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\title{Math 319 Assignment 2}
\author{Due Wednesday February 10}
\date{Winter 2016} % Activate to display a given date or no date
\begin{document}
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\section*{Part A}
From \S5.2 of the textbook, solve the following problems: 1(ace), 5, 8.
\section*{Part B}
\begin{enumerate}[1.]
\item
Consider the wave equation
$$
u_{tt} = c^2 u_{xx} ,
$$
on the whole line, with the initial data
$$
u(x,0) = f(x) ,
\qquad
u_t(x,0) = g(x) .
$$
Suppose that $f$ is fixed, and that we have some freedom in choosing $g$.
For instance, we know that if $g\equiv0$ then the solution $u(x,t)$ is simply the initial profile $f$ split in two halves, travelling in opposite directions with speed $c$.
However, we do not want $f$ to be split. We want the solution $u(x,t)$ to be equal to the initial profile $f$ travelling to the right with speed $c$.
What should $g$ be if this is the case?
What would the choice be if we want the solution $u(x,t)$ to be equal to the initial profile $f$ travelling to the {\em left} with speed $c$?
In each case, provide a concrete example of $f$ and $g$.
\item
Solve the problem
$$
\begin{cases}
u_{tt} = c^2 u_{xx}, \quad 0\leq x\leq L,\,-\infty