# Solving wave equation with Fourier transform

1. Feb 7, 2012

### fluidistic

1. The problem statement, all variables and given/known data
Use Fourier transforms to calculate the motion of an infinitly large stretched string with initial conditions $u(x,0)=f(x)$ and null initial velocity. The displacements satisfy the homogeneous wave equation.

2. Relevant equations

$\frac{\partial ^2 u }{\partial t^2 }-c\frac{\partial ^2 u }{\partial x^2 }=0$.
$\mathbb{F} (u)=F(\omega )=\int _{-\infty }^{\infty }u(t)e^{i\omega t }dt$.
$\mathbb{F} \left ( \frac{d^nf}{dx^n} \right )=(-i\omega )^n \mathbb{F} (u)$.

3. The attempt at a solution
So my idea is to take the Fourier transform of the wave equation. I guess I have the choice to take it with respect to either x or t?
Taking it with respect to x, I obtain $\frac{d^2}{dt^2}\mathbb{F} (u)+\underbrace {c\omega ^2}_{\geq 0 } \mathbb{F}(u)=0$.
So that $\mathbb{F} (u)=A\cos (c\omega ^2 t )+B \sin (c\omega ^2 t )$. Now I don't really know how to proceed.
I don't know if I should take some inverse Fourier transform or use the initial conditions, namely $u(x,0)=f(x)$ and $\frac{\partial u}{\partial t } (x,0)=0$. I'm not confident so far in what I've done... could someone help me?