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{SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT -1 0 "" }}{PARA 200 "" 0 "" 
{TEXT 202 7 "PROBLEM" }{TEXT 203 235 ":  Solve the wave equation for t
he vertical displacements of a circular membrane whose edge is rigidly
 fixed.  Assume that the displacements are independent of the angle th
eta, so that the wave equation in polar coordinates reduces to " }
{TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 0 "" }}{PARA 201 "" 0 "" 
{TEXT 203 2 "  " }{XPPEDIT 2 0 "diff(u(r,t),r,r) + (1/r)*diff(u(r,t),r
) = (1/c^2)*diff(u(r,t),t,t)" "6#/,&-%%diffG6%-%\"uG6$%\"rG%\"tGF+F+\"
\"\"*(F-F-F+!\"\"-F&6$-F)6$F+F,F+F-F-*(F-F-*$%\"cG\"\"#F/-F&6%-F)6$F+F
,F,F,F-" }{TEXT 203 6 "      " }{TEXT 203 0 "" }}{PARA 200 "" 0 "" 
{TEXT 203 0 "" }}{PARA 201 "" 0 "" {TEXT 203 28 "0 < r < 1      and   \+
  t > 0" }{TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 0 "" }}{PARA 
200 "" 0 "" {TEXT 202 19 "Boundary Conditions" }{TEXT 203 0 "" }}
{PARA 200 "" 0 "" {TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 85 "(1)
  The condition \"rigidly fixed\" at the boundary implies       u(1,t)
 = 0    t > 0." }{TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 93 "(2) \+
 Since r = 0 is a singular point of the PDE, we require3:     u(r,t) f
inite as  r ---> 0+" }{TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 0 "
" }}{PARA 200 "" 0 "" {TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 202 18 
"Initial Conditions" }{TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 0 "
" }}{PARA 200 "" 0 "" {TEXT 203 5 "(1)  " }{XPPEDIT 2 0 "u(r,0) = f(r)
" "6#/-%\"uG6$%\"rG\"\"!-%\"fG6#F'" }{TEXT 203 32 "   ,  0 < r < 1  (i
nitial shape)" }{TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 5 "(2)  \+
" }{XPPEDIT 2 0 "u[t](r,0) = g(r)" "6#/-&%\"uG6#%\"tG6$%\"rG\"\"!-%\"g
G6#F*" }{TEXT 203 50 "   ,  0 < r < 1  (initial velocity distribution)
  " }{TEXT 203 0 "" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}
{PARA 203 "" 0 "" {TEXT 205 133 "We now separate variables with the as
sumption u(r,t) = R(r) T(t).  Substitution into the PDE followed by di
vision by R(r) T(t) yields" }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 
205 0 "" }}{PARA 204 "" 0 "" {TEXT 205 5 "     " }{XPPEDIT 2 0 "((Diff
(R(r), `$`(r, 2)))+(Diff(R(r), r))/r)/R = (Diff(T(t), `$`(t, 2)))/(c^2
*T(t))" "6#/*&,&-%%DiffG6$-%\"RG6#%\"rG-%\"$G6$F,\"\"#\"\"\"*&-F'6$-F*
6#F,F,F1F,!\"\"F1F1F*F7*&-F'6$-%\"TG6#%\"tG-F.6$F>F0F1*&%\"cGF0-F<6#F>
F1F7" }{TEXT 205 7 "       " }{TEXT 205 0 "" }}{PARA 203 "" 0 "" 
{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 76 "By the standard argue
ment both sides must be a constant, which we denote by " }{XPPEDIT 2 
0 "-lambda" "6#,$%'lambdaG!\"\"" }{TEXT 205 46 ".  The minus sigh was \+
inserted here to ......." }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 
205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 
"" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 23 "Hence we have two \+
ODES:" }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 204 
"" 0 "" {TEXT 205 3 "   " }{XPPEDIT 2 0 "Diff(R(r),r,r) + (1/r)*Diff(R
(r),r) + lambda * R(r) = 0" "6#/,(-%%DiffG6%-%\"RG6#%\"rGF+F+\"\"\"*(F
,F,F+!\"\"-F&6$-F)6#F+F+F,F,*&%'lambdaGF,-F)6#F+F,F,\"\"!" }{TEXT 205 
2 "  " }{TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 205 0 "" }}{PARA 204 
"" 0 "" {TEXT 205 3 "and" }{TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 
205 0 "" }}{PARA 204 "" 0 "" {XPPEDIT 2 0 "Diff(T(t),t,t) + lambda*c^2
*T(t) =  0" "6#/,&-%%DiffG6%-%\"TG6#%\"tGF+F+\"\"\"*(%'lambdaGF,*$%\"c
G\"\"#F,-F)6#F+F,F,\"\"!" }{TEXT 205 4 "    " }{TEXT 205 0 "" }}{PARA 
204 "" 0 "" {TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "
" }}{PARA 203 "" 0 "" {TEXT 205 170 "Next we must recast the homogeneo
us boundary and initial conditions.  We will consider first the case o
f non-trivial initial shape but zero initial velocity distribution." }
{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 203 "" 0 "" 
{TEXT 206 20 "Boundary Conditions:" }{TEXT 205 0 "" }}{PARA 200 "" 0 "
" {TEXT 203 85 "(1)  The condition      u(1,t) = 0    t > 0           \+
      ===>  R(1) = 0  for t > 0" }{TEXT 203 0 "" }}{PARA 200 "" 0 "" 
{TEXT 203 87 "(2)  The condition      u(r,t) finite as  r ---> 0+     \+
===>   R(r) finite as r ---> 0+" }{TEXT 203 0 "" }}{PARA 200 "" 0 "" 
{TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 202 19 "Initial Conditions:" 
}{TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 5 "(1)  " }{XPPEDIT 2 0 
"u(r,0) = f(r)" "6#/-%\"uG6$%\"rG\"\"!-%\"fG6#F'" }{TEXT 203 56 "   , \+
 0 < r < 1  (initial shape)       (Save for later) " }{TEXT 203 0 "" }
}{PARA 200 "" 0 "" {TEXT 203 5 "(2)  " }{XPPEDIT 2 0 "u[t](r, 0) = 0" 
"6#/-&%\"uG6#%\"tG6$%\"rG\"\"!F+" }{TEXT 203 60 "      ,  0 < r < 1  (
initial velocity)   ===>    T'(0) = 0 ." }{TEXT 203 0 "" }}}{SECT 0 
{PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 37 "Hen
ce our Sturm-Liouville Problem is:" }{TEXT 205 0 "" }}{PARA 204 "" 0 "
" {TEXT 205 3 "   " }{XPPEDIT 2 0 "Diff(R(r),r,r) + (1/r)*Diff(R(r),r)
 + lambda * R(r) = 0" "6#/,(-%%DiffG6%-%\"RG6#%\"rGF+F+\"\"\"*(F,F,F+!
\"\"-F&6$-F)6#F+F+F,F,*&%'lambdaGF,-F)6#F+F,F,\"\"!" }{TEXT 205 0 "" }
}{PARA 204 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 205 45 "R(
1) = 0  ,     R(r) finite at r ---> 0+     " }{TEXT 205 0 "" }}{PARA 
204 "" 0 "" {TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 26 "while the
 T(t) equation is" }{TEXT 205 0 "" }}{PARA 204 "" 0 "" {XPPEDIT 2 0 "D
iff(T(t),t,t) + lambda*c^2*T(t) =  0" "6#/,&-%%DiffG6%-%\"TG6#%\"tGF+F
+\"\"\"*(%'lambdaGF,*$%\"cG\"\"#F,-F)6#F+F,F,\"\"!" }{TEXT 205 4 "    \+
" }{TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 
"" {TEXT 205 9 "T'(0) = 0" }{TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "
" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 117 "The differential e
quation in the SL problem is Bessel's equation of order zero, and we w
ill consider the three cases " }{XPPEDIT 2 0 "lambda < 0" "6#2%'lambda
G\"\"!" }{TEXT 205 3 " , " }{XPPEDIT 2 0 "lambda = 0" "6#/%'lambdaG\"
\"!" }{TEXT 205 6 ", and " }{XPPEDIT 2 0 "lambda > 0" "6#2\"\"!%'lambd
aG" }{TEXT 205 2 " ." }{TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "" 
{TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 9 "Case I:  " }{XPPEDIT 
2 0 "lambda = - omega^2" "6#/%'lambdaG,$*$%&omegaG\"\"#!\"\"" }{TEXT 
205 8 " < 0 ,  " }{XPPEDIT 2 0 "omega > 0" "6#2\"\"!%&omegaG" }{TEXT 
205 41 " .      Then the differential equation is" }{TEXT 205 0 "" }}
{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 "" {XPPEDIT 2 0 "r*(
Diff(R(r), r, r))+(Diff(R(r), r))-omega^2*r*R(r) = 0" "6#/,(*&%\"rG\"
\"\"-%%DiffG6%-%\"RG6#F&F&F&F'F'-F)6$-F,6#F&F&F'*(%&omegaG\"\"#F&F'-F,
6#F&F'!\"\"\"\"!" }{TEXT 205 2 " ." }{TEXT 205 0 "" }}}{SECT 0 {PARA 
202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 126 "This is \+
Bessel's \"modified\" equation of order zero, with the general solutio
n given in terms of the modified Bessel functions " }{XPPEDIT 2 0 "I[0
](omega*r)" "6#-&%\"IG6#\"\"!6#*&%&omegaG\"\"\"%\"rGF+" }{TEXT 205 5 "
 and " }{XPPEDIT 2 0 "K[0](omega*r)" "6#-&%\"KG6#\"\"!6#*&%&omegaG\"\"
\"%\"rGF+" }{TEXT 205 2 " :" }{TEXT 205 0 "" }}{PARA 203 "" 0 "" 
{TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 205 1 " " }{XPPEDIT 2 0 "R(r)
 = A* I[0](omega*r)   +  B*K[0](omega*r)" "6#/-%\"RG6#%\"rG,&*&%\"AG\"
\"\"-&%\"IG6#\"\"!6#*&%&omegaGF+F'F+F+F+*&%\"BGF+-&%\"KG6#F06#*&F3F+F'
F+F+F+" }{TEXT 205 5 "     " }{TEXT 205 0 "" }}{PARA 203 "" 0 "" 
{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 100 "with the boundary co
nditions R(1) = 0  and R(r) finite at r approaches zero from the posit
ive side. " }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}}
{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 
205 53 "The modified Bessel functions are denoted by BesselI(" }
{XPPEDIT 2 0 "nu" "6#%#nuG" }{TEXT 205 62 ",x) (notice the capital I a
t the end of the name) and BesselK(" }{XPPEDIT 2 0 "nu" "6#%#nuG" }
{TEXT 205 98 ",x) , where nu denotes the \"order\" of the equation and
 x is the independent variable.  We need on " }{XPPEDIT 2 0 "nu" "6#%#
nuG" }{TEXT 205 6 " = 0 ." }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 
205 0 "" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 59 "plot([BesselI
(0,x),BesselK(0,x)],x=0..2,color=[red,green]);" }{MPLTEXT 1 207 0 "" }
}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{PARA 203 "" 0 "" 
{TEXT 205 201 "From the plot it is clear that BesselK(0,x) is unbounde
d at x = 0 and hence cannot satisfy the condition that R(r) be finite \+
as r ---> 0+.  Hence we must take B=0 in our solution, which now reduc
es to " }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 
204 "" 0 "" {TEXT 205 1 " " }{XPPEDIT 2 0 "R(r) = A*I[0](omega*r)" "6#
/-%\"RG6#%\"rG*&%\"AG\"\"\"-&%\"IG6#\"\"!6#*&%&omegaGF*F'F*F*" }{TEXT 
205 6 "      " }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 36 "The f
inal condition R(1) = 0 implies" }{TEXT 205 0 "" }}{EXCHG {PARA 205 ">
 " 0 "" {MPLTEXT 1 207 26 "0 = A*evalf(BesselI(0,1));" }{MPLTEXT 1 
207 0 "" }}}{PARA 203 "" 0 "" {TEXT 205 67 "Hence A = 0 and there are \+
no negative eigenvalues for this problem." }{TEXT 205 0 "" }}{EXCHG 
{PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}}{SECT 0 {PARA 202 "" 0 "" 
{TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 206 7 "Case II" }{TEXT 205 3 
":  " }{XPPEDIT 2 0 "lambda = 0" "6#/%'lambdaG\"\"!" }{TEXT 205 56 " .
     The differential equation in this case reduces to" }{TEXT 205 0 "
" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 "" {XPPEDIT 2 0 
"r*(Diff(R(r), r, r))+(Diff(R(r), r)) = 0" "6#/,&*&%\"rG\"\"\"-%%DiffG
6%-%\"RG6#F&F&F&F'F'-F)6$-F,6#F&F&F'\"\"!" }{TEXT 205 2 "  " }{TEXT 
205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 
"" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 12 "Notice that " }
{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{EXCHG {PARA 205 "
> " 0 "" {MPLTEXT 1 207 24 "diff(r*diff(R(r),r),r) ;" }{MPLTEXT 1 207 
0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{PARA 203 "
" 0 "" {TEXT 205 29 "so that the above ODE implies" }{TEXT 205 0 "" }}
{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 "" {XPPEDIT 2 0 "Dif
f(R(r), r) = A/r" "6#/-%%DiffG6$-%\"RG6#%\"rGF**&%\"AG\"\"\"F*!\"\"" }
{TEXT 205 22 "    for some constant " }{TEXT 210 1 "A" }{TEXT 205 0 "
" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" 
{TEXT 205 48 "Another integration yields the general solution:" }
{TEXT 205 0 "" }}{PARA 204 "" 0 "" {XPPEDIT 2 0 "R(r) = A*ln(r) +  B" 
"6#/-%\"RG6#%\"rG,&*&%\"AG\"\"\"-%#lnG6#F'F+F+%\"BGF+" }{TEXT 205 1 " \+
" }{TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 
203 "" 0 "" {TEXT 205 102 "In order for R(r) to be bounded as r--> 0+ \+
we must take A=0, so R(r) = B.  But then R(1) = 0 => B = 0." }{TEXT 
205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 
205 4 "So, " }{XPPEDIT 2 0 "lambda = 0" "6#/%'lambdaG\"\"!" }{TEXT 
205 39 " is NOT an eigenvalue for this problem." }{TEXT 205 0 "" }}
{PARA 203 "" 0 "" {TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 
204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 11 "Case III:  " }{XPPEDIT 2 0 
"lambda = omega^2" "6#/%'lambdaG*$%&omegaG\"\"#" }{TEXT 205 7 " > 0 , \+
" }{XPPEDIT 2 0 "omega > 0" "6#2\"\"!%&omegaG" }{TEXT 205 59 " :  In t
his case the ODE is Bessel's equation of order zero" }{TEXT 205 0 "" }
}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 205 4 "   \+
 " }{XPPEDIT 2 0 "r*Diff(R(r),r,r) + Diff(R(r),r) + omega^2*r*R(r) = 0
" "6#/,(*&%\"rG\"\"\"-%%DiffG6%-%\"RG6#F&F&F&F'F'-F)6$-F,6#F&F&F'*(%&o
megaG\"\"#F&F'-F,6#F&F'F'\"\"!" }{TEXT 205 8 "        " }{TEXT 205 0 "
" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" 
{TEXT 205 4 "The " }{TEXT 211 16 "GENERAL SOLUTION" }{TEXT 205 3 " is
" }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 
"" {XPPEDIT 2 0 "R(r) = A*J[0](omega*r)  +  B*Y[0](omega*r)" "6#/-%\"R
G6#%\"rG,&*&%\"AG\"\"\"-&%\"JG6#\"\"!6#*&%&omegaGF+F'F+F+F+*&%\"BGF+-&
%\"YG6#F06#*&F3F+F'F+F+F+" }{TEXT 205 4 "    " }{TEXT 205 0 "" }}}
{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 
205 16 "and must satisfy" }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 
205 0 "" }}{PARA 204 "" 0 "" {TEXT 212 24 "R(r) bounded as r --> 0+" }
{TEXT 205 1 " " }{TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 205 0 "" }}
{PARA 204 "" 0 "" {TEXT 205 3 "and" }{TEXT 205 0 "" }}{PARA 204 "" 0 "
" {TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 212 9 "R(1)  = 0" }{TEXT 
205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 
"" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 22 "Here are the plots
 of " }{XPPEDIT 2 0 "J[0](x)" "6#-&%\"JG6#\"\"!6#%\"xG" }{TEXT 205 5 "
 and " }{XPPEDIT 2 0 "Y[0](x)" "6#-&%\"YG6#\"\"!6#%\"xG" }{TEXT 205 1 
":" }{TEXT 205 0 "" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 71 "pl
ot([BesselJ(0,x),BesselY(0,x)],x=0..5,color=[red,green],thickness=3);
" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 
"" }}}{PARA 203 "" 0 "" {TEXT 205 8 "Clearly " }{XPPEDIT 2 0 "Y[0](x)
" "6#-&%\"YG6#\"\"!6#%\"xG" }{TEXT 205 61 " is unbounded as x --> 0+, \+
so we must choose the coefficient " }{TEXT 212 1 "B" }{TEXT 205 5 " of
  " }{XPPEDIT 2 0 "Y[0](x)" "6#-&%\"YG6#\"\"!6#%\"xG" }{TEXT 205 30 " \+
 to be zero.  R(r) reduces to" }{TEXT 205 0 "" }}{PARA 203 "" 0 "" 
{TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 205 2 "  " }{XPPEDIT 2 0 "R(r
) = A*J[0](omega*r)" "6#/-%\"RG6#%\"rG*&%\"AG\"\"\"-&%\"JG6#\"\"!6#*&%
&omegaGF*F'F*F*" }{TEXT 205 2 "  " }{TEXT 205 0 "" }}}{SECT 0 {PARA 
202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 71 "Applying \+
the remaining boundary condition R(1) = 0 we find the equation" }
{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{EXCHG {PARA 205 "
> " 0 "" {MPLTEXT 1 207 32 "BC_eqn:= A*BesselJ(0,omega) = 0;" }
{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" 
}}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" 
{TEXT 205 39 "To have a non-trivial solution we need " }{XPPEDIT 2 0 "
A <> 0" "6#0%\"AG\"\"!" }{TEXT 205 40 " . So  the eigenvalues will be \+
given by " }{XPPEDIT 2 0 "lambda = omega^2" "6#/%'lambdaG*$%&omegaG\"
\"#" }{TEXT 205 8 " where  " }{XPPEDIT 2 0 "omega" "6#%&omegaG" }
{TEXT 205 31 " is a solution of the equation " }{XPPEDIT 2 0 "J[0](ome
ga) = 0" "6#/-&%\"JG6#\"\"!6#%&omegaGF(" }{TEXT 205 15 ".   Let's plot
 " }{XPPEDIT 2 0 "J[0](x)" "6#-&%\"JG6#\"\"!6#%\"xG" }{TEXT 205 24 " t
o look at it's zero's." }{TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "" 
{TEXT 204 0 "" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 27 "plot(Be
sselJ(0,x),x=0..15);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 205 "> " 0 
"" {MPLTEXT 1 207 0 "" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}
{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 31 "fsolve(BesselJ(0,x)=0,x=
8..10);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 
1 207 0 "" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "
" 0 "" {TEXT 205 35 "In fact Maple's built in functions " }{TEXT 213 
15 "BesselJZeros( )" }{TEXT 205 6 "  and " }{TEXT 213 15 "BesselYZeros
( )" }{TEXT 205 26 " will do this work for us!" }{TEXT 205 0 "" }}
{PARA 203 "" 0 "" {TEXT 205 0 "" }}{EXCHG {PARA 205 "> " 0 "" 
{MPLTEXT 1 207 18 "BesselJZeros(0,3);" }{MPLTEXT 1 207 0 "" }{MPLTEXT 
1 207 10 "\nevalf(%);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 205 "> " 
0 "" {MPLTEXT 1 207 0 "" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" 
}}{PARA 203 "" 0 "" {TEXT 205 32 "Let's compute the first 10 zero:" }
{TEXT 205 0 "" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 11 "for i t
o 10" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 5 "\n  do" }{MPLTEXT 1 207 
0 "" }{MPLTEXT 1 207 38 "\n     k[i]:= evalf(BesselJZeros(0,i));" }
{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 6 "\n  od;" }{MPLTEXT 1 207 0 "" }
}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}}{SECT 0 {PARA 
202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 25 "Hence, wi
th eigenvalues  " }{XPPEDIT 2 0 "lambda[n] = omega[n]^2" "6#/&%'lambda
G6#%\"nG*$&%&omegaG6#F'\"\"#" }{TEXT 205 42 " we have the correspondin
g eigenfunctions " }{XPPEDIT 2 0 "R[n](r)" "6#-&%\"RG6#%\"nG6#%\"rG" }
{TEXT 205 9 " given by" }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 
0 "" }}{PARA 204 "" 0 "" {XPPEDIT 2 0 "R[n](r) =  J[0](k[n]*r)" "6#/-&
%\"RG6#%\"nG6#%\"rG-&%\"JG6#\"\"!6#*&&%\"kG6#F(\"\"\"F*F5" }{TEXT 205 
2 "  " }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 6 "where " }
{XPPEDIT 2 0 "k[n]" "6#&%\"kG6#%\"nG" }{TEXT 205 65 " , n = 1, 2, 3, .
.. is the nth positive solution of the equation " }{XPPEDIT 2 0 "J[0](
z) = 0" "6#/-&%\"JG6#\"\"!6#%\"zGF(" }{TEXT 205 52 ".  The first 10 ro
ots are listed above to 10 digits." }{TEXT 205 0 "" }}{PARA 203 "" 0 "
" {TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 
203 "" 0 "" {TEXT 205 35 "The T(t) equation now take the form" }{TEXT 
205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 "" 
{XPPEDIT 2 0 "(Diff(T[n](t), t, t))+k[n]^2*c^2*T[n](t) = 0" "6#/,&-%%D
iffG6%-&%\"TG6#%\"nG6#%\"tGF.F.\"\"\"*(&%\"kG6#F,\"\"#%\"cGF4-&F*6#F,6
#F.F/F/\"\"!" }{TEXT 205 4 "    " }{TEXT 205 0 "" }}{PARA 204 "" 0 "" 
{TEXT 205 0 "" }}{PARA 204 "" 0 "" {XPPEDIT 2 0 "Diff(T[n](t),t)[t=0] \+
= 0" "6#/&-%%DiffG6$-&%\"TG6#%\"nG6#%\"tGF.6#/F.\"\"!F1" }{TEXT 205 0 
"" }}{PARA 204 "" 0 "" {TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 
"" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" 
{TEXT 205 46 "The general solution of the ODE is well-known:" }{TEXT 
205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 
205 1 " " }{XPPEDIT 2 0 "T[n](t) = A* sin(k[n]*c*t)   +  B* cos(k[n]*c
*t)" "6#/-&%\"TG6#%\"nG6#%\"tG,&*&%\"AG\"\"\"-%$sinG6#*(&%\"kG6#F(F.%
\"cGF.F*F.F.F.*&%\"BGF.-%$cosG6#*(&F46#F(F.F6F.F*F.F.F." }{TEXT 205 5 
"     " }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{EXCHG 
{PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}}{SECT 0 {PARA 202 "" 0 "" 
{TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 49 "Applying the derivati
ve initial condition we find" }{TEXT 205 0 "" }}{EXCHG {PARA 205 "> " 
0 "" {MPLTEXT 1 207 41 " Diff(A*sin(k[n]*c*t)+B*cos(k[n]*c*t),t);" }
{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 48 "E
qn_1:= diff(A*sin(k[n]*c*t)+B*cos(k[n]*c*t),t);" }{MPLTEXT 1 207 0 "" 
}}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 25 "Eqn_2:=eval(Eqn_1,t=0
)=0;" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 
207 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 205 11 "Since each " }
{XPPEDIT 2 0 "k[n] > 0" "6#2\"\"!&%\"kG6#%\"nG" }{TEXT 205 32 " and c \+
> 0 the only solution is " }{TEXT 214 5 "A = 0" }{TEXT 215 2 ". " }
{TEXT 205 5 " The " }{XPPEDIT 2 0 "T[n](t)" "6#-&%\"TG6#%\"nG6#%\"tG" 
}{TEXT 205 21 " solution reduces to " }{TEXT 205 0 "" }}}{PARA 203 "" 
0 "" {TEXT 205 0 "" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }
}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 
205 2 "  " }{XPPEDIT 2 0 "T[n](t) = B[n]*cos(k[n]*c*t)" "6#/-&%\"TG6#%
\"nG6#%\"tG*&&%\"BG6#F(\"\"\"-%$cosG6#*(&%\"kG6#F(F/%\"cGF/F*F/F/" }
{TEXT 205 2 "  " }{TEXT 205 0 "" }}{EXCHG {PARA 205 "> " 0 "" 
{MPLTEXT 1 207 0 "" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}
{PARA 203 "" 0 "" {TEXT 205 36 "Now we can reconstruct the solution " 
}{XPPEDIT 2 0 "u(r,t) = R(r)*T(t)" "6#/-%\"uG6$%\"rG%\"tG*&-%\"RG6#F'
\"\"\"-%\"TG6#F(F-" }{TEXT 205 58 ".  For each positive integer n = 1,
 2, 3, ... the function" }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 
205 0 "" }}{PARA 204 "" 0 "" {XPPEDIT 2 0 "u[n](r,t)  =  c[n]*cos(k[n]
*c*t)*BesselJ(0,k[n]*r)" "6#/-&%\"uG6#%\"nG6$%\"rG%\"tG*(&%\"cG6#F(\"
\"\"-%$cosG6#*(&%\"kG6#F(F0F.F0F+F0F0-%(BesselJG6$\"\"!*&&F66#F(F0F*F0
F0" }{TEXT 205 2 "  " }{TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "" 
{TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 35 "is a SOLUTION of the \+
wave equation " }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}
{PARA 204 "" 0 "" {TEXT 205 4 "    " }{XPPEDIT 2 0 "(diff(u(r, t), r, \+
r))+(diff(u(r, t), r))/r = (diff(u(r, t), t, t))/(c^2)" "6#/,&-%%diffG
6%-%\"uG6$%\"rG%\"tGF+F+\"\"\"*&-F&6$-F)6$F+F,F+F-F+!\"\"F-*&-F&6%-F)6
$F+F,F,F,F-*$%\"cG\"\"#F3" }{TEXT 205 4 "    " }{TEXT 205 0 "" }}}
{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 
205 35 "that satisfies the three conditions" }{TEXT 205 0 "" }}{PARA 
203 "" 0 "" {TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 13 "        (
1)  " }{XPPEDIT 2 0 "u[n](r,t)" "6#-&%\"uG6#%\"nG6$%\"rG%\"tG" }{TEXT 
205 15 " is bounded as " }{XPPEDIT 2 0 "r -> 0" "6#f*6#%\"rG7\"6$%)ope
ratorG%&arrowG6\"\"\"!F*F*F*" }{TEXT 205 3 " +." }{TEXT 205 0 "" }}
{PARA 203 "" 0 "" {TEXT 205 13 "        (2)  " }{XPPEDIT 2 0 "u[n](1,t
) = 0" "6#/-&%\"uG6#%\"nG6$\"\"\"%\"tG\"\"!" }{TEXT 205 6 "  for " }
{XPPEDIT 2 0 "t >= 0" "6#1\"\"!%\"tG" }{TEXT 205 2 " ." }{TEXT 205 0 "
" }}{PARA 203 "" 0 "" {TEXT 205 13 "        (3)  " }{XPPEDIT 2 0 "Diff
(u[n](r,t),t)" "6#-%%DiffG6$-&%\"uG6#%\"nG6$%\"rG%\"tGF-" }{TEXT 205 
23 " evaluated at t=0 is 0." }{TEXT 205 0 "" }}{PARA 203 "" 0 "" 
{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 279 "The fundamental mode
s of vibration lead to standing waves since the each have a single sha
pe, J[0](k[n]*r), but with a variable amplitude given by the cosine fu
nction.  These modes and the nodes that do not move are illustrated in
 the worksheet and video \"CircularStandingWaves\"." }{TEXT 205 0 "" }
}{PARA 203 "" 0 "" {TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 
204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 38 "To satisfy the initial shap
e condition" }{TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 205 34 "u(r,0) \+
 =  f(r)   ,     0 < r <  1" }{TEXT 205 0 "" }}{PARA 204 "" 0 "" 
{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 59 "we form a formal infi
nite sum of the fundamental solutions " }{XPPEDIT 2 0 "u[n](r,t)" "6#-
&%\"uG6#%\"nG6$%\"rG%\"tG" }{TEXT 205 3 " . " }{TEXT 205 0 "" }}}
{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{EXCHG {PARA 205 "> " 0 "" 
{MPLTEXT 1 207 83 "EigenfunctionExpansion := Sum(c[n]*cos(k[n]*c*t)*Be
sselJ(0, k[n]*r),n=1..infinity);" }{MPLTEXT 1 207 0 "" }}}{EXCHG 
{PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}}{SECT 0 {PARA 202 "" 0 "" 
{TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 38 "Applying the initial \+
condition we find" }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" 
}}{PARA 204 "" 0 "" {XPPEDIT 2 0 "f(r)  =  Sum(c[n]*BesselJ(0,k[n]*r),
 n = 1..infinity)" "6#/-%\"fG6#%\"rG-%$SumG6$*&&%\"cG6#%\"nG\"\"\"-%(B
esselJG6$\"\"!*&&%\"kG6#F/F0F'F0F0/F/;F0%)infinityG" }{TEXT 205 2 "  \+
" }{TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 
203 "" 0 "" {TEXT 205 10 "This is a " }{TEXT 210 21 "Fourier-Bessel se
ries" }{TEXT 205 18 " for the function " }{TEXT 210 4 "f(r)" }{TEXT 
205 61 ".  The general formula for the coefficients are given by the \+
" }{TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 
"" {TEXT 205 2 "  " }{XPPEDIT 2 0 "c[n]  =  (2/J[1](k[n])^2)*Int(r*f(r
)*J[0](k[n]*r) , r=0..1)" "6#/&%\"cG6#%\"nG*(\"\"#\"\"\"*$-&%\"JG6#F*6
#&%\"kG6#F'F)!\"\"-%$IntG6$*(%\"rGF*-%\"fG6#F9F*-&F.6#\"\"!6#*&&F26#F'
F*F9F*F*/F9;F@F*F*" }{TEXT 205 4 "    " }{TEXT 205 0 "" }}{PARA 203 "
" 0 "" {TEXT 205 42 "This result is discussed in lecture class." }
{TEXT 205 0 "" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{PARA 
204 "" 0 "" {TEXT 205 7 "SUMMARY" }{TEXT 205 0 "" }}{PARA 204 "" 0 "" 
{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 203 33 "The solution of the W
ave Equation" }{TEXT 203 0 "" }}{PARA 201 "" 0 "" {TEXT 203 2 "  " }
{XPPEDIT 2 0 "diff(u(r,t),r,r) + (1/r)*diff(u(r,t),r) = (1/c^2)*diff(u
(r,t),t,t)" "6#/,&-%%diffG6%-%\"uG6$%\"rG%\"tGF+F+\"\"\"*(F-F-F+!\"\"-
F&6$-F)6$F+F,F+F-F-*(F-F-*$%\"cG\"\"#F/-F&6%-F)6$F+F,F,F,F-" }{TEXT 
203 6 "      " }{TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 0 "" }}
{PARA 201 "" 0 "" {TEXT 203 28 "0 < r < 1      and     t > 0" }{TEXT 
203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 
203 19 "that satisfies the " }{TEXT 202 19 "Boundary Conditions" }
{TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 0 "" }}{PARA 200 "" 0 "" 
{TEXT 203 85 "(1)  The condition \"rigidly fixed\" at the boundary imp
lies       u(1,t) = 0    t > 0." }{TEXT 203 0 "" }}{PARA 200 "" 0 "" 
{TEXT 203 93 "(2)  Since r = 0 is a singular point of the PDE, we requ
ire3:     u(r,t) finite as  r ---> 0+" }{TEXT 203 0 "" }}{PARA 200 "" 
0 "" {TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 23 "and also satisfi
es the " }{TEXT 202 18 "Initial Conditions" }{TEXT 203 0 "" }}{PARA 
200 "" 0 "" {TEXT 203 0 "" }}{PARA 200 "" 0 "" {TEXT 203 5 "(1)  " }
{XPPEDIT 2 0 "u(r,0) = f(r)" "6#/-%\"uG6$%\"rG\"\"!-%\"fG6#F'" }{TEXT 
203 32 "   ,  0 < r < 1  (initial shape)" }{TEXT 203 0 "" }}{PARA 200 
"" 0 "" {TEXT 203 5 "(2)  " }{XPPEDIT 2 0 "u[t](r, 0) = 0" "6#/-&%\"uG
6#%\"tG6$%\"rG\"\"!F+" }{TEXT 203 55 "   ,  0 < r < 1  (zero initial v
elocity distribution)  " }{TEXT 203 0 "" }}{PARA 204 "" 0 "" {TEXT 
205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 11 "is given by" }{TEXT 205 0 "
" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{EXCHG {PARA 205 "> " 0 "" 
{MPLTEXT 1 207 83 "EigenfunctionExpansion := Sum(c[n]*cos(k[n]*c*t)*Be
sselJ(0, k[n]*r),n=1..infinity);" }{MPLTEXT 1 207 0 "" }}}{EXCHG 
{PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{PARA 203 "" 0 "" {TEXT 
205 6 "where " }{XPPEDIT 2 0 "k[n]" "6#&%\"kG6#%\"nG" }{TEXT 205 29 " \+
is the nth positive root of " }{XPPEDIT 2 0 "J[0](z)=0" "6#/-&%\"JG6#
\"\"!6#%\"zGF(" }{TEXT 205 24 " , and the coefficients " }{XPPEDIT 2 
0 "c[n]" "6#&%\"cG6#%\"nG" }{TEXT 205 14 " are given by " }{TEXT 205 
0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 205 
3 "   " }{XPPEDIT 2 0 "c[n]  =  (2/J[1](k[n])^2)*Int(r*f(r)*J[0](k[n]*
r) , r=0..1)" "6#/&%\"cG6#%\"nG*(\"\"#\"\"\"*$-&%\"JG6#F*6#&%\"kG6#F'F
)!\"\"-%$IntG6$*(%\"rGF*-%\"fG6#F9F*-&F.6#\"\"!6#*&&F26#F'F*F9F*F*/F9;
F@F*F*" }{TEXT 205 8 "        " }{TEXT 205 0 "" }}}{SECT 0 {PARA 202 "
" 0 "" {TEXT 204 0 "" }}{PARA 203 "" 0 "" {TEXT 205 81 "Let us specify
 an initial shape and work out the first ten terms in the series.  " }
{TEXT 205 0 "" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 48 "f:= pie
cewise(0<r and r< 1/2,1,1/2<r and r<1,0);" }{MPLTEXT 1 207 0 "" }}}
{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 59 "plot3d([r*cos(theta),r*s
in(theta),f],r=0..1,theta=0..2*Pi);" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 
207 1 "\n" }}}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 203 "" 0 "" 
{TEXT 205 0 "" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}}
{SECT 0 {PARA 202 "" 0 "" {TEXT 204 0 "" }}{EXCHG {PARA 205 "> " 0 "" 
{MPLTEXT 1 207 11 "for i to 10" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 
6 "\n   do" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 70 "\n      b[i]:= (2
/BesselJ(1,k[i])^2)*int(r*f*BesselJ(0,k[i]*r),r=0..1);" }{MPLTEXT 1 
207 0 "" }{MPLTEXT 1 207 7 "\n   od;" }{MPLTEXT 1 207 0 "" }}}{EXCHG 
{PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{PARA 203 "" 0 "" {TEXT 
205 0 "" }}{PARA 203 "" 0 "" {TEXT 205 0 "" }}{PARA 203 "" 0 "" {TEXT 
205 0 "" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}}{SECT 0 
{PARA 202 "" 0 "" {TEXT 204 0 "" }}{EXCHG {PARA 205 "> " 0 "" 
{MPLTEXT 1 207 7 "c:=0.1;" }{MPLTEXT 1 207 0 "" }}}{PARA 203 "" 0 "" 
{TEXT 205 0 "" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 77 "Eigenfu
nctionExpansion := Sum(b[n]*cos(k[n]*c*t)*BesselJ(0, k[n]*r),n=1..10);
" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 78 "\nEigenfunctionExpansion :=
 sum(b[n]*cos(k[n]*c*t)*BesselJ(0, k[n]*r),n=1..10);" }{MPLTEXT 1 207 
0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 12 "with(plots):" }
{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 102 "\nanimate3d([r*cos(theta),r*s
in(theta),EigenfunctionExpansion],r=0..1,theta=0..2*Pi,t=0..40,frames=
40);" }{MPLTEXT 1 207 0 "" }}}{PARA 203 "" 0 "" {TEXT 205 0 "" }}
{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}}{PARA 208 "" 0 "" 
{TEXT 216 0 "" }}{PARA 208 "" 0 "" {TEXT 216 0 "" }}{PARA 208 "" 0 "" 
{TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 2 1 1805 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }