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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "# membrane problem \+
3.7-5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "# u_\{tt\} = c^2(l
aplacian u), u=0 on the 4 edges\n" }{MPLTEXT 1 0 41 "# position and ve
locity given u=1, u_t=0\n" }{MPLTEXT 1 0 75 "# Separation of variables
 leads to the answer in 3.7, display before thm 1." }}}{EXCHG }{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# we calculate on paper from 3.7 eq
 (8) that B_\{mn\}=0 and from (9) that\n" }{MPLTEXT 1 0 75 "# B_\{mn\}
^* = (16/sqrt(m^2+n^2))(1/(mn))(1/pi^2) for m,n odd, zero otherwise" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Bstar:=(m,n)->(16/sqrt(m^2
+n^2))*(1/(m*n))(1/Pi^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
63 "phi:=(x,y,t,m,n)->sin(m*Pi*x)*sin(n*Pi*y)*sin(sqrt(m^2+n^2)*t);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "N:=20: # Number of partial
 sum terms is N^2=400." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "u
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=0..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot3d(u(x,y,0.5
),x=0..1,y=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(p
lots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "animate(plot3d,[u
(x,y,t),x=0..1,y=0..1],t=0..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}
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