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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 55 "Vibrations of a Circular \+
Membrane With Theta-Dependence" }}{PARA 19 "" 0 "" {TEXT -1 32 "by J. \+
Hebron, SFU, November 1999" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 196 "This Maple worksheet illustrates the normal mode
s of oscillation of a circular membrane, including theta-dependence.  \+
This problem was solved in the Math 314 Lecture of Monday November 29t
h, 1999." }}{PARA 0 "" 0 "" {TEXT -1 312 "In order to get normal modes
 of oscillation, we choose initial conditions which are described by f
unctions which are already members of the orthogonal set.  All the coe
ffieients in the summation therefore go away except for the term corre
sponding to the initial condition.  Our orthogonal set consists of Bes
sel " }{XPPEDIT 18 0 "J[m](lambda[m,n]*r);" "6#-&%\"JG6#%\"mG6#*&&%'la
mbdaG6$F'%\"nG\"\"\"%\"rGF." }{XPPEDIT 18 0 "cos(m*theta);" "6#-%$cosG
6#*&%\"mG\"\"\"%&thetaGF(" }{TEXT -1 56 " functions.  We let the radiu
s of the memberae be 1, so " }{XPPEDIT 18 0 "lambda[m,n];" "6#&%'lambd
aG6$%\"mG%\"nG" }{TEXT -1 0 "" }{TEXT -1 9 " becomes " }{XPPEDIT 18 0 
"alpha[m,n];" "6#&%&alphaG6$%\"mG%\"nG" }{TEXT -1 0 "" }{TEXT -1 0 "" 
}{TEXT -1 1 " " }{TEXT -1 26 ", the zeros of the Bessel " }{XPPEDIT 
18 0 "J[m];" "6#&%\"JG6#%\"mG" }{TEXT -1 0 "" }{TEXT -1 10 " function.
" }}{PARA 0 "" 0 "" {TEXT -1 100 "These animations should be compared \+
to the nodal curves shown in figure 9, page 328 of the textbook." }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Do this first" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Initial con
dition " }{XPPEDIT 18 0 "J[1](alpha[1,1]*r);" "6#-&%\"JG6#\"\"\"6#*&&%
&alphaG6$\"\"\"\"\"\"\"\"\"%\"rGF/" }{XPPEDIT 18 0 "cos(theta);" "6#-%
$cosG6#%&thetaG" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "alpha[1,1]:=BesselJZeros(1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>&%&alphaG6$\"\"\"F'-%-BesselJZerosGF&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "u:=(r,theta,t)->BesselJ(1,alpha[1,1]*r)*cos(theta)*co
s(alpha[1,1]*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6%%\"rG%&th
etaG%\"tG6\"6$%)operatorG%&arrowGF**(-%(BesselJG6$\"\"\"*&&%&alphaG6$F
2F2F29$F2F2-%$cosG6#9%F2-F96#*&F4\"\"\"9&F2F2F*F*F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "
Plot of Initial condition" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
123 "plot3d([r,theta,u(r,theta,0)],r=0..1,theta=-Pi..Pi,coords=cylindr
ical,axes=frame,orientation=[70,65],labels=[`x`,`y`,`u`]);" }}{PARA 
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