Exercise 0 ().

Additional Exercise - Wave Equation

Exercise 0 ().

Consider the wave equation

(1)

\left\{\begin{array}[]{ll}u_{tt}(x,t)=c^{2}u_{xx}(x,t),&\left(x,t\right)\in% \left(0,2\pi\right)\times\left(0,\infty\right),\\ u(0,t)=u(2\pi,t),&t\in(0,\infty),\\ u(x,0)=f(x),&t\in(0,2\pi),\\ u_{t}\left(x,0\right)=0,&x\in\left(0,2\pi\right),\end{array}\right.

where $f$ is a smooth function on $\left[0,2\pi\right]$ with $f(0)=f(2\pi)=0$ . We will find a solution to the (1) by utilising the Fourier series (like the heat equation). We assume that we can write

u(x,t)=\sum_{n\in\mathbb{Z}}a_{n}(t)e^{inx}

with

a_{n}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}u(x,t)e^{-inx}dx

and that the convergence is such that all interchanging of differentiation and integration is allowed (this will happen, for instance, if we seek a solution that is $C^{4}$ on the periodic domain).

(i)

Show that $a_{n}(t)$ satisfies the equation

$a_{n}^{\prime\prime}(t)+n^{2}c^{2}a_{n}(t)=0.$
(ii)

using he boundary conditions show that

$a_{n}(t)=A_{n}\cos\left(nct\right)$

and find an explicit expression for $\left\{A_{n}\right\}_{n\in\mathbb{Z}}$ which depends on $f$ .
(iii)

Write a solution to (1)