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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 259 10 "Lecture 7:" }}{PARA 
258 "" 0 "" {TEXT 256 43 "The Wave Equation & Separation of Variables
" }}{PARA 258 "" 0 "" {TEXT -1 33 "Mihaela B. Vajiac and Juan Tolosa" 
}}{PARA 259 "" 0 "" {TEXT -1 17 "PCMI, Summer 2003" }{TEXT 260 0 "" }
{TEXT 261 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: w
ith(plots):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Two-dimensional e
xamples" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Rectangular oscillating
 membrane" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:=Pi:b:=Pi:c:
=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "u := (m,n,x,y,t) -> \+
16*((-1)^m-1)*((-1)^n-1)/n^3/m^3/Pi^2*cos((m^2+n^2)^(1/2)*t)*sin(m*x)*
sin(n*y):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The first harmonic i
s periodic, with period " }{XPPEDIT 18 0 "sqrt(2)*Pi;" "6#*&-%%sqrtG6#
\"\"#\"\"\"%#PiGF(" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 90 "animate3d(u(1,1,x,y,t),x=0..a,y=0..b,t=0..Pi*sqrt(2),
\naxes=framed,thickness=2, frames=16);" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 18 0 "u(1,3,x,y,t);" "6#-%\"uG6'\"\"\"\"\"$%\"xG%\"yG%\"tG" }
{TEXT -1 12 " has period " }{XPPEDIT 18 0 "2*Pi/sqrt(10);" "6#*(\"\"#
\"\"\"%#PiGF%-%%sqrtG6#\"#5!\"\"" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 92 "animate3d(u(1,3,x,y,t),x=0..a,y=0..b,t=0..2*
Pi/sqrt(10),axes=framed,thickness=2, frames=16);" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 18 0 "u(5,3,x,y,t);" "6#-%\"uG6'\"\"&\"\"$%\"xG%\"yG%
\"tG" }{TEXT -1 12 " has period " }{XPPEDIT 18 0 "2*Pi/sqrt(34);" "6#*
(\"\"#\"\"\"%#PiGF%-%%sqrtG6#\"#M!\"\"" }{TEXT -1 2 " :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "animate3d(u(5,3,x,y,t),x=0..a,y=0..
b,t=0..2*Pi/sqrt(34),axes=framed,thickness=2, frames=16);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 103 "Since the various harmonics have differe
nt frequencies, a linear combination will not be periodic, but " }
{TEXT 265 16 "almost periodic." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 65 "combination := u(1,1,x,y,t) + 10*u(1,3, x,y,t)+ 200*u(5,3,x,y,
t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "animate3d(combinatio
n,x=0..a,y=0..b,t=0..1.35*Pi,axes=framed,thickness=2, frames=16);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 25 "This is a round membrane:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 144 "upartic:= (r, theta, t) -> 3.583422770*3^(1/2)/((1.9
13229428*Pi+105.4984657)*Pi)^(1/2)*cos(2.567811151*t)*BesselJ(2,5.1356
22302*r)*sin(2*theta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "T
period := 2*Pi/2.567811151:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
68 "addcoords(z_cylindrical,[z,r,theta],\n[r*cos(theta),r*sin(theta),z
]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "animate3d(upartic(r
,theta,t),r=0..1,theta=0..2*Pi,t=0..Tperiod,\ncoords=z_cylindrical,axe
s=BOXED, frames = 12);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Dampe
d string" }}{PARA 0 "" 0 "" {TEXT -1 73 "Let us consider a homogeneous
 problem corresponding to the damped string:" }}{PARA 0 "" 0 "" {TEXT 
-1 5 "PDE: " }{XPPEDIT 18 0 "u[tt] = c^2*u[xx]-gamma*u[t];" "6#/&%\"uG
6#%#ttG,&*&%\"cG\"\"#&F%6#%#xxG\"\"\"F/*&%&gammaGF/&F%6#%\"tGF/!\"\"" 
}{TEXT -1 9 " , where " }{XPPEDIT 18 0 "c^2 = T/rho;" "6#/*$%\"cG\"\"#
*&%\"TG\"\"\"%$rhoG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "gamma;" 
"6#%&gammaG" }{TEXT -1 32 " is a damping factor (positive);" }}{PARA 
0 "" 0 "" {TEXT -1 4 "BC: " }{XPPEDIT 18 0 "u(0,t) = 0;" "6#/-%\"uG6$
\"\"!%\"tGF'" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 8 "        \+
" }{XPPEDIT 18 0 "u(l,t) = 0;" "6#/-%\"uG6$%\"lG%\"tG\"\"!" }{TEXT -1 
1 ";" }}{PARA 0 "" 0 "" {TEXT -1 4 "IC: " }{XPPEDIT 18 0 "u(x,0) = f(x
);" "6#/-%\"uG6$%\"xG\"\"!-%\"fG6#F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 
"" {TEXT -1 6 "      " }{XPPEDIT 18 0 "u[t](x,0) = g(x);" "6#/-&%\"uG6
#%\"tG6$%\"xG\"\"!-%\"gG6#F*" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 31 "We seek non trivial solutions (" }{TEXT 266 14 "eigenfu
nctions" }{TEXT -1 47 "), using the method of separation of variables,
" }}{PARA 0 "" 0 "" {TEXT -1 3 "as " }{XPPEDIT 18 0 "u[n](x,t) = X(x)*
T(t);" "6#/-&%\"uG6#%\"nG6$%\"xG%\"tG*&-%\"XG6#F*\"\"\"-%\"TG6#F+F0" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "As before, the given ini
tial conditions yield" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "X[n](x) = sin(
n*Pi*x/l);" "6#/-&%\"XG6#%\"nG6#%\"xG-%$sinG6#**F(\"\"\"%#PiGF/F*F/%\"
lG!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "n = 1;" "6#/%\"nG\"\"\"" }
{TEXT -1 8 ", 2, ..." }}{PARA 0 "" 0 "" {TEXT -1 38 "The time factor i
s solved next, giving" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "T[n](t) = exp(
-gamma*t/2)*(A[n]*cos(alpha[n]*t)+sin(alpha[n]*t));" "6#/-&%\"TG6#%\"n
G6#%\"tG*&-%$expG6#,$*(%&gammaG\"\"\"F*F2\"\"#!\"\"F4F2,&*&&%\"AG6#F(F
2-%$cosG6#*&&%&alphaG6#F(F2F*F2F2F2-%$sinG6#*&&F?6#F(F2F*F2F2F2" }
{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "where" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "alpha[n] = sqrt((c*n*Pi/l)^2-gamma^2/4);" "6#/&%&alphaG
6#%\"nG-%%sqrtG6#,&*$**%\"cG\"\"\"F'F/%#PiGF/%\"lG!\"\"\"\"#F/*&%&gamm
aGF3\"\"%F2F2" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 14 "assuming
 that " }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT -1 72 " is suffici
ently small, so the expression under the radical is positive " }}
{PARA 0 "" 0 "" {TEXT -1 21 "for all n = 1, 2, ..." }}{PARA 0 "" 0 "" 
{TEXT -1 133 "The general solution of the whole problem (including the
 initial conditions) is sought as a linear combination of the eigenfun
ctions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Let us consider a part
icular case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:wit
h(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "l:= 1:   c:= 1
:  rho:= 1:   T:= 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "unp
rotect(gamma): gamma:= 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "assume(n::integer);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "
alpha := n -> sqrt((c*n*Pi/l)^2 - gamma^2/4):" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 17 "Initial position:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "f := x -> x*(1-x):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 17 "Initial velocity:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"g := x -> 0 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:= n -> \+
(2/l)*int(f(x)*sin(n*Pi*x/l), x=0..l):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "B:= n -> (2/alpha(n)/l)*int((gamma*f(x)/2+g(x))*\n   \+
       sin(n*Pi*x/l), x=0..l):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 
"Eigenfunctions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "upart \+
:= (n,x,t) -> exp(-gamma*t/2)*(A(n)*cos(alpha(n)*t)\n                 \+
   + B(n)*sin(alpha(n)*t))*sin(n*Pi*x/l):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 62 "animate(upart(1,x,t), x=0..1, t=0..4, frames=36, th
ickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot3d(upart
(1,x,t), x=0..1, t=0..4, orientation=[-15,77]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "The \+
plucked string" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: w
ith(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "PDE: " }{XPPEDIT 
18 0 "u[tt] = c^2*u[xx];" "6#/&%\"uG6#%#ttG*&%\"cG\"\"#&F%6#%#xxG\"\"
\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "BC: " }{XPPEDIT 18 0 "u(0,t) = 0;" "6#/-%\"uG6$\"\"!%\"tGF
'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "u(L,t) = 0;" "6#/-%\"uG6$%\"LG%\"t
G\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 4 "IC: " }{XPPEDIT 18 0 "u(x,0) = f(x);" "6#/-%\"uG6$%\"
xG\"\"!-%\"fG6#F'" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "u[t](x,0) = 0;
" "6#/-&%\"uG6#%\"tG6$%\"xG\"\"!F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Eigenvalues:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "lambda := n -> n*Pi/l:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 264 16 "Initial position" }{TEXT -1 2 ": \+
" }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 32 "is a triangle
 (plucked string) ." }}{PARA 0 "" 0 "" {TEXT -1 25 "We assume that at \+
a point" }{TEXT 262 1 " " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT -1 42 "
 on (0, l) the string is lifted to height " }{TEXT 263 1 "h" }{TEXT 
-1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 33 "remaining fixed at the endpoin
ts." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "assume(0<p, p<l, 0<h
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f := x -> piecewise(x
 <=p, h*x/p, x > p, h*(l-x)/(l-p)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 61 "We plot the function for particular values of the parameters:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "particular := \{l=1, h=1/
2, p=2/3, c=1\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(su
bs(particular,f(x)), x=0..1,y=0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 28 "Fourier coefficientes, A(n)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "Af := n -> (2/l)*int(f(x)*sin(lambda(n)*x),x=0..l):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "result :=subs(\{cos(n*Pi)
=(-1)^n, sin(n*Pi)=0\}, Af(n)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "A := unapply(result,n):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
63 "We find the amplitudes for particular values of the parameters:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "AA := subs(particular,unap
ply(result,n)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "This is the so
lution (eigenfunction) por the particular values we chose:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "u:= (n, x, t) -> (AA(n)*cos(c*lambd
a(n)*t))\n               *sin(lambda(n)*x):" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 66 "Approximate solution, for our particular choice of the \+
parameters:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ua := (x,t) \+
-> subs(particular, sum(u(n,x,t), n=1..17)):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "animate(ua(x,t),x=0..1,t=0..2,color=red,thicknes
s=2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "This is a plot of the su
rface u(x, t)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot3d(ua
(x,t), x=0..1, t=0..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Localized plucking" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Lo
calized plucking, to better see the traveling waves." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We give particu
lar values to all parameters:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 59 "l:=1:  h:=1/2:  p:=2/3:   c:=1:  \na:= 2/3-0.2:  b:=2/3+0.1:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "fsmall := x -> piecewise(x>
=a and x <=p,h*(x-a)/(p-a), \nx >=p and x<=b, h*(x-b)/(p-b)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(fsmall(x), x=0..1,y=0..
1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Asmall := m -> (2/l)
*int(fsmall(x)*sin(lambda(m)*x),x=0..l):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 86 "usmall:= (n, x, t) -> (Asmall(n)*cos(c*lambda(n)*t))
\n               *sin(lambda(n)*x):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 35 "Approximate solution for this case:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "uaa := (x,t) -> sum(usmall(n,x,t), n=1..23):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "animate(uaa(x,t),x=0..1,t=0.
.2,color=red,thickness=2, frames=22);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 38 "This is a plot of the surface u(x, t)." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "plot3d(uaa(x,t), x=0..1, t=0..2, grid=[60
,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 19 "Musical instruments" }}{PARA 0 "" 0 "" 
{TEXT -1 40 "We have already seen the plucked string." }}{PARA 0 "" 0 
"" {TEXT -1 67 "Let us discuss other problems for the string equation,
 arising from" }}{PARA 0 "" 0 "" {TEXT -1 40 "the way musical instrume
nts are played. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Localized imp
ulse" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "If we hit the string with an \+
impulse " }{TEXT 257 2 "K " }{TEXT -1 24 "concentrated at a point " }
{TEXT 258 2 "p," }}{PARA 0 "" 0 "" {TEXT -1 79 "(say we hit the string
 with the blade of a knife) then the solution is given by" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The nth ar
monic is given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "uh := \+
(n, x, t) -> 2*K/(Pi*c*rho)*(1/n)*\n       sin(Pi*n*p/l)*sin(Pi*n*x/l)
*sin(Pi*n*c*t/l):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "partic
ular:= \{l=1, c=1, K=1, p=2/3, rho = 1\}:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 44 "Approximate solution to the \"impulse start\":" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "uimp:= (x,t) -> sum(uh(n,x,t
),n=1..9):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "animate(subs(
particular,uimp(x,t)), x=0..1, t=0..2, thickness=2);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 63 "plot3d(subs(particular,uimp(x,t)), x=0..1
, t=0..2, axes=BOXED);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Small flat hammer" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 73 "The initial position is zero, the initial
 velocity is constant, equal to " }{XPPEDIT 18 0 "v[0];" "6#&%\"vG6#\"
\"!" }{TEXT -1 21 ", on a small interval" }}{PARA 0 "" 0 "" {TEXT -1 
1 "(" }{XPPEDIT 18 0 "p-delta,p+delta;" "6$,&%\"pG\"\"\"%&deltaG!\"\",
&F$F%F&F%" }{TEXT -1 27 "). The solution is given by" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "u(x,t):= Sum(u[n](x,t), n=1..infinity):" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "where the nth armonic is given \+
by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "uh := (n, x, t) -> 4
*v[0]*l/(Pi^2*c)*(1/n^2)*\n       sin(Pi*n*p/l)*sin(Pi*n*delta/l)*\n  \+
     sin(Pi*n*x/l)*sin(Pi*n*c*t/l):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "particular:= \{l=1, c=1, v[0]=1, p=2/3, rho = 1, delt
a=.1\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Approximate solution t
o the \"impulse start\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 
"uimp:= (x,t) -> sum(uh(n,x,t),n=1..9):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "animate(subs(particular,uimp(x,t)), x=0..1, t=0..2,th
ickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot3d(subs(
particular,uimp(x,t)), x=0..1, t=0..2, axes=BOXED);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "0 2 0" 11 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }