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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Wave Equation Example. " 
}}{PARA 0 "" 0 "" {TEXT -1 28 "Consider the wave equation  " }
{XPPEDIT 18 0 "a^2*u[xx] = u[tt];" "6#/*&%\"aG\"\"#&%\"uG6#%#xxG\"\"\"
&F(6#%#ttG" }{TEXT -1 8 "  where " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "u;
" "6#%\"uG" }{TEXT -1 12 "(0,t)=0 and " }{XPPEDIT 18 0 "u;" "6#%\"uG" 
}{TEXT -1 8 "(L,t)=0," }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "u;" "6#%\"uG" }{TEXT -1 12 "(x,0)=f(x), " }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "u[t];" "6#&%\"uG6#%\"tG" }{TEXT -1 11 "(x,0) \+
= 0. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 65 "Finding the and plotting
 the solution from initial displacement. " }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 74 " We'll first define the initial displacement function f(x
) plotted below. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "L := 10
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG\"#5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 7 "a := 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
aG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f := x -> piecew
ise(x<5,x,x>=5,10-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"x
G6\"6$%)operatorG%&arrowGF(-%*piecewiseG6&29$\"\"&F01F1F0,&\"#5\"\"\"F
0!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(f(x),x
=0..10,thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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w=F9FD7$$\"1MLL3En$4#F9FG7$$\"1nm;/RE&G#F9FJ7$$\"1+++D.&4]#F9FM7$$\"1+
++vB_<FF9FP7$$\"1+++v'Hi#HF9FS7$$\"1nm\"z*ev:JF9FV7$$\"1MLL347TLF9FY7$
$\"1MLLLY.KNF9Ffn7$$\"1++D\"o7Tv$F9Fin7$$\"1LLL$Q*o]RF9F\\o7$$\"1,+D\"
=lj;%F9F_o7$$\"1++vV&R<P%F9Fbo7$$\"1ML$e9Ege%F9Feo7$$\"1MLeR\"3Gy%F9Fh
o7$$\"1+](=7O*))[F9F[p7$$\"1nm;/T1&*\\F9F^p7$$\"1m;/^7I0^F9$\"1M$e*[()
p%*[F97$$\"1nm\"zRQb@&F9$\"1LL3-;Y%y%F97$$\"1++v=>Y2aF9$\"1++D\"3QDf%F
97$$\"1nm;zXu9cF9$\"1LL$3Ub_Q%F97$$\"1+++]y))GeF9$\"1+++]@6rTF97$$\"1+
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*\\Pz\"[%H#F97$$\"1LLL3s?6zF9$\"1nmm\"z#z)3#F97$$\"1++DJXaE\")F9$\"1++
voaXt=F97$$\"1ommm*RRL)F9$\"1KLLL+1m;F97$$\"1om;a<.Y&)F9$\"1KL$eCoRX\"
F97$$\"1NLe9tOc()F9$\"1lmT&oKOC\"F97$$\"1,++]Qk\\*)F9$\"1******\\hN]5F
97$$\"1NL$3dg6<*F9$\"1_mm\"H%R)G)F,7$$\"1ommmxGp$*F9$\"1ALLLB72jF,7$$
\"1++D\"oK0e*F9$\"1)***\\(=tY>%F,7$$\"1,+v=5s#y*F9$\"1*)**\\7)*ys@F,7$
$\"#5F(F(-%'COLOURG6&%$RGBG$Fjw!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"%!G-%*T
HICKNESSG6#\"\"$-%%VIEWG6$;F(Fiw%(DEFAULTG" 1 2 0 1 3 2 9 1 4 2 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 
"Now we'll plot the odd extension of f(x) and define it as oddext(x).
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "oddext := x -> piecewis
e(x<0,-f(-x),x>=0,g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'oddextG
R6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6&29$\"\"!,$-%\"fG6#,$F
0!\"\"F71F1F0-%\"gG6#F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "plot(oddext(x),x=-10..10,thickness=3);" }}{PARA 13 "" 1 "" 
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t7$$\"1ML$3A_1?&F.$\"1mm;zxM*z%F.7$$\"1,+]7k.6aF.$\"1****\\(ej*)e%F.7$
$\"1mmm;WTAeF.$\"1MLL$e&exTF.7$$\"1****\\i!*3`iF.$\"1,+]P4\"pu$F.7$$\"
1NLLL*zym'F.$\"1lmmm+7KLF.7$$\"1OLL3N1#4(F.$\"1kmm\"\\Oz!HF.7$$\"1pm;H
Yt7vF.$\"1JL$3Pls[#F.7$$\"1-+++xG**yF.$\"1)******H725#F.7$$\"1qmmT6KU$
)F.$\"1ILLe)ywl\"F.7$$\"1OLLLbdQ()F.$\"1kmmmWUh7F.7$$\"1++]i`1h\"*F.$
\"1&****\\PY$*Q)F17$$\"1-+]P?Wl&*F.$\"1y***\\izbM%F17$$\"#5F*F*-%'COLO
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45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "No we'll extend th
is over three periods with fundamental period = 20. " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 68 "h := x -> piecewise(x<-10,oddext(x+20),x>
10,oddext(x-20),oddext(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR
6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6'29$!#5-%'oddextG6#,&F0
\"\"\"\"#?F62\"#5F0-F36#,&F0F6!#?F6-F36#F0F(F(F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 33 "plot(h(x),x=-20..20,thickness=3);" }}{PARA 
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7$$\"10+++(fa<*F7$\"1`******HSX#)F17$$\"1KLLeg`!)**F7$\"1@ommTRY>Fes7$
$\"1++]#G2A3\"F.$!1H++]#G2A)F17$$\"1LLL$)G[k6F.$!1KLLL)G[k\"F77$$\"1++
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#?F*F*-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$Q\"x6\"%!G-%*THI
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"Now that we have the odd extension over three periods, we define the \+
solution by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sol := (x,t
) -> 1/2 * (h(x-a*t) + h(x+a*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$solGR6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF),&-%\"hG6#,&9$\"\"\"*&%\"
aGF39%F3!\"\"#F3\"\"#-F/6#,&F2F3F4F3F8F)F)F)" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 78 "and plot it over the domain of intereste x = 0..10. You
 can change the time.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "
plot(sol(x,10),x=0..10,thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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{TEXT -1 174 "With regards to time, the Fourier solution shows that th
ere is a fundamental period of 2 L / a.  We can animate the solutions \+
over this one period with the animate commands. " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
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