It's been forever since I've done a course in PDE's, so I'd really appreciate a step by step process for solving this

$\displaystyle u_{tt}-\alpha^2 u_{xx} $ $\displaystyle 0<x<L, t>0$

$\displaystyle u(0,t)=u(L,t)=0 $ $\displaystyle t \geq 0$

$\displaystyle u(x,0)=f(x)$ $\displaystyle 0 \leq x \leq L$

$\displaystyle u_t(x,0)=g(x) $ $\displaystyle 0 \leq x \leq L$

leads to a solution of the form

$\displaystyle u(x,t) = \sum_{n=1}^\infty \left[a_n \cos\left(\frac{n\pi \alpha}{L}t\right)+b_n \sin\left(\frac{n\pi \alpha}{L}t\right)\right]\sin\left(\frac{n\pi x}{L}\right)$

If $\displaystyle \alpha=3$,$\displaystyle L=\pi$, $\displaystyle f(x)=6\sin(2x)+2\sin(6x)$ and $\displaystyle g(x)=11\sin(9x)-14\sin(15x)$.

I want to say the $\displaystyle a_n=0$ due to the boundary condition, but I'm not sure.

I'm also not sure what the integral's should look like when I'm trying to find the fourier coefficents of u,f and g.

Thanks in advance.