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\noindent
Math 325

\noindent
Spring, 2001

\bigskip

\begin{center}
GROUP PROJECT 2, due Wednesday, April 25
\end{center}

\bigskip

You have a choice of projects I, II or III.  If you choose a project
which involves making  \lq\lq movies," your group must make an
appointment to show me the movies.  Movies may be made with Maple
or Matlab.   In Maple use the command animate (for a two dimensional
plot) or animate3d (for a three dimensional plot).  These commands
are in the plots package, so you have to give the command with(plots)
first.  The OIT handout on Matlab probably explain how to make movies,
or you can use the Matlab helpdesk to look up movie. 
 The time variable  $t\/$
should be used as the parameter for the frames of the movies.

\bigskip
{\bf I.  Vibrating strings}

\medskip

The purpose of this project is to study the one dimensional wave
equation
in greater detail and to make movies of vibrating strings.
Part B is independent of part A.

\medskip
A.  Do pp.\ 601-603 \#13,14,17-19.  
In \#14, make a movie of  $y=\phi(x-at)\/$
and also of  $\phi(x+at)\/$, taking  $a=1\/$.
In 17(d), use the function $f$ from 17(c).


\medskip
B. For each of the following initial positions,
 make a  movie of a vibrating string
of length  $2\pi\/$ (thought of as the interval  $0\le x \le 2\pi\/$ 
when it is at rest), initial velocity  $0\/$  and the given initial position:
\begin{enumerate}
\item  $\sin x/2\/$;
\item   $\sin x\/$;
\item  $\sin 10x\/$;
\item the function 
$$
f(x) = 
\left\{
\begin{array}{cl}
x & 0\le x \le \pi
\\
2\pi-x & \pi \le x \le 2\pi
\end{array}
\right. ;
$$
\item the  function 
$$
g(x) = \left\{ 
\begin{array}{cl} 
f(2x) &0\le x\le \pi \\
          -f(2(x-\pi)) & \pi\le x\le 2\pi 
\end{array} 
\right. ; 
$$
where  $f\/$
is the function in 4;
\item several initial positions of your choice.
\end{enumerate}

\bigskip
{\bf II.  Vibrating drumheads}

\medskip
The purpose of this problem is to study the two dimensional 
wave equation in the disk  $D=\{x^2+y^2\le 1\}\/$
and to make movies of some vibrating drumheads.
The two dimensional wave equation is 
$$
a^2(u_{xx} + u_{yy})=u_{tt}
$$
where  $u=u(x,y,t)\/$ is the displacement of the point  $(x,y)\/$
at time  $t\/$.
The condition that the edge of the drumhead is fixed is expressed
as 
$$
u(x,y,t)=0 \qquad \mbox{ if }\qquad x^2+y^2=1.
$$

\medskip
A.  
{\it The wave equation in polar coordinates}:  Suppose  $u(x,y,t)\/$
solves the wave equation and  $v(r,\theta, t)=u(r\cos \theta,r\sin
\theta, t)\/$.  Show that  $v\/$
satisfies 
\begin{eqnarray}
v_{rr}+\frac 1r v_r +\frac 1{r^2}v_{\theta \theta} = a^{-2} v_{tt}.
\label{polar-wave}
\end{eqnarray}

\smallskip
B. {\it Separation of variables}:
Suppose  $v(r,\theta,t)=R(r)\Theta(\theta)T(t)\/$
solves (\ref{polar-wave}) and that  $v\/$ satisfies
the boundary condition  $v(1,\theta,t)=0\/$.
Show that 
\begin{eqnarray}
\Theta ''&+&m^2 \Theta =0 \qquad \mbox{for some } m=0,1,2,3,\dots;
\vspace{2ex}
\\
T'' &+& \lambda a^2 T=0 \qquad \mbox{for some }\lambda;
\vspace{2ex}
\\
R\,''&+& \frac 1r R\,' 
+ \left(\lambda - \frac {m^2}{r^2}\right)R=0, \qquad R(1)=0.
\label{bessel}
\end{eqnarray}
{\bf Hint}:   $\Theta\/$
must be periodic with period  $2\pi\/$.

\smallskip
C. {\it Bessel's equation}:  
Equation 
(\ref{bessel}) is Bessel's equation.  It has a regular singular
point at the origin, so you expect that there will only be one
independent solution which remains bounded as  $r\to 0\/$.  
This solution is (up to a constant): 
$$
y(r)=\sum _{n=0}^\infty\frac {(-\lambda )^n 
  r^{2n+m}}{2^{2n}n!(m+n)!}.
$$
Show that 
if  $y(1)=0\/$ 
then  $\lambda >0\/$.

\smallskip
D.  From now on, assume that  $\lambda >0\/$.  The Bessel function
 $J_m\/$
is defined by 
$$
J_m(r)=\sum _{n=0}^\infty\frac {(-1 )^n 
  r^{2n+m}}{2^{2n+m}n!(m+n)!}.
$$
Verify that   $R(r)=J_m(r)\/$
satisfies the  differential equation
$$
R\,''+\frac 1r R\,' +\left(1-\frac {m^2}{r^2}\right)
R =0.  
$$
The function  $J_m(r) \/$
is known to Maple as {\bf BesselJ}(m,r).

\smallskip
E.  Show that  $J_m(r\sqrt \lambda\,) \/$
solves (\ref{bessel}).  

\smallskip
F.  Use the results of B and E
to show that the separated solutions of the wave equation in polar
coordinates which satisfy the boundary condition  $v(1,\theta,t)=0\/$  
are of the form 
$$
v_{m,n}(r,\theta,t)=
J_m(rx^{(m)}_n)(A \cos m\theta + B\sin m\theta)(\widetilde A \cos
tx^{(m)}_n + \widetilde B \sin tx^{(m)}_n) 
$$
where  $\{x^{(m)}_n:n=1,2,...\}\/$
are the positive zeros of  $J_m\/$, listed in increasing order.  

\smallskip 
G.  Find the first two positive zeros of  $J_0\/$,  $J_1\/$ and  $J_2\/$.
({\bf Hint}: Use Maple to graph these functions.  From the graph,
find a range of  $x\/$  containing only the zero you
want to compute.
  Then use the Maple command {\bf fsolve},
giving as an option  the range for the variable,  e.g.,
fsolve(BesselJ(0,x),x,x=2..3) to find the first positive zero of $J_0(x)\/$.)    

\smallskip
H.  
For  $m=0,1,2\/$ 
and  $n=1,2\/$
make a movie of the vibrating drumheads $v_{m,n}\/$ 
with  $A=\widetilde A=1\/$,  $B=\widetilde B=0\/$.
({\bf Warning}:  The Maple option for graphing in cylindrical coordinates
graphs  $r\/$ 
as a function of  $\theta\/$ 
and  $z\/$.  This is not what you want to do.  You want to plot
  $(r\cos \theta, r\sin \theta, v_{m,n}(r,\theta,t))\/$ 
as a function of  $r\/$ 
and  $\theta\/$ 
for
fixed t.)

\bigskip

{\bf III.  The Gibbs Phenomenon}

\medskip
The purpose of this project is to study the
behavior of a Fourier series at a jump 
discontinuity.

\smallskip
A.  Calculate the Fourier series of the functions
$$
f(x) =
\left\{
\begin{array}{rr}
-1 & -\pi\le x<0
\\
1 & 0\le x \le \pi
\end{array}
\right. 
$$
and 
$$
g(x)=
x ,\qquad -\pi\le x\le \pi.
$$
Also calculate the Fourier series of
at least one other function with a jump
discontinuity.

\smallskip
B.  For each of the functions in A, use Maple to graph the function  
and the partial sums of the Fourier series with terms of index
up to  $10,20,50,100\/$.  In the graphs, you should see that 
the partial sum overshoots the function near a jump discontinuity.
Click the mouse at a peak to estimate the overshoot.  

\smallskip
C.  For the function  $f\/$
of A, you should have found that the partial sums 
of the Fourier series are 
$$
f_{2n-1}(x)=\frac 4\pi \left( \sin x + \frac{\sin 3x}3+\dots
+\frac{\sin (2n-1)x}{2n-1}\right).
$$
The overshoot of the partial sum  $f_{2n-1}\/$
at  $x=0\/$
is the  value of  $f_{2n-1}\/$
at its first positive maximum.  To find the location of this 
maximum,  first show that 
\begin{eqnarray}
f'_{2n-1}(x)=\frac 4\pi \left(\cos x + \cos 3x +\dots +\cos
(2n-1)x\right).
\label{gibbs-deriv}
\end{eqnarray}
The maximum occurs at a point where  $f'_{2n-1}(x)=0.\/$
To solve this, multiply (\ref{gibbs-deriv}) by  $\sin x\/$
and use the identity 
$$
\sin x \cos kx =\frac 12 (\sin (k+1)x-\sin (k-1)x)
$$
to show that 
$$
\pi \sin x f'_{2n-1}(x)=2\sin 2nx.
$$
Conclude that the extrema occur at 
$$
2nx =\pm \pi , \pm 2\pi, \dots, \pm (2n-1)\pi
$$
and that 
the first positive maximum occurs at 
$$
x=\frac \pi{2n}.
$$

\smallskip
D.  From C, you know where the overshoot occurs.  Now you need
to estimate its behavior as  $n\to \infty\/$.  
The overshoot is
$$
f_{2n-1}\left(\frac \pi{2n}\right)
=\frac 4\pi\left(\sin \frac \pi{2n}+\frac 13 \sin \frac {3\pi}{2n}+\dots+
\frac 1{2n-1}\sin \frac{(2n-1)\pi}{2n}\right).
$$
Show that this is an approximating sum for 
\begin{eqnarray}
\frac 2\pi\int _0^\pi
\frac {\sin x}x \, dx.
\label{gibbs-integral}
\end{eqnarray}
({\bf Hint}:  Divide the interval  $0\le x\le \pi\/$
into  $n\/$
subintervals of length  $\frac \pi n\/$, with endpoints
 $0,\frac \pi n, \frac {2\pi }n, \dots ,\frac {n\pi }n\/$.  Evaluate 
the function  $\frac 2\pi\frac {\sin x}x\/$
at the midpoints of the subintervals to form the approximating sums.) 
Conclude that the limiting value of the overshoot is
$$
\lim _{n\to \infty} f_{2n-1}\left(\frac \pi{2n}\right)=\frac 2\pi 
\int _0^\pi \frac {\sin x}x \, dx. 
$$

\smallskip
E.  Evaluate 
(\ref{gibbs-integral})
numerically (for example, by using Maple to calculate
the integral, then giving the command {\bf evalf}(\%) to 
evaluate it as a floating point number).
This gives the limiting value of the overshoot
as  $n\to \infty\/$.  Your answer should be about  $9\%\/$
of the jump made at the discontinuity. 

\smallskip
F.  Does the answer you obtained in E fit with what you 
observed in the plots in B?

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