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{SECT 0 {EXCHG {PARA 208 "" 0 "" {TEXT 217 37 "Standing Waves on a Cir
cular Membrane" }{TEXT 217 0 "" }}{PARA 209 "" 0 "" {TEXT 218 43 "(Bas
ed on the work in the Maple worksheet \"" }{TEXT 219 27 "Circular Vibr
ating Membrane" }{TEXT 218 2 "\")" }{TEXT 218 0 "" }}{PARA 210 "" 0 ""
 {TEXT 220 0 "" }}{PARA 210 "" 0 "" {TEXT 220 105 "We investigate the \+
\"Standing Waves\" for a vibrating circular membrane.  Each of the fun
damental solutions" }{TEXT 220 0 "" }}{PARA 210 "" 0 "" {TEXT 220 0 ""
 }}{PARA 209 "" 0 "" {XPPEDIT 2 0 "u[n](r,t)" "6#-&I\"uG6\"6#I\"nGF&6$
I\"rGF&I\"tGF&" }{TEXT 218 7 "   =   " }{XPPEDIT 2 0 "cos(k[n]*c*t)*Be
sselJ(0, k[n]*r)" "6#*&-I$cosG6\"6#*(&I\"kGF&6#I\"nGF&\"\"\"I\"cGF&F-I
\"tGF&F-F--I(BesselJGF&6$\"\"!*&F)F-I\"rGF&F-F-" }{TEXT 218 3 "   " }
{TEXT 218 0 "" }}{PARA 210 "" 0 "" {TEXT 220 0 "" }}{PARA 211 "" 0 "" 
{TEXT 218 24 "may be considered as a \"" }{TEXT 221 13 "standing wave"
 }{TEXT 218 36 "\" as each has a fixed shape, namely " }{XPPEDIT 2 0 "
BesselJ(0,k[n]*r)" "6#-I(BesselJG6$I*protectedGF&I(_syslibG6\"6$\"\"!*
&&I\"kGF(6#I\"nGF(\"\"\"I\"rGF(F0" }{TEXT 218 28 ", with a variable am
plitude " }{XPPEDIT 2 0 "cos(k[n]*c*t)" "6#-I$cosG6\"6#*(&I\"kGF%6#I\"
nGF%\"\"\"I\"cGF%F,I\"tGF%F," }{TEXT 218 150 " .  Since the Bessel fun
ctions oscillate and cross the r axes many times, there must be points
 that do not move, and these points are referred to as \"" }{TEXT 222 
5 "nodes" }{TEXT 218 215 "\".  To show the nodes for each fundamental \+
mode (mode n has n-1 nodes, n=1,2,...) we plot the above equation in t
he u-r plane.  In the actual motion these nodes will appear as circles
 that stay put in the z=0 plane." }}{PARA 211 "" 0 "" {TEXT 218 0 "" }
}{PARA 211 "" 0 "" {TEXT 218 7 "So now " }{TEXT 223 41 "execute the pr
ocedure in the next section" }{TEXT 218 44 " and then scroll down  to \+
use the procedure." }{TEXT 218 0 "" }}{PARA 210 "" 0 "" {TEXT 220 0 ""
 }}{PARA 210 "" 0 "" {TEXT 220 0 "" }}}{SECT 0 {PARA 212 "" 0 "" 
{TEXT -1 33 "View the nodes via cross-sections" }{TEXT -1 0 "" }}
{PARA 210 "" 0 "" {TEXT 220 0 "" }}{PARA 210 "" 0 "" {TEXT 220 0 "" }}
{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 8 "restart;" }{MPLTEXT 1 
224 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 51 "CrossSection
OfFundamentalMode:= proc(TheModeNumber)" }{MPLTEXT 1 224 0 "" }
{MPLTEXT 1 224 23 "\n  global TheAnimation;" }{MPLTEXT 1 224 0 "" }
{MPLTEXT 1 224 18 "\n  local NN,kk,cc;" }{MPLTEXT 1 224 0 "" }
{MPLTEXT 1 224 1 "\n" }{MPLTEXT 1 224 22 "\n  NN:= TheModeNumber;" }
{MPLTEXT 1 224 0 "" }{MPLTEXT 1 224 1 "\n" }{MPLTEXT 1 224 17 "\n  if \+
NN = 0 then" }{MPLTEXT 1 224 0 "" }{MPLTEXT 1 224 136 "\n     TheAnima
tion :=print(\"You have not entered 0 for the Mode Number. Please redo
 by entering a positive integer for TheModeNumber.\");" }{MPLTEXT 1 
224 0 "" }{MPLTEXT 1 224 9 "\n  else  " }{MPLTEXT 1 224 0 "" }
{MPLTEXT 1 224 18 "\n    cc:= 0.05/NN;" }{MPLTEXT 1 224 0 "" }
{MPLTEXT 1 224 36 "\n    kk:= evalf(BesselJZeros(0,NN));" }{MPLTEXT 1 
224 0 "" }{MPLTEXT 1 224 258 "\n    TheAnimation:=  plots[animate](cos
(kk*cc*t)*BesselJ(0,kk*r),r=0..1,t=0..60,frames=50,title=sprintf(\" Fu
ndamental mode %a \\n has %a node(s) or fixed point(s), not counting \+
\\n the fixed point at the boundary.\",NN,(NN-1)),titlefont=[HELVETICA
,OBLIQUE,24]):" }{MPLTEXT 1 224 0 "" }{MPLTEXT 1 224 4 "\nfi;" }
{MPLTEXT 1 224 0 "" }{MPLTEXT 1 224 1 "\n" }{MPLTEXT 1 224 14 "\nTheAn
imation;" }{MPLTEXT 1 224 0 "" }{MPLTEXT 1 224 1 "\n" }{MPLTEXT 1 224 
1 "\n" }{MPLTEXT 1 224 10 "\nend proc:" }{MPLTEXT 1 224 0 "" }
{MPLTEXT 1 224 1 "\n" }{MPLTEXT 1 224 1 "\n" }{MPLTEXT 1 224 1 "\n" }
{MPLTEXT 1 224 1 "\n" }{MPLTEXT 1 224 1 "\n" }}}{EXCHG {PARA 213 "> " 
0 "" {MPLTEXT 1 224 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 220 67 "You
 are free to insert any number in place of the 2 in the command " }
{TEXT 225 32 "CrossSectionOfFundamentalMode(2)" }{TEXT 220 22 " that a
ppears below.  " }{TEXT 220 223 "After executing the next command, cli
ck on the plot and then use the \"play movie\" tools that will appear \+
in the context menu above to play the animation.  Then change the numb
er an see the nodes for other fundamental modes." }{TEXT 220 0 "" }}
{PARA 210 "" 0 "" {TEXT 220 0 "" }}}{EXCHG {PARA 213 "> " 0 "" 
{MPLTEXT 1 224 33 "CrossSectionOfFundamentalMode(2);" }{MPLTEXT 1 224 
1 "\n" }{MPLTEXT 1 224 1 "\n" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 
1 224 0 "" }}}{PARA 210 "" 0 "" {TEXT 220 0 "" }}{PARA 210 "" 0 "" 
{TEXT 220 0 "" }}{PARA 210 "" 0 "" {TEXT 220 0 "" }}{PARA 210 "" 0 "" 
{TEXT 220 0 "" }}{PARA 210 "" 0 "" {TEXT 220 0 "" }}{EXCHG {PARA 213 "
> " 0 "" {MPLTEXT 1 224 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 
224 0 "" }}}}{PARA 214 "" 0 "" {TEXT 204 0 "" }}{PARA 214 "" 0 "" 
{TEXT 204 0 "" }}{PARA 214 "" 0 "" {TEXT 204 0 "" }}{PARA 214 "" 0 "" 
{TEXT 204 0 "" }}{PARA 214 "" 0 "" {TEXT 204 0 "" }}{PARA 215 "" 0 "" 
{TEXT 226 0 "" }}{PARA 216 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }
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