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{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 31 "lib
name:=libname,`/user/sound`:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rea
d `/home/sound/sound.mpl`; " }}}{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. Vanderkooy" }}{PARA 
297 "" 0 "" {TEXT 261 60 "University of Waterloo, Waterloo, Ontario, N
2L 3G1 - Canada." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 299 "" 0 "" 
{TEXT -1 0 "" }{TEXT 260 8 "Abstract" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 252 "The propagation of sound waves in sphe
rical cavities is revisited from the viewpoint of a computer algebra s
ystem. This allows the easy generation and visualization of solutions \+
with different initial conditions for this conceptually difficult prob
lem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 320 "We analyze a clas
sical problem in mathematical physics from the perspective of a comput
er algebra system. The problem consists in the analysis of the propaga
tion of sound waves in spherical cavities using the Fourier method to \+
solve the wave equation.  This paper has an associated Maple worksheet
 and Maple program file" }{TEXT -1 152 " which allow the reader to re-
perform all calculations and display the animations of the physical qu
antities using the Maple symbolic computation system" }{TEXT -1 468 ".
 The physical parameters like the cavity radius, sound velocity and th
e initial conditions can be changed. The calculation is performed anal
ytically as far as possible, and with success for many types of initia
l conditions. If it is not possible to solve the problem exactly, the \+
solution is performed numerically. When the Fourier coefficients can b
e found exactly, the time spent to perform the animation is much small
er 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 waves are:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 
0 "rho*Diff(v,t)+grad(p)=0" "/,&*&%$rhoG\"\"\"-%%DiffG6$%\"vG%\"tGF&F&
-%%gradG6#%\"pGF&\"\"!" }{TEXT -1 22 "  (Euler's equation), " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Diff(rho,t)+div(rho*v)=0" "/,&-%%DiffG6$%$rhoG%\"tG\"\"\"-%$divG
6#*&F'F)%\"vGF)F)\"\"!" }{TEXT -1 24 "  (continuity 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])^g
amma" "/*&%\"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
" "I\"vG6\"" }{TEXT -1 34 " is the particle velocity vector, " }
{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 23 " the absolute density, \+
" }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 43 " the absolute pressure, \+
and the quantities " }{XPPEDIT 18 0 "rho[0]" "&%$rhoG6#\"\"!" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "p[0]" "&%\"pG6#\"\"!" }{TEXT -1 83 " are \+
the equilibrium density and pressure of the medium respectively. The c
onstant " }{XPPEDIT 18 0 "gamma" "I&gammaG6\"" }{TEXT -1 29 " is defin
ed by the equation: " }}{PARA 271 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "gamma=C[p]/C[v]" "/%&gammaG*&&%\"CG6#%\"pG\"\"\"&F&6#%\"vG!\"\"" }
{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 7 "where  " }{XPPEDIT 18 0 
"C[p]" "&%\"CG6#%\"pG" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "C[v]" "&%
\"CG6#%\"vG" }{TEXT -1 90 "  are the specific heats (at constant press
ure and constant volume respectively). For air " }{XPPEDIT 18 0 "gamma
" "I&gammaG6\"" }{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 
263 8 "acoustic" }{TEXT -1 65 "quantities, which we shall henceforth d
enote by the same symbols " }{TEXT 264 1 "p" }{TEXT -1 4 "and " }
{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 20 "[note These are the " }
{TEXT 265 10 "deviations" }{TEXT -1 49 " from equilibrium; the acousti
c pressure is thus " }{XPPEDIT 18 0 "p-p[0]" ",&%\"pG\"\"\"&F#6#\"\"!!
\"\"" }{TEXT -1 26 " and the acoustic density " }{XPPEDIT 18 0 "rho-rh
o[0]" ",&%$rhoG\"\"\"&F#6#\"\"!!\"\"" }{TEXT -1 3 ".]:" }}{PARA 272 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p=rho[0]*Diff(Phi,t)" "/%\"pG*&&
%$rhoG6#\"\"!\"\"\"-%%DiffG6$%$PhiG%\"tGF)" }{TEXT -1 3 " , " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Diff(p,t)-rho[0]*c^2*Delta^2*Phi=0" "/,&-%%DiffG6$%\"pG%\"tG\"\"
\"**&%$rhoG6#\"\"!F)*$%\"cG\"\"#F)%&DeltaG\"\"#%$PhiGF)!\"\"F." }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 273 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "p=c^2*rho" "/%\"pG*&%\"cG\"\"#%$rhoG
\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Phi" "I$PhiG6\"" }{TEXT -1 
28 " is the velocity potential (" }{XPPEDIT 18 0 "(v=-grad(Phi)" "/%\"
vG,$-%%gradG6#%$PhiG!\"\"" }{TEXT -1 7 ") and  " }{XPPEDIT 18 0 "c" "I
\"cG6\"" }{TEXT -1 48 " is the speed of sound  in the medium, given by
:" }}{PARA 294 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c^2=gamma*p[0]/
rho[0]" "/*$%\"cG\"\"#*(%&gammaG\"\"\"&%\"pG6#\"\"!F(&%$rhoG6#F,!\"\"
" }{TEXT -1 4 "  . " }}{PARA 0 "" 0 "" {TEXT -1 13 "Eliminating  " }
{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "rho" 
"I$rhoG6\"" }{TEXT -1 69 " from the differential equations above, we g
et the wave equation for " }{XPPEDIT 18 0 "Phi" "I$PhiG6\"" }{TEXT -1 
1 ":" }}{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta^2*Phi -
Diff(Phi,t,t)/c^2=0" "/,&*&%&DeltaG\"\"#%$PhiG\"\"\"F(*&-%%DiffG6%F'%
\"tGF-F(*$%\"cG\"\"#!\"\"F1\"\"!" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "It is easy to verify that
 the density " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 17 " and the
 pressure" }{TEXT 266 0 "" }{TEXT -1 1 " " }{TEXT 267 1 "p" }{TEXT -1 
86 "obey the same partial differential equation, and so do the compone
nts of the velocity " }{XPPEDIT 18 0 "v" "I\"vG6\"" }{TEXT -1 130 " if
 the velocity field is irrotational, which would be the case for invis
cid 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 cavities it is advisable to use s
pherical coordinates, so that  " }{XPPEDIT 18 0 "Phi=Phi(r,theta,phi,t
)" "/%$PhiG-F#6&%\"rG%&thetaG%$phiG%\"tG" }{TEXT -1 7 ". Let  " }
{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 63 " be the radius of the cavit
y. The rigid boundary condition is: " }{XPPEDIT 18 0 "Diff(Phi,r)=0" "
/-%%DiffG6$%$PhiG%\"rG\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "r=a" 
"/%\"rG%\"aG" }{TEXT -1 48 ". For sinusoidal solutions of angular freq
uency " }{XPPEDIT 18 0 "omega" "I&omegaG6\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "Phi(r,theta,phi,t) = Psi(r,theta,phi)*exp(-i*omega*t);
" "/-%$PhiG6&%\"rG%&thetaG%$phiG%\"tG*&-%$PsiG6%F&F'F(\"\"\"-%$expG6#,
$*(%\"iGF.%&omegaGF.F)F.!\"\"F." }{TEXT -1 12 ",   we have:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 " Delta^2*Psi +k^2*Psi=0" "/,&*&%&DeltaG\"\"#%$PsiG\"\"\"F(*&%\"k
G\"\"#F'F(F(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "k" "I\"kG6\"" }
{TEXT -1 23 " is the wavenumber and " }{XPPEDIT 18 0 "omega=k*c" "/%&o
megaG*&%\"kG\"\"\"%\"cGF&" }{TEXT -1 254 ".  This last equation is cal
led Helmholtz's equation; its solutions in spherical coordinates invol
ve the well-known spherical harmonics. The series solution of the wave
 equation can thus be obtained from the solution of the Helmholtz equa
tion, and it is:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$PhiG6&%\"rG%&t
hetaG%$phiG%\"tG,&*&,&&%\"AG6#\"\"!\"\"\"*&&%\"BGF0F2F*F2F2F2-%\"FG6%F
'F(F)F2F2-%$SumG6$-F:6$-F:6$,&*(-&%\"RG6#%\"lG6#F'F2,&*&&%#A1G6%FF%\"m
G%\"nGF2-&%#Y1G6$FFFM6$F(F)F2F2*&&%#A2GFLF2-&%#Y2GFRFSF2F2F2-%$cosG6#*
&&%&omegaG6$FFFNF2F*F2F2F2*(FBF2,&*&&%#B1GFLF2FOF2F2*&&%#B2GFLF2FWF2F2
F2-%$sinGFfnF2F2/FN;F2%)infinityG/FM;F1FF/FF;F1FgoF2" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "
R[l](r)" "-&%\"RG6#%\"lG6#%\"rG" }{TEXT -1 37 " are the spherical Bess
el functions, " }{XPPEDIT 18 0 "Y1" "I#Y1G6\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "Y2" "I#Y2G6\"" }{TEXT -1 35 " are the  spherical harmon
ics, and " }{XPPEDIT 18 0 "A1" "I#A1G6\"" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "A2" "I#A2G6\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "B1" "I#B1G6\"" }
{TEXT -1 6 ", and " }{XPPEDIT 18 0 "B2" "I#B2G6\"" }{TEXT -1 68 " are \+
constants which are determined once the initial conditions for " }
{XPPEDIT 18 0 "Phi" "I$PhiG6\"" }{TEXT -1 25 " are specified. The term
 " }{XPPEDIT 18 0 "(A[0]+B[0]*t)*F(r,theta,phi)" "*&,&&%\"AG6#\"\"!\"
\"\"*&&%\"BG6#F'F(%\"tGF(F(F(-%\"FG6%%\"rG%&thetaG%$phiGF(" }{TEXT -1 
87 "  is introduced in order that we have the most general solution of
 the wave equation.  " }{XPPEDIT 18 0 "F(r,theta,phi)" "-%\"FG6%%\"rG%
&thetaG%$phiG" }{TEXT -1 25 " is a harmonic function: " }{XPPEDIT 18 
0 "Delta^2*F=0" "/*&%&DeltaG\"\"#%\"FG\"\"\"\"\"!" }{TEXT -1 70 ". For
 the boundary conditions we are going to analyze, we must impose " }
{XPPEDIT 18 0 "F(r,theta,phi)=1" "/-%\"FG6%%\"rG%&thetaG%$phiG\"\"\"" 
}{TEXT -1 22 ". The remaining part (" }{XPPEDIT 18 0 "A[0]+B[0]*t;" ",
&&%\"AG6#\"\"!\"\"\"*&&%\"BG6#F&F'%\"tGF'F'" }{TEXT -1 70 ") is the ze
ro-frequency term in the series expansion. The coefficient " }
{XPPEDIT 18 0 "A[0]" "&%\"AG6#\"\"!" }{TEXT -1 135 " has no physical s
ignificance, since the physical quantities are obtained from derivativ
es of the velocity potential.  The coefficient " }{XPPEDIT 18 0 "B[0]
" "&%\"BG6#\"\"!" }{TEXT -1 105 " is related to the total mass inside \+
the cavity, and must be zero for the equilibrium mass density to be " 
}{XPPEDIT 18 0 "rho[0]" "&%$rhoG6#\"\"!" }{TEXT -1 70 ". These facts w
ill be clarified in the next few sections of the paper." }}{PARA 0 "" 
0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 219 "        We want to
 perform the animation of the normal modes of the physical quantities \+
discussed above, and their time development subject to initial conditi
ons. A density plot animation is a suitable way to represent " }
{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 529 ", since this type of pl
ot shows the variation of density directly. The white part of the dens
ity plot output  represents the highest density, while the black part \+
 represents the lowest density. The intermediate gray tones represent \+
intermediate values of the density. This type of plot can similarly be
 used to show the pressure and the velocity potential, the white part \+
representing high pressure or high values of the velocity potential, a
nd the black part representing low pressure or low values of the veloc
ity potential.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT -1 49 "The Angular Solutions and the Spherical Harmonics
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "To solve the wave equation we \+
suppose that " }{XPPEDIT 18 0 "Phi(r,theta,phi,t)=R(r)*Y(theta,phi)*T(
t)" "/-%$PhiG6&%\"rG%&thetaG%$phiG%\"tG*(-%\"RG6#F&\"\"\"-%\"YG6$F'F(F
.-%\"TG6#F)F." }{TEXT -1 118 ". Using the standard method of separatio
n of variables, we obtain one differential equation for each of the fu
nctions " }{XPPEDIT 18 0 "R(r)" "-%\"RG6#%\"rG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "Y(theta,phi)" "-%\"YG6$%&thetaG%$phiG" }{TEXT -1 6 ", a
nd " }{XPPEDIT 18 0 "T(t)" "-%\"TG6#%\"tG" }{TEXT -1 20 ".  The equati
on for " }{XPPEDIT 18 0 "Y(theta,phi)" "-%\"YG6$%&thetaG%$phiG" }
{TEXT -1 4 " is:" }}{PARA 275 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s
in(theta)*Diff(sin(theta)*Diff(Y,theta),theta)+Diff(Y,phi,phi)+l*(l+1)
*sin(theta)^2*Y=0" "/,(*&-%$sinG6#%&thetaG\"\"\"-%%DiffG6$*&-F&6#F(F)-
F+6$%\"YGF(F)F(F)F)-F+6%F2%$phiGF5F)**%\"lGF),&F7F)\"\"\"F)F)-F&6#F(\"
\"#F2F)F)\"\"!" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 51 "In ord
er for the solutions be finite in the region " }{XPPEDIT 18 0 "0<=thet
a*`<`*2*Pi" "1\"\"!**%&thetaG\"\"\"%\"<GF&\"\"#F&%#PiGF&" }{TEXT -1 
22 ", we must impose that " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 
152 " is a non-negative integer. In this case the solutions are the sp
herical harmonics, whose definition  in terms of the associated Legend
re polynomial is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "Y1[l,m](theta,phi)=sqrt((2*l+1)*facto
rial(l-m)/2/Pi/factorial(l+m))*P[l,m](cos(theta))*cos(m*phi)" "/-&%#Y1
G6$%\"lG%\"mG6$%&thetaG%$phiG*(-%%sqrtG6#*,,&*&\"\"#\"\"\"F'F4F4\"\"\"
F4F4-%*factorialG6#,&F'F4F(!\"\"F4\"\"#F:%#PiGF:-F76#,&F'F4F(F4F:F4-&%
\"PG6$F'F(6#-%$cosG6#F*F4-FF6#*&F(F4F+F4F4" }{TEXT -1 2 ", " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 
276 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Y2[l,m](theta,phi)=sqrt((2*
l+1)*factorial(l-m)/2/Pi/factorial(l+m))*P[l,m](cos(theta))*sin(m*phi)
" "/-&%#Y2G6$%\"lG%\"mG6$%&thetaG%$phiG*(-%%sqrtG6#*,,&*&\"\"#\"\"\"F'
F4F4\"\"\"F4F4-%*factorialG6#,&F'F4F(!\"\"F4\"\"#F:%#PiGF:-F76#,&F'F4F
(F4F:F4-&%\"PG6$F'F(6#-%$cosG6#F*F4-%$sinG6#*&F(F4F+F4F4" }{TEXT -1 2 
", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Y1" "I#Y1G6
\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y2" "I#Y2G6\"" }{TEXT -1 5 " a
re " }{XPPEDIT 18 0 "Y^`+`" ")%\"YG%\"+G" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "Y^`-`" ")%\"YG%\"-G" }{TEXT -1 53 " respectively in the
 standard notation. The constant " }{XPPEDIT 18 0 "m" "I\"mG6\"" }
{TEXT -1 67 " can assume only non-negative integer values less than or
 equal to " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 86 ". This ensures
 that the solutions are rotationally invariant under the transformatio
n " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 57 "     The associated Legendre functions can be defined as:
" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P[l,m](x)=(1-x^2)
^(m/2)*``(d^m*P[l]/d/x^m)" "/-&%\"PG6$%\"lG%\"mG6#%\"xG*&),&\"\"\"\"\"
\"*$F*\"\"#!\"\"*&F(F/\"\"#F2F/-%!G6#**)%\"dGF(F/&F%6#F'F/F:F2)F*F(F2F
/" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{XPPEDIT 18 0 "P[l]" "&%\"PG6#%\"lG" }{TEXT -1 8 
" is the " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 146 "th Legendre po
lynomial. These functions are implemented in the Maple library. We hav
e implemented the associated Legendre functions with the name " }
{XPPEDIT 18 0 "P" "I\"PG6\"" }{TEXT -1 30 ", and having three argument
s: " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "m
" "I\"mG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT 
-1 21 ". Here is an example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "P(5,1,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "When the second
 argument is zero we obtain the Legendre polynomials. For example: " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P(3,0,x);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 64 "   The spherical harmonics are  implemented her
e with the names " }{XPPEDIT 18 0 "Y1" "I#Y1G6\"" }{TEXT -1 5 " and " 
}{XPPEDIT 18 0 "Y2" "I#Y2G6\"" }{TEXT -1 26 ", and have two arguments:
 " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m
" "I\"mG6\"" }{TEXT -1 16 ". The variables " }{XPPEDIT 18 0 "theta" "I
&thetaG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }
{TEXT -1 13 ", as well as " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 5 
" and " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 157 ", are global vari
ables throughout this worksheet. This means that they are not argument
s of any function. 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 a
nd the Boundary Condition" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "    T
he differential equation for " }{XPPEDIT 18 0 "R(r)" "-%\"RG6#%\"rG" }
{TEXT -1 4 " is:" }}{PARA 278 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "D
iff(r^2*Diff(R,r),r)+(k^2*r^2-l*(l+1))*R=0" "/,&-%%DiffG6$*&%\"rG\"\"#
-F%6$%\"RGF(\"\"\"F(F-*&,&*&%\"kG\"\"#F(\"\"#F-*&%\"lGF-,&F5F-\"\"\"F-
F-!\"\"F-F,F-F-\"\"!" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 25 "
Its general solution for " }{XPPEDIT 18 0 "k<>0" "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)" "/-&%\"RG6#%\"lG6#%\"rG,&*&&%\"CG6#\"
\"\"\"\"\"-&%\"jG6#F'6#*&%\"kGF0F)F0F0F0*&&F-6#\"\"#F0-&%\"nG6#F'6#*&F
7F0F)F0F0F0" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "j[l](k*r)" "-&%\"jG6#%\"lG6#*&%\"kG\"\"\"%\"rGF*" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "n[l](k*r)" "-&%\"nG6#%\"lG6#*&%\"kG
\"\"\"%\"rGF*" }{TEXT -1 92 " are respectively the spherical Bessel fu
nction and the spherical Neumann function of order " }{XPPEDIT 18 0 "l
" "I\"lG6\"" }{TEXT -1 7 ". When " }{XPPEDIT 18 0 "k=0" "/%\"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)" "/-&%\"RG6#%\"lG6#%\"rG,
&*&&%\"CG6#\"\"\"\"\"\")F)F'F0F0*&&F-6#\"\"#F0)F),&F'F0\"\"\"F0!\"\"F0
" }{TEXT -1 5 " .   " }}{PARA 0 "" 0 "" {TEXT -1 20 "Since the functio
ns " }{XPPEDIT 18 0 "n[l]" "&%\"nG6#%\"lG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "1/r^(l+1)" "*&\"\"\"\"\"\")%\"rG,&%\"lGF$\"\"\"F$!\"\"
" }{TEXT -1 21 " are not analytic at " }{XPPEDIT 18 0 "r=0" "/%\"rG\"
\"!" }{TEXT -1 32 ", we require that the constants " }{XPPEDIT 18 0 "C
[2]" "&%\"CG6#\"\"#" }{TEXT -1 76 " be zero. Without loss of generalit
y, we can choose the radial solution for " }{XPPEDIT 18 0 "k<>0" "0%\"
kG\"\"!" }{TEXT -1 4 " as:" }}{PARA 280 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "R[l](r)=j[l](k*r)" "/-&%\"RG6#%\"lG6#%\"rG-&%\"jG6#F'6#
*&%\"kG\"\"\"F)F1" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 59 "We implement the spherical Bessel functio
ns using the name " }{XPPEDIT 18 0 "SphericalBesselJ" "I1SphericalBess
elJG6\"" }{TEXT -1 68 ", and use an alias to simplify the notation. He
re are some examples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "al
ias(j=SphericalBesselJ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "
j(0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "simp := x -> sor
t(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" "/-%%DiffG6$%$PhiG%\"rG\"\"!" }{TEXT -1 
5 " for " }{XPPEDIT 18 0 "r=a" "/%\"rG%\"aG" }{TEXT -1 13 ".  Therefor
e " }{XPPEDIT 18 0 "Diff(R[l](r),r)=0" "/-%%DiffG6$-&%\"RG6#%\"lG6#%\"
rGF,\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "r=a" "/%\"rG%\"aG" }
{TEXT -1 18 ", or equivalently " }{XPPEDIT 18 0 "Diff(j(l,k[](l,n)*r),
r) = 0;" "/-%%DiffG6$-%\"jG6$%\"lG*&-&%\"kG6\"6$F)%\"nG\"\"\"%\"rGF1F2
\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "r=a" "/%\"rG%\"aG" }{TEXT 
-1 16 ". The constants " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 10 " \+
must be: " }{XPPEDIT 18 0 "k[](l,n)=jSB(l,n)/a" "/-&%\"kG6\"6$%\"lG%\"
nG*&-%$jSBG6$F(F)\"\"\"%\"aG!\"\"" }{TEXT -1 7 " where " }{XPPEDIT 18 
0 "jSB(l,n)" "-%$jSBG6$%\"lG%\"nG" }{TEXT -1 8 " is the " }{XPPEDIT 
18 0 "n" "I\"nG6\"" }{TEXT -1 68 "th zero of the derivative of the sph
erical Bessel function of order " }{XPPEDIT 18 0 "l" "I\"lG6\"" }
{TEXT -1 2 " (" }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 38 " a positiv
e integer).  The wavenumber " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 
27 " and the angular frequency " }{XPPEDIT 18 0 "omega" "I&omegaG6\"" 
}{TEXT -1 36 " are implemented with two arguments " }{XPPEDIT 18 0 "l
" "I\"lG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT 
-1 55 ". To obtain a numerical evaluation, we have to use the " }
{XPPEDIT 18 0 "evalf" "I&evalfG6\"" }{TEXT -1 87 " command. For exampl
e, the fundamental frequency and the corresponding  wavenumber are:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(omega(0,1));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(k(0,1));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The radia
l part of the solution is described by the function " }{XPPEDIT 18 0 "
R" "I\"RG6\"" }{TEXT -1 26 " which has two arguments: " }{XPPEDIT 18 
0 "l" "I\"lG6\"" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "n" "I\"nG6\"" }
{TEXT -1 14 ". For example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
7 "R(0,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "     When " }
{XPPEDIT 18 0 "k=0" "/%\"kG\"\"!" }{TEXT -1 27 " the boundary conditio
n at " }{XPPEDIT 18 0 "r=a" "/%\"rG%\"aG" }{TEXT -1 14 " implies that \+
" }{XPPEDIT 18 0 "l=0" "/%\"lG\"\"!" }{TEXT -1 12 ". Therefore " }
{XPPEDIT 18 0 "F(r,theta,phi)" "-%\"FG6%%\"rG%&thetaG%$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" "I$rhoG6\"" }{TEXT -1 25 " is the same as that for " }{XPPEDIT 
18 0 "Phi" "I$PhiG6\"" }{TEXT -1 9 ", since  " }{XPPEDIT 18 0 "Diff(Ph
i,r)=0" "/-%%DiffG6$%$PhiG%\"rG\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 
18 0 "r=a" "/%\"rG%\"aG" }{TEXT -1 15 " implies that  " }{XPPEDIT 18 
0 "Diff(rho,r)=0" "/-%%DiffG6$%$rhoG%\"rG\"\"!" }{TEXT -1 5 " for " }
{XPPEDIT 18 0 "r=a" "/%\"rG%\"aG" }{TEXT -1 46 ". The same conclusion \+
applies to the pressure " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 64 "
, but only the radial component of the velocity need be zero at " }
{XPPEDIT 18 0 "r = a" "/%\"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 modes and perfor
m their animation. The normal modes are solutions of the wave equation
 characterized by a single frequency, and are mutually orthogonal. The
y form a basis, which allows us to express any solution of the  wave e
quation as a linear combination of the normal modes. If two different \+
normal modes have the same frequency, they are called degenerate. In t
he case of the propagation of sound waves in spherical cavities, we ca
n define the following modes:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 281 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "normalmode11(l,m,n) =
 R(l,n)*Y1(l,m)*cos(omega(l,n)*t);" "/-%-normalmode11G6%%\"lG%\"mG%\"n
G*(-%\"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 "normalmode12(l,m,n) =
 R(l,n)*Y1(l,m)*sin(omega(l,n)*t);" "/-%-normalmode12G6%%\"lG%\"mG%\"n
G*(-%\"RG6$F&F(\"\"\"-%#Y1G6$F&F'F--%$sinG6#*&-%&omegaG6$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 "normalmode21(l,m,n) =
 R(l,n)*Y2(l,m)*cos(omega(l,n)*t);" "/-%-normalmode21G6%%\"lG%\"mG%\"n
G*(-%\"RG6$F&F(\"\"\"-%#Y2G6$F&F'F--%$cosG6#*&-%&omegaG6$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);" "/-%-normalmode22G6%%\"lG%\"mG%\"n
G*(-%\"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 th
e numbers " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 12 ". If we fix " }{XPPEDIT 18 
0 " l" "I\"lG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n" "I\"nG6\"" }
{TEXT -1 13 " we can have " }{XPPEDIT 18 0 "l+1" ",&%\"lG\"\"\"\"\"\"F
$" }{TEXT -1 12 " values for " }{XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT 
-1 72 ". In addition, for each frequency we can have four different mo
des when " }{XPPEDIT 18 0 "m<>0" "0%\"mG\"\"!" }{TEXT -1 15 ", and two
 when " }{XPPEDIT 18 0 "m=0" "/%\"mG\"\"!" }{TEXT -1 36 ". So  the num
ber of degeneracies is " }{XPPEDIT 18 0 "4*l+2" ",&*&\"\"%\"\"\"%\"lGF
%F%\"\"#F%" }{TEXT -1 17 ". The modes with " }{XPPEDIT 18 0 "l=0" "/%
\"lG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m=0" "/%\"mG\"\"!" }
{TEXT -1 74 " have two degeneracies and are spherically symmetric. Her
e is one example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "normal
mode11(l=0,m=0,n=3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "After a c
oordinate transformation, we can use the " }{TEXT 257 11 "densityplot
" }{TEXT -1 31 " command to plot this function:" }{MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "radius_a := 1: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "densityplot(subs(t=0,r=sqrt(x^2+y^
2),normalmode11(l=0, m=0, n=3)),x=-radius_a..radius_a,y=-radius_a..rad
ius_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" "0%\"lG\"\"!" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "m=0" "/%\"
mG\"\"!" }{TEXT -1 69 " have two degeneracies and are axially symmetri
c. Here is an example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "n
ormalmode11(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),normalmode
11(l=1,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 22 "       The procedure  " }{XPPEDIT 18 0 "a
nimate_mode" "I-animate_modeG6\"" }{TEXT -1 37 "  performs the animati
on of the mode " }{XPPEDIT 18 0 "normalmode11" "I-normalmode11G6\"" }
{TEXT -1 54 "[note The other normal modes can be animated with the " }
{XPPEDIT 18 0 "animate_ic" "I+animate_icG6\"" }{TEXT -1 47 "  function
 described in the next section.] for " }{XPPEDIT 18 0 "phi=0" "/%$phiG
\"\"!" }{TEXT -1 51 ". Let us see the animation of the two normal mode
s:" }}}{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" "/%\"mG\"\"!
" }{TEXT -1 164 ", the normal modes have axial symmetry, and therefore
 the three-dimensional picture can be easily visualized by rotating th
e picture around the vertical axis. When " }{XPPEDIT 18 0 "m<>0" "0%\"
mG\"\"!" }{TEXT -1 58 ", it is possible to animate the modes for other
 values of " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 11 " using the
 " }{XPPEDIT 18 0 "animate_ic" "I+animate_icG6\"" }{TEXT -1 34 " funct
ion (see the next section )." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 
"Animation with Initial Conditions" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 53 "Solution of the Wave Equation with Initial Conditions" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 " For simplicity, we impose the res
triction that the initial conditions do not depend on " }{XPPEDIT 18 
0 "phi" "I$phiG6\"" }{TEXT -1 15 ". In this case " }{XPPEDIT 18 0 "m=0
" "/%\"mG\"\"!" }{TEXT -1 41 ", and the general 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)*Y1[l,0](theta,phi)*(A[l,n]*cos(o
mega[l,n]*t) + B[l,n]*sin(omega[l,n]*t)),n=1..infinity),l=0..infinity)
" "/-%$PhiG6&%\"rG%&thetaG%$phiG%\"tG,(&%\"AG6#\"\"!\"\"\"*&&%\"BG6#F.
F/F)F/F/-%$SumG6$-F56$*(-&%\"jG6#%\"lG6#*&&%\"kG6$F>%\"nGF/F&F/F/-&%#Y
1G6$F>F.6$F'F(F/,&*&&F,6$F>FDF/-%$cosG6#*&&%&omegaG6$F>FDF/F)F/F/F/*&&
F26$F>FDF/-%$sinG6#*&&FS6$F>FDF/F)F/F/F/F//FD;\"\"\"%)infinityG/F>;F.F
[oF/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 28 "The initial conditions are: " }{XPPEDIT 18 0 "Phi(r,th
eta,phi,0)=Phi[0](r,theta)" "/-%$PhiG6&%\"rG%&thetaG%$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)" "/--&%\"DG6#%\"tG6#%$PhiG6&%\"rG%&thetaG%$phiG\"
\"!-&%)PhiprimeG6#F/6$F,F-" }{TEXT -1 4 ".   " }{XPPEDIT 18 0 "Phiprim
e[0]" "&%)PhiprimeG6#\"\"!" }{TEXT -1 25 " is proportional to both " }
{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p
" "I\"pG6\"" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "t=0" "/%\"tG\"\"!" }
{TEXT -1 109 ". Inverting the Fourier series[note We  use the identiti
es 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" "/-%$IntG6$*(%\"rG\"\"#-&%\"jG6#%\"lG6#*&&%\"kG6$F-%
\"nG\"\"\"F'F4F4-&F+6#F-6#*&&F16$F-%#n'GF4F'F4F4/F';\"\"!%\"aGF?" }
{TEXT -1 12 "    when    " }{XPPEDIT 18 0 "n<>`n'`" "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*int(r^2*j[l](k[l,n]*r)*int(Phi[0]*Y1[l,0]*sin(thet
a),theta=0..Pi),r=0..a)/int(r^2*j[l](k[l,n]*r)^2,r=0..a)" "/&%\"AG6$%
\"lG%\"nG**\"\"#\"\"\"%#PiGF*-%$intG6$*(%\"rG\"\"#-&%\"jG6#F&6#*&&%\"k
G6$F&F'F*F0F*F*-F-6$*(&%$PhiG6#\"\"!F*&%#Y1G6$F&FAF*-%$sinG6#%&thetaGF
*/FH;FAF+F*/F0;FA%\"aGF*-F-6$*&F0\"\"#-&F46#F&6#*&&F96$F&F'F*F0F*\"\"#
/F0;FAFM!\"\"" }{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)*int(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]" "/&%\"BG6$%\"lG%\"nG*,
\"\"#\"\"\"%#PiGF*-%$intG6$*(%\"rG\"\"#-&%\"jG6#F&6#*&&%\"kG6$F&F'F*F0
F*F*-F-6$*(&%)PhiprimeG6#\"\"!F*&%#Y1G6$F&FAF*-%$sinG6#%&thetaGF*/FH;F
AF+F*/F0;FA%\"aGF*-F-6$*&F0\"\"#-&F46#F&6#*&&F96$F&F'F*F0F*\"\"#/F0;FA
FM!\"\"&%&omegaG6$F&F'Ffn" }{TEXT -1 3 "  ." }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
17 "The coefficients " }{XPPEDIT 18 0 "A[0]" "&%\"AG6#\"\"!" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "B[0]" "&%\"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" "/&%\"AG6#\"\"!**\"\"$\"\"\"\"\"#!\"\"-%$intG6$*&%\"rG
\"\"#-F-6$*&-%$sinG6#%&thetaGF)&%$PhiG6#F&F)/F8;F&%#PiGF)/F0;F&%\"aGF)
*$FA\"\"$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" "/&%\"BG6#\"\"!**\"\"$\"\"\"\"\"#!\"\"-%$intG6$*&%
\"rG\"\"#-F-6$*&-%$sinG6#%&thetaGF)&%)PhiprimeG6#F&F)/F8;F&%#PiGF)/F0;
F&%\"aGF)*$FA\"\"$F+" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 46 "      Total mass conservation is expr
essed by:" }}{PARA 270 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Int(
Int(r^2*(rho+rho[0])*sin(theta),phi=0..2*Pi),theta=0..Pi),r=0..a)=cons
tant" "/-%$IntG6$-F$6$-F$6$*(%\"rG\"\"#,&%$rhoG\"\"\"&F.6#\"\"!F/F/-%$
sinG6#%&thetaGF//%$phiG;F2*&\"\"#F/%#PiGF//F6;F2F</F+;F2%\"aG%)constan
tG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 
18 0 "rho=rho[0]*Diff(Phi,t)/c^2" "/%$rhoG*(&F#6#\"\"!\"\"\"-%%DiffG6$
%$PhiG%\"tGF(*$%\"cG\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "in
t(r^2*j[l](k[l,n]*r),r=0..a)=0" "/-%$intG6$*&%\"rG\"\"#-&%\"jG6#%\"lG6
#*&&%\"kG6$F-%\"nG\"\"\"F'F4F4/F';\"\"!%\"aGF7" }{TEXT -1 18 ", the to
tal mass (" }{XPPEDIT 18 0 "M" "I\"MG6\"" }{TEXT -1 61 ") is conserved
 for all initial conditions and it is given by:" }}{PARA 274 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "M=4*Pi*a^3*rho[0]*(1+B[0]/c^2)/3" "/%\"
MG*.\"\"%\"\"\"%#PiGF&%\"aG\"\"$&%$rhoG6#\"\"!F&,&\"\"\"F&*&&%\"BG6#F-
F&*$%\"cG\"\"#!\"\"F&F&\"\"$F7" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 28 "For the equilibrium density " }{XPPEDIT 18 0 "rho[0]" "&%
$rhoG6#\"\"!" }{TEXT -1 22 " to remain unchanged: " }{XPPEDIT 18 0 "In
t(rho,V=`V `..``)=0" "/-%$IntG6$%$rhoG/%\"VG;%#V~G%!G\"\"!" }{TEXT -1 
17 ", we must impose " }{XPPEDIT 18 0 "B[0]=0" "/&%\"BG6#\"\"!F&" }
{TEXT -1 8 " (since " }{XPPEDIT 18 0 "B[0]=(M-M[0])*c^2/M[0]" "/&%\"BG
6#\"\"!*(,&%\"MG\"\"\"&F)6#F&!\"\"F**$%\"cG\"\"#F*&F)6#F&F-" }{TEXT 
-1 2 ")." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 93 "An approximate solution of the wave equation can be repre
sented by a truncated double series:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$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-%$sinGFJF2F2F2/F,;F2F,/F);F1F)F2" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The terms " }{XPPEDIT 18 
0 "A[0]" "&%\"AG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B[0]" "&%
\"BG6#\"\"!" }{TEXT -1 14 " are given by " }{XPPEDIT 18 0 "A(0,0)" "-%
\"AG6$\"\"!F%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B(0,0)" "-%\"BG6$\"
\"!F%" }{TEXT -1 103 " which are treated separately from the other coe
fficients, because the calculation of the general term " }{XPPEDIT 18 
0 "A(0,n)" "-%\"AG6$\"\"!%\"nG" }{TEXT -1 15 " does not give " }
{XPPEDIT 18 0 "A(0,0)" "-%\"AG6$\"\"!F%" }{TEXT -1 26 " as a limiting \+
case when  " }{XPPEDIT 18 0 "n=0" "/%\"nG\"\"!" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " The proc
edure " }{TEXT 258 10 "animate_ic" }{TEXT -1 40 " performs the animati
on of the solution " }{XPPEDIT 18 0 "Phi(trunc_l=l,trunc_n=n)" "-%$Phi
G6$/%(trunc_lG%\"lG/%(trunc_nG%\"nG" }{TEXT -1 35 " subject to the ini
tial conditions " }{XPPEDIT 18 0 "Phi[0]" "&%$PhiG6#\"\"!" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "Phiprime[0]" "&%)PhiprimeG6#\"\"!" }{TEXT 
-1 80 ". The details about its arguments can be obtained from 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 ge
neral example, suppose that the initial conditions are spherically sym
metric:" }}}{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]" "&%\"AG6$%\"lG%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B[
l,n]" "&%\"BG6$%\"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=radius_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 " The \+
general solution for the velocity potential with just the first two te
rms 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 not depe
nd on " }{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 113 ", the results can be di
splayed as a two-dimensional plot of the physical quantity of interest
 against the radius " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 96 ". Th
e actual three-dimensional picture can easily be visualized by using t
he spherical symmetry." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 242 "        As a first physical example, we want to per
form the animation of the physical quantities for the solution of the \+
wave equation corresponding to a localized shell-like pulse initially \+
traveling inward. This pulse will reflect first at " }{XPPEDIT 18 0 "r
=0" "/%\"rG\"\"!" }{TEXT -1 78 ", changing its propagation to the outw
ard direction, and will then reflect at " }{XPPEDIT 18 0 "r=a" "/%\"rG
%\"aG" }{TEXT -1 152 ", again changing its direction, and so on.  This
 example will allow us to analyze the sign changes in the physical qua
ntities during these reflections. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 30 "      To begin, let us define " }
{XPPEDIT 18 0 "Psi" "I$PsiG6\"" }{TEXT -1 4 " by:" }}{PARA 287 "" 0 "
" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Phi(r,t)=Psi(r+c*t)/r" "/-%$PhiG6$%
\"rG%\"tG*&-%$PsiG6#,&F&\"\"\"*&%\"cGF-F'F-F-F-F&!\"\"" }{TEXT -1 2 " \+
." }}{PARA 0 "" 0 "" {TEXT -1 13 "We know that " }{XPPEDIT 18 0 "Psi(r
,t)" "-%$PsiG6$%\"rG%\"tG" }{TEXT -1 76 " is a solution of the one-dim
ensional wave equation, and so the argument of " }{XPPEDIT 18 0 "Psi" 
"I$PsiG6\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "r+c*t" ",&%\"rG\"\"\"*&
%\"cGF$%\"tGF$F$" }{TEXT -1 76 ". The initial conditions that simulate
 the inwardly-directed pulse are thus:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 288 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Phi[0]=``(Psi(r+c
*t)/r)*``[``][t=0]" "/&%$PhiG6#\"\"!*&-%!G6#*&-%$PsiG6#,&%\"rG\"\"\"*&
%\"cGF1%\"tGF1F1F1F0!\"\"F1&&F)6#F)6#/F4F&F1" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 3 "and" }}{PARA 289 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Phiprime[0]=``(Diff(Psi(r+c*t),t)/r)*``[``][t=0]" "/&%)
PhiprimeG6#\"\"!*&-%!G6#*&-%%DiffG6$-%$PsiG6#,&%\"rG\"\"\"*&%\"cGF4%\"
tGF4F4F7F4F3!\"\"F4&&F)6#F)6#/F7F&F4" }{TEXT -1 3 " , " }}{PARA 0 "" 
0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Psi(r)" "-%$PsiG6#%\"rG" }
{TEXT -1 63 " is a function which is nonzero in a sub-interval smaller
 than " }{XPPEDIT 18 0 "[0,a]" "7$\"\"!%\"aG" }{TEXT -1 50 ", and drop
s off to zero outside this sub-interval." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "    If we suppose that the average
 density inside the cavity remains " }{XPPEDIT 18 0 "rho[0]" "&%$rhoG6
#\"\"!" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "Psi" "I$PsiG6\"" }{TEXT 
-1 37 " must satisfy the following equation:" }}{PARA 290 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Psi(r),r=0..a)=a*Psi(a)" "/-%$IntG6
$-%$PsiG6#%\"rG/F);\"\"!%\"aG*&F-\"\"\"-F'6#F-F/" }{TEXT -1 2 " ." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "This resu
lt can be proved by performing an integration by parts with respect to
 the variable " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 17 " in the eq
uation " }{XPPEDIT 18 0 "B[0]=0" "/&%\"BG6#\"\"!F&" }{TEXT -1 26 ", an
d using the fact that " }{XPPEDIT 18 0 "Psi" "I$PsiG6\"" }{TEXT -1 14 
" is finite at " }{XPPEDIT 18 0 "r=0" "/%\"rG\"\"!" }{TEXT -1 37 ". Fo
r a localized pulse we have that " }{XPPEDIT 18 0 "Psi(a)=0" "/-%$PsiG
6#%\"aG\"\"!" }{TEXT -1 28 ", and so the area under the " }{XPPEDIT 
18 0 "Psi" "I$PsiG6\"" }{TEXT -1 21 " curve must be zero. " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "     Let us choos
e 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" "I&pulseG6\"" }
{TEXT -1 32 " helps in the definition of the " }{XPPEDIT 18 0 "Psi" "I
$PsiG6\"" }{TEXT -1 10 " function:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "pulse := (center,width) -> " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "piecewise(r<center-width/2,0,r<cent
er+width/2,sin(Pi/width*(r+width/2-center))^2,0); " }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 139 " The first argument is the position of the cente
r of the pulse and the second is its width. The pulse shape is represe
nted by the function " }{XPPEDIT 18 0 "sine" "I%sineG6\"" }{TEXT -1 
47 " squared. Here is one possible definition  for " }{XPPEDIT 18 0 "P
si" "I$PsiG6\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 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" "I$PsiG6\"" }{TEXT -1 24 " satisfies the equation " }{XPPEDIT 18 
0 "Int(Psi(r),r=0..a)=0" "/-%$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]" "&%$rhoG6#\"\"!" }{TEXT -1 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "B(0,0);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 65 " The initial conditions can be visualized by the followin
g plots:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "plot(Phi0,r=0..radius_a,title='Phi0');" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 76 "plot(Phiprime0,r=0..radius_a,title='Phiprime0',ytic
kmarks=[-10000,0,10000]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 " Fi
rst let us calculate the velocity potential with enough terms in the F
ourier expansion, and store the result in a variable to save time:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "vel_pot
 := evalf(Phi(trunc_n=20)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Th
e first 3 terms are:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "evalf(Phi(trunc_n=2),2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 82 " The animation of the velocity potential can be obtained \+
by the following command:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 120 "animate_ic(function=vel_pot,color=black,frames=
9,xtickmarks=[radius_a],ytickmarks=[1],r=0..a,verticalrange=-5.51..5.5
1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 " The animation of  " }
{XPPEDIT 18 0 "rho/rho[0]" "*&%$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 "animat
e_ic(function=dens,color=black,frames=9,xtickmarks=[radius_a],ytickmar
ks=[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 a
nimate the function " }{XPPEDIT 18 0 "r^2*rho/rho[0]" "*(%\"rG\"\"#%$r
hoG\"\"\"&F%6#\"\"!!\"\"" }{TEXT -1 12 " instead of " }{XPPEDIT 18 0 "
rho/rho[0]" "*&%$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=[radius_a],ytickmarks=
[1],verticalrange=-0.1..0.1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 
"From this last animation we can more easily see that the total mass i
s conserved even though the sign of the density inverts upon reflectio
n at " }{XPPEDIT 18 0 "r=0" "/%\"rG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " The " }{TEXT 262 
10 "animate_ic" }{TEXT -1 136 " command allows one to animate more tha
n one function at the same time, like the radial velocity together wit
h the density, for example." }{MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "
" 0 "" {TEXT -1 27 "Analysis of the Reflections" }}{PARA 0 "" 0 "" 
{TEXT -1 27 "        We have shown that " }{XPPEDIT 18 0 "Phi" "I$PhiG
6\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 6 ", \+
and " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 140 " obey the same diff
erential equation with the same boundary conditions; therefore they ha
ve the same behavior concerning the reflections at " }{XPPEDIT 18 0 "r
=a" "/%\"rG%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r=0" "/%\"rG\"\"
!" }{TEXT -1 46 ".   Let us begin by analyzing what happens at " }
{XPPEDIT 18 0 "r=0" "/%\"rG\"\"!" }{TEXT -1 26 ". We define the functi
ons " }{XPPEDIT 18 0 "Psi[Phi]" "&%$PsiG6#%$PhiG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "Psi[rho]" "&%$PsiG6#%$rhoG" }{TEXT -1 6 ", and " }
{XPPEDIT 18 0 "Psi[p]" "&%$PsiG6#%\"pG" }{TEXT -1 4 " as " }{XPPEDIT 
18 0 "r*Phi" "*&%\"rG\"\"\"%$PhiGF$" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "
r*rho" "*&%\"rG\"\"\"%$rhoGF$" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "r*
p" "*&%\"rG\"\"\"%\"pGF$" }{TEXT -1 67 " respectively.   For the spher
ically-symmetric case, we know that  " }{XPPEDIT 18 0 "Psi[Phi]" "&%$P
siG6#%$PhiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Psi[rho]" "&%$PsiG6#%$rh
oG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "Psi[p]" "&%$PsiG6#%\"pG" }
{TEXT -1 91 " satisfy the one-dimensional wave equation. Therefore, th
e analysis of the behavior of the " }{XPPEDIT 18 0 "Psi" "I$PsiG6\"" }
{TEXT -1 35 " functions is easier than that of  " }{XPPEDIT 18 0 "Phi
" "I$PhiG6\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }
{TEXT -1 5 ", or " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 7 ".  The \+
" }{XPPEDIT 18 0 "Psi" "I$PsiG6\"" }{TEXT -1 66 " expressions can be r
eadily obtained from the following equations:" }}{PARA 291 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Phi(r,t)=(A[0]+B[0]*t)+Sum(j[0](k[0,n]*
r)*T[n](t),n=1..infinity)" "/-%$PhiG6$%\"rG%\"tG,(&%\"AG6#\"\"!\"\"\"*
&&%\"BG6#F,F-F'F-F--%$SumG6$*&-&%\"jG6#F,6#*&&%\"kG6$F,%\"nGF-F&F-F--&
%\"TG6#F?6#F'F-/F?;\"\"\"%)infinityGF-" }{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..i
nfinity)" "/-%\"pG6$%\"rG%\"tG,&*&&%$rhoG6#\"\"!\"\"\"&%\"BG6#F-F.F.*&
&F+6#F-F.-%$SumG6$*&-&%\"jG6#F-6#*&&%\"kG6$F-%\"nGF.F&F.F.-%%DiffG6$-&
%\"TG6#FB6#F'F'F./FB;\"\"\"%)infinityGF.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(omega[0,n]*t)+B[0,n]*sin(omega[0,n]
*t))/2/sqrt(Pi)" "/-&%\"TG6#%\"nG6#%\"tG*(,&*&&%\"AG6$\"\"!F'\"\"\"-%$
cosG6#*&&%&omegaG6$F0F'F1F)F1F1F1*&&%\"BG6$F0F'F1-%$sinG6#*&&F76$F0F'F
1F)F1F1F1F1\"\"#!\"\"-%%sqrtG6#%#PiGFD" }{TEXT -1 2 " ." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "It is easy to show t
hat " }{XPPEDIT 18 0 "Psi[Phi]" "&%$PsiG6#%$PhiG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "Psi[rho]" "&%$PsiG6#%$rhoG" }{TEXT -1 6 ", and " }
{XPPEDIT 18 0 "Psi[p]" "&%$PsiG6#%\"pG" }{TEXT -1 43 " satisfy a Diric
hlet boundary condition at " }{XPPEDIT 18 0 "r=0" "/%\"rG\"\"!" }
{TEXT -1 91 ", and therefore these quantities undergo a simple polarit
y inversion after a reflection at " }{XPPEDIT 18 0 "r=0" "/%\"rG\"\"!
" }{TEXT -1 143 ".  We can thus conclude that the velocity potential, \+
density, and pressure must also undergo a simple polarity inversion af
ter a reflection at " }{XPPEDIT 18 0 "r=0" "/%\"rG\"\"!" }{TEXT -1 51 
", their wave-shapes being maintained. We note that " }{XPPEDIT 18 0 "
Phi" "I$PhiG6\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }
{TEXT -1 6 ", and " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 27 " ( in \+
contradistinction to " }{XPPEDIT 18 0 "Psi[Phi]" "&%$PsiG6#%$PhiG" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "Psi[rho]" "&%$PsiG6#%$rhoG" }{TEXT -1 
6 ", and " }{XPPEDIT 18 0 "Psi[p]" "&%$PsiG6#%\"pG" }{TEXT -1 17 ") ha
ve values at " }{XPPEDIT 18 0 "r=0" "/%\"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" "/%\"rG%\"aG" }{TEXT -1 26 " is straightforward si
nce " }{XPPEDIT 18 0 "Psi[Phi]=r*Phi" "/&%$PsiG6#%$PhiG*&%\"rG\"\"\"F&
F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(Phi(r,t),r)=0" "/-%%DiffG
6$-%$PhiG6$%\"rG%\"tGF)\"\"!" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "r=a" 
"/%\"rG%\"aG" }{TEXT -1 11 " imply that" }}{PARA 295 "" 0 "" {TEXT -1 
2 "  " }{XPPEDIT 18 0 "Psi[Phi](a,t)=a*Diff(Psi[Phi](a,t),r)" "/-&%$Ps
iG6#%$PhiG6$%\"aG%\"tG*&F)\"\"\"-%%DiffG6$-&F%6#F'6$F)F*%\"rGF," }
{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 45 "Similar boundary condit
ions are satisfied by " }{XPPEDIT 18 0 "Psi[rho]" "&%$PsiG6#%$rhoG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "Psi[p]" "&%$PsiG6#%\"pG" }{TEXT -1 
6 ". The " }{XPPEDIT 18 0 "Psi" "I$PsiG6\"" }{TEXT -1 115 "'s thus obe
y Robin-type boundary conditions. These conditions do not maintain the
 wave-shape after a reflection at " }{XPPEDIT 18 0 "r=a" "/%\"rG%\"aG
" }{TEXT -1 72 ", but the deviations from a simple non-polarity-invert
ing reflection at " }{XPPEDIT 18 0 "r=a" "/%\"rG%\"aG" }{TEXT -1 110 "
 are relatively small in this case. After many reflections, however, t
he 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)" "/&%\"vG6#%\"rG,$-%%DiffG6$%$Phi
GF&!\"\"" }{TEXT -1 99 " under reflection can be deduced by the follow
ing analysis. If we have a pulse moving inward, then " }{XPPEDIT 18 0 
"v[r]=-Diff(Psi[Phi](r+c*t)/r,r)" "/&%\"vG6#%\"rG,$-%%DiffG6$*&-&%$Psi
G6#%$PhiG6#,&F&\"\"\"*&%\"cGF3%\"tGF3F3F3F&!\"\"F&F7" }{TEXT -1 26 ". \+
It is easy to show that " }{XPPEDIT 18 0 "v[r]=Phi/r-c*rho/rho[0]" "/&
%\"vG6#%\"rG,&*&%$PhiG\"\"\"F&!\"\"F**(%\"cGF*%$rhoGF*&F.6#\"\"!F+F+" 
}{TEXT -1 42 ". If the wave is moving outward, we have  " }{XPPEDIT 
18 0 "v[r]=Phi/r+c*rho/rho[0]" "/&%\"vG6#%\"rG,&*&%$PhiG\"\"\"F&!\"\"F
**(%\"cGF*%$rhoGF*&F.6#\"\"!F+F*" }{TEXT -1 29 ".  In the beginning we
 have  " }{XPPEDIT 18 0 "c*rho/rho[0]" "*(%\"cG\"\"\"%$rhoGF$&F%6#\"\"
!!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`<<`" "I#<<G6\"" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "Phi/r" "*&%$PhiG\"\"\"%\"rG!\"\"" }{TEXT -1 14 " ; \+
therefore  " }{XPPEDIT 18 0 "v[r]=-c*rho/rho[0]" "/&%\"vG6#%\"rG,$*(%
\"cG\"\"\"%$rhoGF*&F+6#\"\"!!\"\"F/" }{TEXT -1 42 "   when the pulse i
s moving inward, and   " }{XPPEDIT 18 0 "v[r]=c*rho/rho[0]" "/&%\"vG6#
%\"rG*(%\"cG\"\"\"%$rhoGF)&F*6#\"\"!!\"\"" }{TEXT -1 42 "  when the pu
lse is moving outward. Since " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }
{TEXT -1 25 " inverts its polarity at " }{XPPEDIT 18 0 "r=0" "/%\"rG\"
\"!" }{TEXT -1 24 " and does not invert at " }{XPPEDIT 18 0 "r=a" "/%
\"rG%\"aG" }{TEXT -1 39 ", we conclude that the same happens to " }
{XPPEDIT 18 0 "v[r]" "&%\"vG6#%\"rG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "      For completeness,
 we also mention that  " }{XPPEDIT 18 0 "v[r](0,t)=0" "/-&%\"vG6#%\"rG
6$\"\"!%\"tGF)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v[r](a,t)=0" "/-&%
\"vG6#%\"rG6$%\"aG%\"tG\"\"!" }{TEXT -1 9 " for all " }{XPPEDIT 18 0 "
t" "I\"tG6\"" }{TEXT -1 37 ", as can be seen from the expression:" }}
{PARA 296 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v[r](r,t)=Sum(k[0,n]*
j[1](k[0,n]*r)*T[n](t),n=1..infinity)" "/-&%\"vG6#%\"rG6$F'%\"tG-%$Sum
G6$*(&%\"kG6$\"\"!%\"nG\"\"\"-&%\"jG6#\"\"\"6#*&&F/6$F1F2F3F'F3F3-&%\"
TG6#F26#F)F3/F2;\"\"\"%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 " Terms which describe the
 acoustic pressure such a " }{XPPEDIT 18 0 "f(t-r/c)/r;" "*&-%\"fG6#,&
%\"tG\"\"\"*&%\"rGF(%\"cG!\"\"F,F(F*F," }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "g(t+r/c);" "-%\"gG6#,&%\"tG\"\"\"*&%\"rGF'%\"cG!\"\"F'" }{TEXT 
-1 207 " represent respectively outgoing and incoming spherical pressu
re waves.  Each of these terms individually becomes infinite at the or
igin, and is respectively associated with a source or a sink at the or
igin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "
A source of volume-velocity strength " }{XPPEDIT 18 0 "S(t);" "-%\"SG6
#%\"tG" }{TEXT -1 2 " [" }{XPPEDIT 18 0 "m^3/s;" "*&%\"mG\"\"$%\"sG!\"
\"" }{TEXT -1 89 "] at the origin, results in an outgoing pressure wav
e related to the volume acceleration " }{XPPEDIT 18 0 "`S'`(t);" "-%#S
'G6#%\"tG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "p(r,t) = rho[0]*`S'`(t-r
/c)/(4*Pi*r);" "/-%\"pG6$%\"rG%\"tG*(&%$rhoG6#\"\"!\"\"\"-%#S'G6#,&F'F
-*&F&F-%\"cG!\"\"F4F-*(\"\"%F-%#PiGF-F&F-F4" }{TEXT -1 78 ", where S' \+
is the first derivative of S.  For this outgoing wave we must have " }
{XPPEDIT 18 0 "`S'`(t) = 4*Pi*f(t)/rho[0];" "/-%#S'G6#%\"tG**\"\"%\"\"
\"%#PiGF)-%\"fG6#F&F)&%$rhoG6#\"\"!!\"\"" }{TEXT -1 78 ".  Similarly a
n appropriate sink can represent the incoming wave described by " }
{XPPEDIT 18 0 "g(t+r/c);" "-%\"gG6#,&%\"tG\"\"\"*&%\"rGF'%\"cG!\"\"F'
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 210 "In our spherical cavity, there is no actual source or si
nk at the origin, and so the sink term must exactly cancel the source \+
term, so that there is zero net flow into or out of the origin.  Hence
 we must have " }{XPPEDIT 18 0 "g(t) = -f(t);" "/-%\"gG6#%\"tG,$-%\"fG
6#F&!\"\"" }{TEXT -1 557 ".  This matching of the incoming and outgoin
g solutions in order to maintain mass conservation is the physical exp
lanation for the simple polarity inversion experienced by an incoming \+
spherical wave upon \"reflection\" at the origin. This result may be s
omewhat unexpected, since a naive physical argument might have predict
ed that an incoming pressure wave would experience a \"rigid wall\"-ty
pe boundary condition at the origin (from which the air cannot escape)
, producing a singularity in the pressure, and reflecting the wave wit
hout a polarity inversion. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 34 "Writing the total pressure then as" }}{PARA 
298 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(r,t) = (f(t-r/c)-f(t+r/c)
)/r;" "/-%\"pG6$%\"rG%\"tG*&,&-%\"fG6#,&F'\"\"\"*&F&F.%\"cG!\"\"F1F.-F
+6#,&F'F.*&F&F.F0F1F.F1F.F&F1" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 83 "it follows that, as we let r approach zero, the pressure \+
at the origin is given by " }{XPPEDIT 18 0 "p(0,t) = -2*`f'`(t)/c;" "/
-%\"pG6$\"\"!%\"tG,$*(\"\"#\"\"\"-%#f'G6#F'F+%\"cG!\"\"F0" }{TEXT -1 
60 ", which is finite and not identically zero for well-behaved " }
{XPPEDIT 18 0 "f(t);" "-%\"fG6#%\"tG" }{TEXT -1 37 ", and shows no hin
t of a singularity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 297 "The particle velocity at the origin has earlier been s
hown to be zero.  This result can be understood intuitively as a conse
quence of the fact that, were it nonzero, there would be outflow or in
flow of air at the origin, contradicting the assumption that the origi
n is neither a source nor a sink." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 11 "Conclusions" }}{PARA 0 "" 0 "" {TEXT -1 237 "We have analyzed t
he propagation of sound waves in spherical cavities by means of a comp
uter algebra system. With the aid of the Maple system, we investigated
 in some detail the reflection properties of the sound waves at both t
he point " }{XPPEDIT 18 0 "r=0" "/%\"rG\"\"!" }{TEXT -1 21 " and at th
e boundary " }{XPPEDIT 18 0 "r=a" "/%\"rG%\"aG" }{TEXT -1 54 ". We con
firmed that there is no polarity inversion at " }{XPPEDIT 18 0 "r=a" "
/%\"rG%\"aG" }{TEXT -1 67 ", and were able to ``see'' the non-intuitiv
e polarity inversion at " }{XPPEDIT 18 0 "r=0" "/%\"rG\"\"!" }{TEXT 
-1 158 ". Although we have shown only a few plots here, it is easy to \+
generate many more to amplify details, to test different parameters, o
r to perform simulations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 259 "At some stages of the analysis of the problem, t
he development of new routines was required; for example, functions fo
r representing the spherical harmonics and the Fourier coefficients, a
nd for performing animations. All of them are available from a web sit
e" }{TEXT -1 380 ", and can be easily modified and adapted to differen
t situations. The Maple programming language follows the standard styl
e of the structured programming languages like C and Pascal. The prope
r way for the user to interact with these routines, and to analyze the
 problem, is through the  ``worksheet''. The complete text of this pap
er is in the worksheet provided on the web site." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 517 "This  way of addressing \+
the problem has pedagogical benefits. The plots and the animations pro
vided here can be used as an aid in teaching the propagation of sound \+
waves in udergraduate couses. Our experience shows that the use of com
puter a algebra system in the classroom increases student motivation a
nd allows the analysis of more realistic problems. The tools used by t
he teacher to show animations are available to the students, if they h
ave a personal computer with the computer algebra system installed on \+
it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{MARK "0 2 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 }