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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 58 "Vibrations of a Circular \+
Membrane With No Theta-Dependence" }}{PARA 19 "" 0 "" {TEXT -1 32 "by \+
J. Hebron, SFU, November 1999" }}{PARA 0 "" 0 "" {TEXT -1 193 "This Ma
ple worksheet illustrates the normal modes of oscillation of a circula
r membrane with no theta-dependence.  This problem was solved in the M
ath 314 Lecture of Friday November 26th, 1999." }}{PARA 0 "" 0 "" 
{TEXT -1 312 "In order to get normal modes of oscillation, we choose i
nitial conditions which are described by functions which are already m
embers of the orthogonal set.  All the coeffieients in the summation t
herefore go away except for the term corresponding to the initial cond
ition.  Our orthogonal set consists of Bessel " }{XPPEDIT 18 0 "J[0](l
ambda[n]*r);" "6#-&%\"JG6#\"\"!6#*&&%'lambdaG6#%\"nG\"\"\"%\"rGF." }
{TEXT -1 56 " functions.  We let the radius of the memberae be 1, so \+
" }{XPPEDIT 18 0 "lambda[n];" "6#&%'lambdaG6#%\"nG" }{TEXT -1 9 " beco
mes " }{XPPEDIT 18 0 "alpha[n];" "6#&%&alphaG6#%\"nG" }{TEXT -1 35 ", \+
the zeros of the Bessel function." }}}{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 condition " }{XPPEDIT 18 0 "J[0](
alpha[1]*r);" "6#-&%\"JG6#\"\"!6#*&&%&alphaG6#\"\"\"\"\"\"%\"rGF." }
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "alpha1:=Bess
elJZeros(0,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'alpha1G-%-BesselJ
ZerosG6$\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "v:=(
r,t)->BesselJ(0,alpha1*r)*cos(alpha1*t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"vGR6$%\"rG%\"tG6\"6$%)operatorG%&arrowGF)*&-%(BesselJG6$\"\"
!*&%'alpha1G\"\"\"9$F4F4-%$cosG6#*&F3\"\"\"9%F4F4F)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 76 "plot3d([r,theta,v(r,0)],r=0..1,theta=-Pi..Pi,coords=cylindrica
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