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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "# drumhead problem,
 General case, Example 2 in Asmar 4.3\n# u_\{tt\} = c^2(laplacian u), \+
u=0 on clamped edge\n# circle radius=a, c=wave speed, \n# position = f
(r,theta) = (a^2-r^2)*r*sin(theta), \n# velocity = g(r,theta) = 0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a:=1;c:=1;N:=5;f:=(r,theta)-
>(a^2-r^2)*r*sin(theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 
"alpha1:=evalf([BesselJZeros(1,1..N)]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "b1:=n->2*2*(a^5)*4/(alpha1[n]^3*BesselJ(2,alpha1[n]))
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "u:=(r,theta,t)->sin(th
eta)*sum(b1(n)*BesselJ(1,alpha1[n]*r)*cos(alpha1[n]*c*t),n=1..N);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "addcoords(z_cylindrical,[z,r
,theta],[r*cos(theta),r*sin(theta),z]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "plot3d(f(r,theta),r=0..a,theta=0..2*Pi,coords=z_cylin
drical,orientation=[13,81]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 64 "#plot3d(u(r,theta,0),r=0..a,theta=0..2*Pi,coords=z_cylindrical);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "plots[animate](plot3d,
\n[u(r,theta,t),r=0..a,theta=0..2*Pi,coords=z_cylindrical,orientation=
[13,81]],t=0..3.1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "# b1
 can also be defined by integration directly." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 150 "b2:=n->(2/(Pi*BesselJ(2,alpha1[n])^2))*\nint(in
t(f(r,theta)*BesselJ(1,alpha1[n]*r)*r*sin(theta),theta=0..2*Pi),r=0..a
);\nseq(evalf(b2(j)-b1(j)),j=1..N);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "12" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }
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