facts will be clarified in the next few s ections of the paper." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 219 " We want to perform the animation of the nor mal modes of the physical quantities discussed above, and their time d evelopment subject to initial conditions. A density plot animation is \+ a suitable way to represent " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 529 ", since this type of plot shows the variation of density direc tly. The white part of the density plot output represents the highest density, while the black part represents the lowest density. The int ermediate gray tones represent intermediate values of the density. Thi s type of plot can similarly be used to show the pressure and the velo city potential, the white part representing high pressure or high valu es of the velocity potential, and the black part representing low pres sure or low values of the velocity potential. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "The Angular Solu tions and the Spherical Harmonics" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "To solve the wave equation we suppose that " }{XPPEDIT 18 0 "Phi(r ,theta,phi,t)=R(r)*Y(theta,phi)*T(t)" "6#/-%$PhiG6&%\"rG%&thetaG%$phiG %\"tG*(-%\"RG6#F'\"\"\"-%\"YG6$F(F)F/-%\"TG6#F*F/" }{TEXT -1 118 ". Us ing the standard method of separation of variables, we obtain one diff erential equation for each of the functions " }{XPPEDIT 18 0 "R(r)" "6 #-%\"RG6#%\"rG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Y(theta,phi)" "6#-%\" YG6$%&thetaG%$phiG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "T(t)" "6#-%\" TG6#%\"tG" }{TEXT -1 20 ". The equation for " }{XPPEDIT 18 0 "Y(theta ,phi)" "6#-%\"YG6$%&thetaG%$phiG" }{TEXT -1 4 " is:" }}{PARA 275 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(theta)*Diff(sin(theta)*Diff(Y,th eta),theta)+Diff(Y,phi,phi)+l*(l+1)*sin(theta)^2*Y=0" "6#/,(*&-%$sinG6 #%&thetaG\"\"\"-%%DiffG6$*&-F'6#F)F*-F,6$%\"YGF)F*F)F*F*-F,6%F3%$phiGF 6F***%\"lGF*,&F8F*\"\"\"F*F*-F'6#F)\"\"#F3F*F*\"\"!" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 51 "In order for the solutions be finite in the region " }{XPPEDIT 18 0 "0<=theta*`<`*2*Pi" "6#1\"\"!**%&thetaG\" \"\"%\"phi+2*Pi " "6#R6#%$phiG7\"6$%)operatorG%&arrowG6\",&F%\"\"\"*&\"\"#F,%#PiGF,F,F *F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 " The associated Legendre functions can be defined \+ as:" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P[l,m](x)=(1-x ^2)^(m/2)*``(d^m*P[l]/d/x^m)" "6#/-&%\"PG6$%\"lG%\"mG6#%\"xG*&),&\"\" \"\"\"\"*$F+\"\"#!\"\"*&F)F0\"\"#F3F0-%!G6#**)%\"dGF)F0&F&6#F(F0F;F3)F +F)F3F0" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "P[l]" "6#&%\"PG6#%\"lG" } {TEXT -1 8 " is the " }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 146 "th L egendre polynomial. These functions are implemented in the Maple libra ry. We have implemented the associated Legendre functions with the nam e " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 30 ", and having three argu ments: " }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 21 ". Here is an example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P(5,1,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "When the second argument is zero we obtain the Legendre polynomials. For example: " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P(3,0,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 " The spherical harmonics are implemented her e with the names " }{XPPEDIT 18 0 "Y1" "6#%#Y1G" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "Y2" "6#%#Y2G" }{TEXT -1 26 ", and have two arguments: \+ " }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 16 ". The variables " }{XPPEDIT 18 0 "theta" "6#%& thetaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 13 ", as well as " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 157 ", are global variables thro ughout this worksheet. This means that they are not arguments of any f unction. Here are some examples of some spherical harmonics:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Y1(0,0);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "Y2(2,1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 47 "The Radial Solutions and the Bou ndary Condition" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 " The differe ntial equation for " }{XPPEDIT 18 0 "R(r)" "6#-%\"RG6#%\"rG" }{TEXT -1 4 " is:" }}{PARA 278 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(r^ 2*Diff(R,r),r)+(k^2*r^2-l*(l+1))*R=0" "6#/,&-%%DiffG6$*&%\"rG\"\"#-F&6 $%\"RGF)\"\"\"F)F.*&,&*&%\"kG\"\"#F)\"\"#F.*&%\"lGF.,&F6F.\"\"\"F.F.! \"\"F.F-F.F.\"\"!" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 25 "Its general solution for " }{XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" }{TEXT -1 4 " is:" }}{PARA 279 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[l](r) =C[1]*j[l](k*r)+C[2]*n[l](k*r)" "6#/-&%\"RG6#%\"lG6#%\"rG,&*&&%\"CG6# \"\"\"\"\"\"-&%\"jG6#F(6#*&%\"kGF1F*F1F1F1*&&F.6#\"\"#F1-&%\"nG6#F(6#* &F8F1F*F1F1F1" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "j[l](k*r)" "6#-&%\"jG6#%\"lG6#*&%\"kG\"\"\"%\"rGF+" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "n[l](k*r)" "6#-&%\"nG6#%\"lG6#*&%\" kG\"\"\"%\"rGF+" }{TEXT -1 92 " are respectively the spherical Bessel \+ function and the spherical Neumann function of order " }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 7 ". When " }{XPPEDIT 18 0 "k=0" "6#/%\"kG\"\" !" }{TEXT -1 10 ", we have:" }}{PARA 285 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R[l](r)=C[1]*r^l+C[2]/r^(l+1)" "6#/-&%\"RG6#%\"lG6#%\"r G,&*&&%\"CG6#\"\"\"\"\"\")F*F(F1F1*&&F.6#\"\"#F1)F*,&F(F1\"\"\"F1!\"\" F1" }{TEXT -1 5 " . " }}{PARA 0 "" 0 "" {TEXT -1 20 "Since the funct ions " }{XPPEDIT 18 0 "n[l]" "6#&%\"nG6#%\"lG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "1/r^(l+1)" "6#*&\"\"\"\"\"\")%\"rG,&%\"lGF%\"\"\"F%!\" \"" }{TEXT -1 21 " are not analytic at " }{XPPEDIT 18 0 "r=0" "6#/%\"r G\"\"!" }{TEXT -1 32 ", we require that the constants " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 76 " be zero. Without loss of gen erality, we can choose the radial solution for " }{XPPEDIT 18 0 "k<>0 " "6#0%\"kG\"\"!" }{TEXT -1 4 " as:" }}{PARA 280 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "R[l](r)=j[l](k*r)" "6#/-&%\"RG6#%\"lG6#%\"rG-&%\"jG6 #F(6#*&%\"kG\"\"\"F*F2" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "We implement the spherical Bessel fu nctions using the name " }{XPPEDIT 18 0 "SphericalBesselJ" "6#%1Spheri calBesselJG" }{TEXT -1 68 ", and use an alias to simplify the notation . Here are some examples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "alias(j=SphericalBesselJ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "j(0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "simp := x -> sort(collect(expand(x),[cos, sin]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simp(j(6,x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "The boundary condition is " } {XPPEDIT 18 0 "Diff(Phi,r)=0" "6#/-%%DiffG6$%$PhiG%\"rG\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "r=a" "6#/%\"rG%\"aG" }{TEXT -1 13 ". The refore " }{XPPEDIT 18 0 "Diff(R[l](r),r)=0" "6#/-%%DiffG6$-&%\"RG6#%\" lG6#%\"rGF-\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "r=a" "6#/%\"rG% \"aG" }{TEXT -1 18 ", or equivalently " }{XPPEDIT 18 0 "Diff(j(l,k[](l ,n)*r),r) = 0;" "6#/-%%DiffG6$-%\"jG6$%\"lG*&-&%\"kG6\"6$F*%\"nG\"\"\" %\"rGF2F3\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "r=a" "6#/%\"rG%\"a G" }{TEXT -1 16 ". The constants " }{XPPEDIT 18 0 "k" "6#%\"kG" } {TEXT -1 10 " must be: " }{XPPEDIT 18 0 "k[](l,n)=jSB(l,n)/a" "6#/-&% \"kG6\"6$%\"lG%\"nG*&-%$jSBG6$F)F*\"\"\"%\"aG!\"\"" }{TEXT -1 7 " wher e " }{XPPEDIT 18 0 "jSB(l,n)" "6#-%$jSBG6$%\"lG%\"nG" }{TEXT -1 8 " is the " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 68 "th zero of the deriv ative of the spherical Bessel function of order " }{XPPEDIT 18 0 "l" " 6#%\"lG" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 38 " \+ a positive integer). The wavenumber " }{XPPEDIT 18 0 "k" "6#%\"kG" } {TEXT -1 27 " and the angular frequency " }{XPPEDIT 18 0 "omega" "6#%& omegaG" }{TEXT -1 36 " are implemented with two arguments " }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n" "6#%\"nG" } {TEXT -1 55 ". To obtain a numerical evaluation, we have to use the " }{XPPEDIT 18 0 "evalf" "6#%&evalfG" }{TEXT -1 87 " command. For exampl e, the fundamental frequency and the corresponding wavenumber are:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(omega(0,1));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(k(0,1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The radia l part of the solution is described by the function " }{XPPEDIT 18 0 " R" "6#%\"RG" }{TEXT -1 26 " which has two arguments: " }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "n" "6#%\"nG" } {TEXT -1 14 ". For example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "R(0,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 " When " } {XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"!" }{TEXT -1 27 " the boundary condit ion at " }{XPPEDIT 18 0 "r=a" "6#/%\"rG%\"aG" }{TEXT -1 14 " implies t hat " }{XPPEDIT 18 0 "l=0" "6#/%\"lG\"\"!" }{TEXT -1 12 ". Therefore \+ " }{XPPEDIT 18 0 "F(r,theta,phi)" "6#-%\"FG6%%\"rG%&thetaG%$phiG" } {TEXT -1 53 " can be chosen equal to 1 without loss of generality." }} {PARA 0 "" 0 "" {TEXT -1 29 " The boundary condition for " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 25 " is the same as that for " } {XPPEDIT 18 0 "Phi" "6#%$PhiG" }{TEXT -1 9 ", since " }{XPPEDIT 18 0 "Diff(Phi,r)=0" "6#/-%%DiffG6$%$PhiG%\"rG\"\"!" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "r=a" "6#/%\"rG%\"aG" }{TEXT -1 15 " implies that " } {XPPEDIT 18 0 "Diff(rho,r)=0" "6#/-%%DiffG6$%$rhoG%\"rG\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "r=a" "6#/%\"rG%\"aG" }{TEXT -1 46 ". The \+ same conclusion applies to the pressure " }{XPPEDIT 18 0 "p" "6#%\"pG " }{TEXT -1 64 ", but only the radial component of the velocity need b e zero at " }{XPPEDIT 18 0 "r = a" "6#/%\"rG%\"aG" }{TEXT -1 1 "." }}} }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "Normal Mode Animation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 503 "In this section we analyze the normal mo des and perform their animation. The normal modes are solutions of the wave equation characterized by a single frequency, and are mutually o rthogonal. They form a basis, which allows us to express any solution \+ of the wave equation as a linear combination of the normal modes. If \+ two different normal modes have the same frequency, they are called de generate. In the case of the propagation of sound waves in spherical c avities, we can define the following modes:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 281 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "normalmode1 1(l,m,n) = R(l,n)*Y1(l,m)*cos(omega(l,n)*t);" "6#/-%-normalmode11G6%% \"lG%\"mG%\"nG*(-%\"RG6$F'F)\"\"\"-%#Y1G6$F'F(F.-%$cosG6#*&-%&omegaG6$ F'F)F.%\"tGF.F." }{TEXT -1 0 "" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 283 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "norma lmode12(l,m,n) = R(l,n)*Y1(l,m)*sin(omega(l,n)*t);" "6#/-%-normalmode1 2G6%%\"lG%\"mG%\"nG*(-%\"RG6$F'F)\"\"\"-%#Y1G6$F'F(F.-%$sinG6#*&-%&ome gaG6$F'F)F.%\"tGF.F." }{TEXT -1 0 "" }{TEXT -1 2 " ," }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 284 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "nor malmode21(l,m,n) = R(l,n)*Y2(l,m)*cos(omega(l,n)*t);" "6#/-%-normalmod e21G6%%\"lG%\"mG%\"nG*(-%\"RG6$F'F)\"\"\"-%#Y2G6$F'F(F.-%$cosG6#*&-%&o megaG6$F'F)F.%\"tGF.F." }{TEXT -1 0 "" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 282 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " normalmode22(l,m,n) = R(l,n)*Y2(l,m)*sin(omega(l,n)*t);" "6#/-%-normal mode22G6%%\"lG%\"mG%\"nG*(-%\"RG6$F'F)\"\"\"-%#Y2G6$F'F(F.-%$sinG6#*&- %&omegaG6$F'F)F.%\"tGF.F." }{TEXT -1 0 "" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 " The frequency is characterized by the numbers " }{XPPEDIT 18 0 "l" "6#%\"lG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 12 ". If we \+ fix " }{XPPEDIT 18 0 " l" "6#%\"lG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 13 " we can have " }{XPPEDIT 18 0 "l+1" "6#, &%\"lG\"\"\"\"\"\"F%" }{TEXT -1 12 " values for " }{XPPEDIT 18 0 "m" " 6#%\"mG" }{TEXT -1 72 ". In addition, for each frequency we can have f our different modes when " }{XPPEDIT 18 0 "m<>0" "6#0%\"mG\"\"!" } {TEXT -1 15 ", and two when " }{XPPEDIT 18 0 "m=0" "6#/%\"mG\"\"!" } {TEXT -1 36 ". So the number of degeneracies is " }{XPPEDIT 18 0 "4*l +2" "6#,&*&\"\"%\"\"\"%\"lGF&F&\"\"#F&" }{TEXT -1 17 ". The modes with " }{XPPEDIT 18 0 "l=0" "6#/%\"lG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m=0" "6#/%\"mG\"\"!" }{TEXT -1 74 " have two degeneracies and ar e spherically symmetric. Here is one example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "normalmode11(l=0,m=0,n=3);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 50 "After a coordinate transformation, we can use the \+ " }{TEXT 257 11 "densityplot" }{TEXT -1 31 " command to plot this func tion:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "radius_a := 1: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "densit yplot(subs(t=0,r=sqrt(x^2+y^2),normalmode11(l=0, m=0, n=3)),x=-radius_ a..radius_a,y=-radius_a..radius_a,axes=none,style=patchnogrid,grid=[50 ,50],scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The modes with " }{XPPEDIT 18 0 "l<>0" "6#0%\"lG\"\"!" }{TEXT -1 6 " and \+ " }{XPPEDIT 18 0 "m=0" "6#/%\"mG\"\"!" }{TEXT -1 69 " have two degene racies and are axially symmetric. Here is an example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "normalmode11(l=1, m=0, n=3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "densityplot(subs(t=0,r=sqrt (x^2+y^2),theta=arctan(x/y),normalmode11(l=1,m=0,n=3)),x=-radius_a..ra dius_a,y=-radius_a..radius_a,axes=none,style=patchnogrid,grid=[50,50], scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 " T he procedure " }{XPPEDIT 18 0 "animate_mode" "6#%-animate_modeG" } {TEXT -1 37 " performs the animation of the mode " }{XPPEDIT 18 0 "no rmalmode11" "6#%-normalmode11G" }{TEXT -1 54 "[note The other normal m odes can be animated with the " }{XPPEDIT 18 0 "animate_ic" "6#%+anima te_icG" }{TEXT -1 47 " function described in the next section.] for \+ " }{XPPEDIT 18 0 "phi=0" "6#/%$phiG\"\"!" }{TEXT -1 51 ". Let us see t he animation of the two normal modes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "vel_c := 320:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "animate_mode(l=0, m=0, n=3, frames=2, grid=[50,50]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "animate_mode(l=1, m=0, n=2, \+ frames=2, grid=[50,50]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 " When \+ " }{XPPEDIT 18 0 "m=0" "6#/%\"mG\"\"!" }{TEXT -1 164 ", the normal mod es have axial symmetry, and therefore the three-dimensional picture ca n be easily visualized by rotating the picture around the vertical axi s. When " }{XPPEDIT 18 0 "m<>0" "6#0%\"mG\"\"!" }{TEXT -1 58 ", it is \+ possible to animate the modes for other values of " }{XPPEDIT 18 0 "ph i" "6#%$phiG" }{TEXT -1 11 " using the " }{XPPEDIT 18 0 "animate_ic" " 6#%+animate_icG" }{TEXT -1 34 " function (see the next section )." }}} }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Animation with Initial Condition s" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 53 "Solution of the Wave Equatio n with Initial Conditions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 " For \+ simplicity, we impose the restriction that the initial conditions do n ot depend on " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 15 ". In this case " }{XPPEDIT 18 0 "m=0" "6#/%\"mG\"\"!" }{TEXT -1 41 ", and the g eneral solution simplifies to:" }}{PARA 260 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Phi(r,theta,phi,t)=A[0]+B[0]*t+Sum(Sum(j[l](k[l,n]*r)*Y 1[l,0](theta,phi)*(A[l,n]*cos(omega[l,n]*t) + B[l,n]*sin(omega[l,n]*t) ),n=1..infinity),l=0..infinity)" "6#/-%$PhiG6&%\"rG%&thetaG%$phiG%\"tG ,(&%\"AG6#\"\"!\"\"\"*&&%\"BG6#F/F0F*F0F0-%$SumG6$-F66$*(-&%\"jG6#%\"l G6#*&&%\"kG6$F?%\"nGF0F'F0F0-&%#Y1G6$F?F/6$F(F)F0,&*&&F-6$F?FEF0-%$cos G6#*&&%&omegaG6$F?FEF0F*F0F0F0*&&F36$F?FEF0-%$sinG6#*&&FT6$F?FEF0F*F0F 0F0F0/FE;\"\"\"%)infinityG/F?;F/F\\oF0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The initial conditio ns are: " }{XPPEDIT 18 0 "Phi(r,theta,phi,0)=Phi[0](r,theta)" "6#/-%$P hiG6&%\"rG%&thetaG%$phiG\"\"!-&F%6#F*6$F'F(" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "D[t](Phi)(r,theta,phi,0)=Phiprime[0](r,theta)" "6#/--&% \"DG6#%\"tG6#%$PhiG6&%\"rG%&thetaG%$phiG\"\"!-&%)PhiprimeG6#F06$F-F." }{TEXT -1 4 ". " }{XPPEDIT 18 0 "Phiprime[0]" "6#&%)PhiprimeG6#\"\"! " }{TEXT -1 25 " is proportional to both " }{XPPEDIT 18 0 "rho" "6#%$r hoG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT -1 109 ". Inverting the Fourier series[note We use the identities that come from the general Sturm-Liouville problem" }}{PARA 286 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(r^2*j[l](k[l,n]*r)*j[l](k[l,`n'`]*r),r=0..a)=0" "6# /-%$IntG6$*(%\"rG\"\"#-&%\"jG6#%\"lG6#*&&%\"kG6$F.%\"nG\"\"\"F(F5F5-&F ,6#F.6#*&&F26$F.%#n'GF5F(F5F5/F(;\"\"!%\"aGF@" }{TEXT -1 12 " when \+ " }{XPPEDIT 18 0 "n<>`n'`" "6#0%\"nG%#n'G" }{TEXT -1 5 " .] " }} {PARA 0 "" 0 "" {TEXT -1 10 "we obtain:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[l,n] = 2*Pi*in t(r^2*j[l](k[l,n]*r)*int(Phi[0]*Y1[l,0]*sin(theta),theta=0..Pi),r=0..a )/int(r^2*j[l](k[l,n]*r)^2,r=0..a)" "6#/&%\"AG6$%\"lG%\"nG**\"\"#\"\" \"%#PiGF+-%$intG6$*(%\"rG\"\"#-&%\"jG6#F'6#*&&%\"kG6$F'F(F+F1F+F+-F.6$ *(&%$PhiG6#\"\"!F+&%#Y1G6$F'FBF+-%$sinG6#%&thetaGF+/FI;FBF,F+/F1;FB%\" aGF+-F.6$*&F1\"\"#-&F56#F'6#*&&F:6$F'F(F+F1F+\"\"#/F1;FBFN!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B[l,n] = 2*Pi*int(r^2*j[l](k[l,n]*r)*in t(Phiprime[0]*Y1[l,0]*sin(theta),theta=0..Pi),r=0..a)/int(r^2*j[l](k[l ,n]*r)^2,r=0..a)/omega[l,n]" "6#/&%\"BG6$%\"lG%\"nG*,\"\"#\"\"\"%#PiGF +-%$intG6$*(%\"rG\"\"#-&%\"jG6#F'6#*&&%\"kG6$F'F(F+F1F+F+-F.6$*(&%)Phi primeG6#\"\"!F+&%#Y1G6$F'FBF+-%$sinG6#%&thetaGF+/FI;FBF,F+/F1;FB%\"aGF +-F.6$*&F1\"\"#-&F56#F'6#*&&F:6$F'F(F+F1F+\"\"#/F1;FBFN!\"\"&%&omegaG6 $F'F(Fgn" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The coefficient s " }{XPPEDIT 18 0 "A[0]" "6#&%\"AG6#\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "B[0]" "6#&%\"BG6#\"\"!" }{TEXT -1 5 " are:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "A[0] = 3/2*int(r^2*int(sin(theta)*(Phi[0]),theta=0..Pi),r=0..a)/a^3 " "6#/&%\"AG6#\"\"!**\"\"$\"\"\"\"\"#!\"\"-%$intG6$*&%\"rG\"\"#-F.6$*& -%$sinG6#%&thetaGF*&%$PhiG6#F'F*/F9;F'%#PiGF*/F1;F'%\"aGF**$FB\"\"$F, " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 268 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B[ 0] = 3/2*int(r^2*int(sin(theta)*(Phiprime[0]),theta=0..Pi),r=0..a)/a^3 " "6#/&%\"BG6#\"\"!**\"\"$\"\"\"\"\"#!\"\"-%$intG6$*&%\"rG\"\"#-F.6$*& -%$sinG6#%&thetaGF*&%)PhiprimeG6#F'F*/F9;F'%#PiGF*/F1;F'%\"aGF**$FB\" \"$F," }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 " Total mass conservation is expressed by:" }} {PARA 270 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Int(Int(r^2*(rho+ rho[0])*sin(theta),phi=0..2*Pi),theta=0..Pi),r=0..a)=constant" "6#/-%$ IntG6$-F%6$-F%6$*(%\"rG\"\"#,&%$rhoG\"\"\"&F/6#\"\"!F0F0-%$sinG6#%&the taGF0/%$phiG;F3*&\"\"#F0%#PiGF0/F7;F3F=/F,;F3%\"aG%)constantG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "rho=rh o[0]*Diff(Phi,t)/c^2" "6#/%$rhoG*(&F$6#\"\"!\"\"\"-%%DiffG6$%$PhiG%\"t GF)*$%\"cG\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "int(r^2*j[l] (k[l,n]*r),r=0..a)=0" "6#/-%$intG6$*&%\"rG\"\"#-&%\"jG6#%\"lG6#*&&%\"k G6$F.%\"nG\"\"\"F(F5F5/F(;\"\"!%\"aGF8" }{TEXT -1 18 ", the total mass (" }{XPPEDIT 18 0 "M" "6#%\"MG" }{TEXT -1 61 ") is conserved for all \+ initial conditions and it is given by:" }}{PARA 274 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M=4*Pi*a^3*rho[0]*(1+B[0]/c^2)/3" "6#/%\"MG*.\" \"%\"\"\"%#PiGF'%\"aG\"\"$&%$rhoG6#\"\"!F',&\"\"\"F'*&&%\"BG6#F.F'*$% \"cG\"\"#!\"\"F'F'\"\"$F8" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 28 "For the equilibrium density " }{XPPEDIT 18 0 "rho[0]" "6#&%$rho G6#\"\"!" }{TEXT -1 22 " to remain unchanged: " }{XPPEDIT 18 0 "Int(rh o,V=`V `..``)=0" "6#/-%$IntG6$%$rhoG/%\"VG;%#V~G%!G\"\"!" }{TEXT -1 17 ", we must impose " }{XPPEDIT 18 0 "B[0]=0" "6#/&%\"BG6#\"\"!F'" } {TEXT -1 8 " (since " }{XPPEDIT 18 0 "B[0]=(M-M[0])*c^2/M[0]" "6#/&%\" BG6#\"\"!*(,&%\"MG\"\"\"&F*6#F'!\"\"F+*$%\"cG\"\"#F+&F*6#F'F." }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "An approximate solution of the wave equation can be repre sented by a truncated double series:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$PhiG6$/%(trunc_lG%\"lG/%(trunc_ nG%\"nG,(-%\"AG6$\"\"!F1\"\"\"*&-%\"BGF0F2%\"tGF2F2-%$sumG6$.-F86$.*(- %\"RG6$F)F,F2-%#Y1G6$F)F1F2,&*&-F/FAF2-%$cosG6#*&-%&omegaGFAF2F6F2F2F2 *&-F5FAF2-%$sinGFJF2F2F2/F,;F2F,/F);F1F)F2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The terms " }{XPPEDIT 18 0 "A[0]" "6#&%\"AG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B[0]" " 6#&%\"BG6#\"\"!" }{TEXT -1 14 " are given by " }{XPPEDIT 18 0 "A(0,0) " "6#-%\"AG6$\"\"!F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B(0,0)" "6#- %\"BG6$\"\"!F&" }{TEXT -1 103 " which are treated separately from the \+ other coefficients, because the calculation of the general term " } {XPPEDIT 18 0 "A(0,n)" "6#-%\"AG6$\"\"!%\"nG" }{TEXT -1 15 " does not \+ give " }{XPPEDIT 18 0 "A(0,0)" "6#-%\"AG6$\"\"!F&" }{TEXT -1 26 " as a limiting case when " }{XPPEDIT 18 0 "n=0" "6#/%\"nG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " The procedure " }{TEXT 258 10 "animate_ic" }{TEXT -1 40 " performs th e animation of the solution " }{XPPEDIT 18 0 "Phi(trunc_l=l,trunc_n=n) " "6#-%$PhiG6$/%(trunc_lG%\"lG/%(trunc_nG%\"nG" }{TEXT -1 35 " subject to the initial conditions " }{XPPEDIT 18 0 "Phi[0]" "6#&%$PhiG6#\"\"! " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Phiprime[0]" "6#&%)PhiprimeG6#\" \"!" }{TEXT -1 80 ". The details about its arguments can be obtained f rom the help pages (command: " }{TEXT 259 11 "?animate_ic" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 " As a general example, suppose that the initial conditions are spheri cally symmetric:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Phi0 := f(r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Phiprime0 := 0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 " The coefficients " }{XPPEDIT 18 0 "A[l,n]" "6#&%\"AG6$%\"lG%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B[l,n]" "6#&%\"BG6$%\"lG%\"nG" }{TEXT -1 9 " will be:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "unassign('radius_a , vel_c'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "alias(a=radiu s_a, c=vel_c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "A(0,0);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "B(0,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "expand(A(l,n));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "B(l,n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 " Th e general solution for the velocity potential with just the first two \+ terms in the Fourier expansion is:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Phi(trunc_n=1);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 42 "Initial Conditions with Spherical Symmetry" }} {PARA 0 "" 0 "" {TEXT -1 53 " When the initial conditions do no t depend on " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 113 ", the results can be di splayed as a two-dimensional plot of the physical quantity of interest against the radius " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 96 ". The actual three-dimensional picture can easily be visualized by using th e spherical symmetry." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 242 " As a first physical example, we want to perfo rm the animation of the physical quantities for the solution of the wa ve equation corresponding to a localized shell-like pulse initially tr aveling inward. This pulse will reflect first at " }{XPPEDIT 18 0 "r=0 " "6#/%\"rG\"\"!" }{TEXT -1 78 ", changing its propagation to the outw ard direction, and will then reflect at " }{XPPEDIT 18 0 "r=a" "6#/%\" rG%\"aG" }{TEXT -1 152 ", again changing its direction, and so on. Th is example will allow us to analyze the sign changes in the physical q uantities during these reflections. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 " To begin, let us define " } {XPPEDIT 18 0 "Psi" "6#%$PsiG" }{TEXT -1 4 " by:" }}{PARA 287 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Phi(r,t)=Psi(r+c*t)/r" "6#/-%$PhiG6$% \"rG%\"tG*&-%$PsiG6#,&F'\"\"\"*&%\"cGF.F(F.F.F.F'!\"\"" }{TEXT -1 2 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 13 "We know that " }{XPPEDIT 18 0 "Psi(r ,t)" "6#-%$PsiG6$%\"rG%\"tG" }{TEXT -1 76 " is a solution of the one-d imensional wave equation, and so the argument of " }{XPPEDIT 18 0 "Psi " "6#%$PsiG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "r+c*t" "6#,&%\"rG\"\" \"*&%\"cGF%%\"tGF%F%" }{TEXT -1 76 ". The initial conditions that simu late the inwardly-directed pulse are thus:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 288 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Phi[0]=``(P si(r+c*t)/r)*``[``][t=0]" "6#/&%$PhiG6#\"\"!*&-%!G6#*&-%$PsiG6#,&%\"rG \"\"\"*&%\"cGF2%\"tGF2F2F2F1!\"\"F2&&F*6#F*6#/F5F'F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 289 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Phiprime[0]=``(Diff(Psi(r+c*t),t)/r)*``[``][t=0]" "6#/ &%)PhiprimeG6#\"\"!*&-%!G6#*&-%%DiffG6$-%$PsiG6#,&%\"rG\"\"\"*&%\"cGF5 %\"tGF5F5F8F5F4!\"\"F5&&F*6#F*6#/F8F'F5" }{TEXT -1 3 " , " }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Psi(r)" "6#-%$PsiG6#%\"rG" }{TEXT -1 63 " is a function which is nonzero in a sub-interval smalle r than " }{XPPEDIT 18 0 "[0,a]" "6#7$\"\"!%\"aG" }{TEXT -1 50 ", and d rops off to zero outside this sub-interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 " If we suppose that the aver age density inside the cavity remains " }{XPPEDIT 18 0 "rho[0]" "6#&%$ rhoG6#\"\"!" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "Psi" "6#%$PsiG" } {TEXT -1 37 " must satisfy the following equation:" }}{PARA 290 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Psi(r),r=0..a)=a*Psi(a)" "6#/-%$I ntG6$-%$PsiG6#%\"rG/F*;\"\"!%\"aG*&F.\"\"\"-F(6#F.F0" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "This \+ result can be proved by performing an integration by parts with respec t to the variable " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "B[0]=0" "6#/&%\"BG6#\"\"!F'" }{TEXT -1 26 ", and using the fact that " }{XPPEDIT 18 0 "Psi" "6#%$PsiG" }{TEXT -1 14 " is finite at " }{XPPEDIT 18 0 "r=0" "6#/%\"rG\"\"!" }{TEXT -1 37 ". For a localized pulse we have that " }{XPPEDIT 18 0 "Psi(a)=0" " 6#/-%$PsiG6#%\"aG\"\"!" }{TEXT -1 28 ", and so the area under the " } {XPPEDIT 18 0 "Psi" "6#%$PsiG" }{TEXT -1 21 " curve must be zero. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 " Let \+ us choose numerical values for sound speed and cavity radius:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "vel_c := 340;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "radius_a := 1;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "alias('a=radius_a'):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 " The function " }{XPPEDIT 18 0 "pulse" "6#%&pulseG" }{TEXT -1 32 " helps in the definition of the " }{XPPEDIT 18 0 "Psi" " 6#%$PsiG" }{TEXT -1 10 " function:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "pulse := (center,width) -> " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "piecewise(r " 0 "" {MPLTEXT 1 0 73 "PSI := subs(r=r+vel_c*t,simplify(pulse(4*a/9,2*a/9)-pulse(5*a/9,2* a/9)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Psi " "6#%$PsiG" }{TEXT -1 24 " satisfies the equation " }{XPPEDIT 18 0 "I nt(Psi(r),r=0..a)=0" "6#/-%$IntG6$-%$PsiG6#%\"rG/F*;\"\"!%\"aGF-" } {TEXT -1 48 " as we can see by issuing the following command:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "int(sub s(t=0,PSI),r=0..a); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 " The init ial conditions are:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Phi0 := simplify(subs(t=0,PSI/r));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Phiprime0 := simplify(subs(t=0,diff (PSI/r,t)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 " Now we can confi rm that the average density inside the cavity is " }{XPPEDIT 18 0 "rho [0]" "6#&%$rhoG6#\"\"!" }{TEXT -1 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "B(0,0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 " The initial conditions can be visualized by the followin g plots:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(Phi0,r=0..radius_a,title='Phi0');" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 76 "plot(Phiprime0,r=0..radius_a,title='Phiprime0',ytic kmarks=[-10000,0,10000]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 " Fi rst let us calculate the velocity potential with enough terms in the F ourier expansion, and store the result in a variable to save time:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "vel_pot := evalf(Phi(trunc_n=20)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Th e first 3 terms are:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(Phi(trunc_n=2),2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 " The animation of the velocity potential can be obtained \+ by the following command:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "animate_ic(function=vel_pot,color=black,frames= 9,xtickmarks=[radius_a],ytickmarks=[1],r=0..a,verticalrange=-5.51..5.5 1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 " The animation of " } {XPPEDIT 18 0 "rho/rho[0]" "6#*&%$rhoG\"\"\"&F$6#\"\"!!\"\"" }{TEXT -1 4 " is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dens := diff( vel_pot,t)/vel_c^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "ani mate_ic(function=dens,color=black,frames=9,xtickmarks=[radius_a],ytick marks=[1],verticalrange=-1..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 " To understand why it is that mass is conserved, it is more useful to animate the function " }{XPPEDIT 18 0 "r^2*rho/rho[0]" "6#*(%\"rG \"\"#%$rhoG\"\"\"&F&6#\"\"!!\"\"" }{TEXT -1 12 " instead of " } {XPPEDIT 18 0 "rho/rho[0]" "6#*&%$rhoG\"\"\"&F$6#\"\"!!\"\"" }{TEXT -1 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "r2dens := expand(r^2*dens):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "animate_ic(function=r2dens,color=black,frames=9,xtickmarks=[r adius_a],ytickmarks=[1],verticalrange=-0.1..0.1);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 144 "From this last animation we can more easily see t hat the total mass is conserved even though the sign of the density in verts upon reflection at " }{XPPEDIT 18 0 "r=0" "6#/%\"rG\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " The " }{TEXT 264 10 "animate_ic" }{TEXT -1 136 " command \+ allows one to animate more than one function at the same time, like th e radial velocity together with the density, for example." }{MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Analysis of the Ref lections" }}{PARA 0 "" 0 "" {TEXT -1 27 " We have shown that " }{XPPEDIT 18 0 "Phi" "6#%$PhiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 140 " obey the same differential equation with the same boundary co nditions; therefore they have the same behavior concerning the reflect ions at " }{XPPEDIT 18 0 "r=a" "6#/%\"rG%\"aG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "r=0" "6#/%\"rG\"\"!" }{TEXT -1 46 ". Let us begin by \+ analyzing what happens at " }{XPPEDIT 18 0 "r=0" "6#/%\"rG\"\"!" } {TEXT -1 26 ". We define the functions " }{XPPEDIT 18 0 "Psi[Phi]" "6# &%$PsiG6#%$PhiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Psi[rho]" "6#&%$PsiG 6#%$rhoG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "Psi[p]" "6#&%$PsiG6#%\" pG" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "r*Phi" "6#*&%\"rG\"\"\"%$PhiGF% " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "r*rho" "6#*&%\"rG\"\"\"%$rhoGF%" } {TEXT -1 6 ", and " }{XPPEDIT 18 0 "r*p" "6#*&%\"rG\"\"\"%\"pGF%" } {TEXT -1 67 " respectively. For the spherically-symmetric case, we k now that " }{XPPEDIT 18 0 "Psi[Phi]" "6#&%$PsiG6#%$PhiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Psi[rho]" "6#&%$PsiG6#%$rhoG" }{TEXT -1 6 ", and \+ " }{XPPEDIT 18 0 "Psi[p]" "6#&%$PsiG6#%\"pG" }{TEXT -1 91 " satisfy th e one-dimensional wave equation. Therefore, the analysis of the behavi or of the " }{XPPEDIT 18 0 "Psi" "6#%$PsiG" }{TEXT -1 35 " functions i s easier than that of " }{XPPEDIT 18 0 "Phi" "6#%$PhiG" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 5 ", or " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 7 ". The " }{XPPEDIT 18 0 "Psi" "6#%$PsiG" }{TEXT -1 66 " expressions can be readily obtained from the following \+ equations:" }}{PARA 291 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Phi(r,t )=(A[0]+B[0]*t)+Sum(j[0](k[0,n]*r)*T[n](t),n=1..infinity)" "6#/-%$PhiG 6$%\"rG%\"tG,(&%\"AG6#\"\"!\"\"\"*&&%\"BG6#F-F.F(F.F.-%$SumG6$*&-&%\"j G6#F-6#*&&%\"kG6$F-%\"nGF.F'F.F.-&%\"TG6#F@6#F(F./F@;\"\"\"%)infinityG F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 293 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(r,t)=rho[0]*B[0]+rho[0]*Sum(j[0] (k[0,n]*r)*Diff(T[n](t),t),n=1..infinity)" "6#/-%\"pG6$%\"rG%\"tG,&*&& %$rhoG6#\"\"!\"\"\"&%\"BG6#F.F/F/*&&F,6#F.F/-%$SumG6$*&-&%\"jG6#F.6#*& &%\"kG6$F.%\"nGF/F'F/F/-%%DiffG6$-&%\"TG6#FC6#F(F(F//FC;\"\"\"%)infini tyGF/F/" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 5 "where" }} {PARA 292 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "T[n](t)=(A[0,n]*cos(o mega[0,n]*t)+B[0,n]*sin(omega[0,n]*t))/2/sqrt(Pi)" "6#/-&%\"TG6#%\"nG6 #%\"tG*(,&*&&%\"AG6$\"\"!F(\"\"\"-%$cosG6#*&&%&omegaG6$F1F(F2F*F2F2F2* &&%\"BG6$F1F(F2-%$sinG6#*&&F86$F1F(F2F*F2F2F2F2\"\"#!\"\"-%%sqrtG6#%#P iGFE" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "It is easy to show that " }{XPPEDIT 18 0 "Psi[Phi]" "6 #&%$PsiG6#%$PhiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Psi[rho]" "6#&%$Psi G6#%$rhoG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "Psi[p]" "6#&%$PsiG6#% \"pG" }{TEXT -1 43 " satisfy a Dirichlet boundary condition at " } {XPPEDIT 18 0 "r=0" "6#/%\"rG\"\"!" }{TEXT -1 91 ", and therefore thes e quantities undergo a simple polarity inversion after a reflection at " }{XPPEDIT 18 0 "r=0" "6#/%\"rG\"\"!" }{TEXT -1 143 ". We can thus \+ conclude that the velocity potential, density, and pressure must also \+ undergo a simple polarity inversion after a reflection at " }{XPPEDIT 18 0 "r=0" "6#/%\"rG\"\"!" }{TEXT -1 51 ", their wave-shapes being mai ntained. We note that " }{XPPEDIT 18 0 "Phi" "6#%$PhiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 27 " ( in contradistinction to " }{XPPEDIT 18 0 "Psi[Phi]" "6#&%$PsiG6#%$PhiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "P si[rho]" "6#&%$PsiG6#%$rhoG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "Psi[ p]" "6#&%$PsiG6#%\"pG" }{TEXT -1 17 ") have values at " }{XPPEDIT 18 0 "r=0" "6#/%\"rG\"\"!" }{TEXT -1 54 " which are in general finite and not identically zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " The analysis at " }{XPPEDIT 18 0 "r=a" "6#/%\"r G%\"aG" }{TEXT -1 26 " is straightforward since " }{XPPEDIT 18 0 "Psi[ Phi]=r*Phi" "6#/&%$PsiG6#%$PhiG*&%\"rG\"\"\"F'F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(Phi(r,t),r)=0" "6#/-%%DiffG6$-%$PhiG6$%\"rG%\"tGF *\"\"!" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "r=a" "6#/%\"rG%\"aG" } {TEXT -1 11 " imply that" }}{PARA 295 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Psi[Phi](a,t)=a*Diff(Psi[Phi](a,t),r)" "6#/-&%$PsiG6#%$ PhiG6$%\"aG%\"tG*&F*\"\"\"-%%DiffG6$-&F&6#F(6$F*F+%\"rGF-" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 45 "Similar boundary conditions are \+ satisfied by " }{XPPEDIT 18 0 "Psi[rho]" "6#&%$PsiG6#%$rhoG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Psi[p]" "6#&%$PsiG6#%\"pG" }{TEXT -1 6 ". The " }{XPPEDIT 18 0 "Psi" "6#%$PsiG" }{TEXT -1 115 "'s thus obey Rob in-type boundary conditions. These conditions do not maintain the wave -shape after a reflection at " }{XPPEDIT 18 0 "r=a" "6#/%\"rG%\"aG" } {TEXT -1 72 ", but the deviations from a simple non-polarity-inverting reflection at " }{XPPEDIT 18 0 "r=a" "6#/%\"rG%\"aG" }{TEXT -1 110 " \+ are relatively small in this case. After many reflections, however, th e initial localized shape will be lost." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 " The behavior of the radial velocity \+ " }{XPPEDIT 18 0 "v[r]=-Diff(Phi,r)" "6#/&%\"vG6#%\"rG,$-%%DiffG6$%$Ph iGF'!\"\"" }{TEXT -1 99 " under reflection can be deduced by the follo wing analysis. If we have a pulse moving inward, then " }{XPPEDIT 18 0 "v[r]=-Diff(Psi[Phi](r+c*t)/r,r)" "6#/&%\"vG6#%\"rG,$-%%DiffG6$*&-&% $PsiG6#%$PhiG6#,&F'\"\"\"*&%\"cGF4%\"tGF4F4F4F'!\"\"F'F8" }{TEXT -1 26 ". It is easy to show that " }{XPPEDIT 18 0 "v[r]=Phi/r-c*rho/rho[0 ]" "6#/&%\"vG6#%\"rG,&*&%$PhiG\"\"\"F'!\"\"F+*(%\"cGF+%$rhoGF+&F/6#\" \"!F,F," }{TEXT -1 42 ". If the wave is moving outward, we have " } {XPPEDIT 18 0 "v[r]=Phi/r+c*rho/rho[0]" "6#/&%\"vG6#%\"rG,&*&%$PhiG\" \"\"F'!\"\"F+*(%\"cGF+%$rhoGF+&F/6#\"\"!F,F+" }{TEXT -1 29 ". In the \+ beginning we have " }{XPPEDIT 18 0 "c*rho/rho[0]" "6#*(%\"cG\"\"\"%$r hoGF%&F&6#\"\"!!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`<<`" "6#%#<