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\centerline{Mathematics 373 -- Applied Mathematics}
\centerline{Individual Problem Set 2 -- D'Alembert's Solution of the Wave 
Equation}
\centerline{{\it Due:} Tuesday February 6, 2001}
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\noindent
A)  Solve $u_{tt} = c^2u_{xx}$ for $x \in {\bf R}$, 
$u(x,0) = e^x$, $u_t(x,0) = \sin(x)$.  Sketch (or use Maple to plot)
the solution ($u$ versus $x$, with $t$ fixed) for $t = 0,1,2,3$.  
What are the two travelling waves that are superposed to get the solution
$u(x,t)$?
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\noindent
B)  A string originally at rest with $u(x,0) = f(x) = 0$ is struck
with a hammer at time $t=0$.  The hammer blow imparts an initial velocity 
$$u_t(x,0) = g(x) = \cases{ 1 & if $|x| \le a$\cr
                            0 & if $|x| > a$.\cr}$$              
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\item{1)} Show that the solution of $u_{tt} = c^2 u_{xx}$
with these initial conditions is 
$$u(x,t) = {1\over 2c} \int_{x-ct}^{x+ct} g(s)\ ds,$$
and explain why this says $u(x,t)$ is ${1\over 2c}$ times
the length of the interval
$(x - ct,x + ct) \cap (-a,a)$ for all $t,x$.
\item{2)}  Plot $u$ versus $x$ ($t$ fixed) for $t = {a\over 2c},
{a\over c}, {3a\over 2c}, {2a\over c}, {5a\over 2c}, {5a\over c}$,
and describe what the string is doing in words.
\item{3)}  As a function of $t$, what is the maximum displacement
$\max_{x\in{\bf R}} u(x,t)$?
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\noindent
C)  Recall that we derived D'Alembert's solution of the wave equation
by factoring the operator 
${\partial^2 \over \partial t^2} - c^2 {\partial^2 \over \partial x^2}$.
Apply the same method to solve
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\item{1)}  $u_{xx} - 3u_{xt} - 4u_{tt} = 0$, $u(x,0) = x^2$, $u_t(x,0) = e^x$.
\item{2)}  $u_{xx} + u_{xt} - 20u_{tt} = 0$, $u(x,0) = f(x)$, $u_t(x,0) = g(x)$,
$f,g$ arbitrary.
\bye