So, I have the general wave equation \begin{equation} c^2u_{xx}=u_{tt} \end{equation}
with given I.C. :
$u(x,0)=g(x)$ and $u_t(x,0)=h(x)$
I have to use D'Alemberts formmula on the solution.
Separating variables, and equalling the two ODEs to $k^2$, I get:
The solution is for $k^2<0$
$u(x,t)=g(x)\cosh\big(\frac{k}{c^2}\big)x$
The solution for $k^2>0$:
\begin{equation} u(x,t)=\begin{cases} A\cos \frac{k}{c^2}x\cos\omega t \\ A\cos \frac{k}{c^2}x \sin\omega t\\ B\sin \frac{k}{c^2}x\cos\omega t\\ B\sin \frac{k}{c^2}x\sin\omega t \end{cases} \end{equation}
D'Alemberts formula is
\begin{equation} u(x,t)=\frac{1}{2}\bigg[g(x-ct)+g(x+ct)\bigg]+\frac{1}{2C}\int_{x-ct}^{x+ct}h(x)dx \end{equation}
How is this used on the solutions given above?
Thanks