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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Simulation der Wellenausbr
eitung in dem zweidimensionalen Quadrat (0,Pi)x(0,Pi). " }}{PARA 0 "" 
0 "" {TEXT -1 88 "Phi ist das Anfangsdatum fuer die Auslenkung u, psi \+
fuer die Anfangsgeschwindigkeit u_t." }}{PARA 0 "" 0 "" {TEXT -1 84 "E
s werden homogene Dirichletranddaten vorgeschrieben. Man beachte, dass
 die Anfangs-" }}{PARA 0 "" 0 "" {TEXT -1 90 "daten ausserhalb eines k
leieneren Quadrats durch 0 ersetzt werden, so dass die Integration" }}
{PARA 0 "" 0 "" {TEXT -1 90 "nur auf (1.5,2)x(1.5,2) ausgefuehrt wird.
 Die Anfangsdaten sind also raeumlich lokalisiert" }}{PARA 0 "" 0 "" 
{TEXT -1 68 "und man kann so die Wellenausbreitung und Reflektion gut \+
beobachten." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "phi:=(x,y)->
(x-1.5)*(2-x)*(y-1.5)*(2-y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "plot3d(phi(x,y),x=0..Pi,y=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "b:=array(1..20,1..20);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "psi:=(x,y)->0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "B:=array(1..20,1..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "for n from 1 by 1 to 20" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " \+
 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "     for m from 1 by 1 to 20
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "        do" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 86 "            b[m,n]:=(4/Pi^2)*int(int(phi(x,y)*sin(m
*x)*sin(n*y) ,x=1.5..2) ,y=1.5..2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "        od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign('n');" }}}{EXCHG {PARA 0 "
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for n f
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"> " 0 "" {MPLTEXT 1 0 28 "     for m from 1 by 1 to 20" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "        do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
84 "            B[m,n]:=(4/Pi^2)*int(int(psi(x,y)*sin(m*x)*sin(n*y) ,x
=0..Pi) ,y=0..Pi);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "        od;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "unassign('n'):unassign('m');" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 154 "u:=(t,x,y)->sum(sum((b[m,n]*sin(m*x) *sin(n*y
)*cos(sqrt(m^2+n^2)*t)+(B[m,n]/sqrt(m^2+n^2) )*sin(m*x) *sin(n*y)*sin(
sqrt(m^2+n^2)*t)   ),n=1..20),m=1..20);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "animate3d(u(t,x,y),x=0..Pi,y=0..Pi,t=0..2,frames=20);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Es folgt ein zweiter Durchlauf, in
 dem der Nachhalleffekt beobachtet werden kann. Dieser Effekt" }}
{PARA 0 "" 0 "" {TEXT -1 92 "ist typisch fuer zwei (geradzahlige) Dime
nsion, waehrend man in ungeradzahligen Dimensionen " }}{PARA 0 "" 0 "
" {TEXT -1 51 "ab  3 scharfe Wellenfronten hat (Huyghens Prinzip)." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "phi:=(x,y)->0;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "b:=array(1..20,1..20);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "psi:=(x,y)->5000*(x-1.5)*(2-x)*(y-1
.5)*(2-y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B:=array(1..2
0,1..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for n from 1 b
y 1 to 20" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "  do" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 28 "     for m from 1 by 1 to 20" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "        do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "    \+
        b[m,n]:=(4/Pi^2)*int(int(phi(x,y)*sin(m*x)*sin(n*y) ,x=0..Pi) \+
,y=0..Pi);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "        od;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 5 "  od;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "unassign('n');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
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{MPLTEXT 1 0 28 "     for m from 1 by 1 to 20" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "        do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "    \+
        B[m,n]:=(4/Pi^2)*int(int(psi(x,y)*sin(m*x)*sin(n*y) ,x=1.5..2)
 ,y=1.5..2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "        od;" }}
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{MPLTEXT 1 0 28 "unassign('n'):unassign('m');" }}}{EXCHG {PARA 0 "> " 
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