Consider a wave equation $$\psi_{tt} - c^2 \psi_{zz} = 0$$ under the bounded domain $-z_0<z<z_0$ and an unbounded domain $t>0$, with the following boundary conditions: $\psi(z,0)=0, \psi(-z_0,t) = \psi(z_0, t)=0$ (fixed end). If I solve this, I get the following expression:
$$\psi(z,t)=\sum_{n=0}^{\infty} A_n \cos{\frac{(2n-1) \pi z}{2z_o}} \sin{\frac{(2n-1) \pi ct}{2z_o}}$$
If I got the final boundary condition that can be expressed in a Fourier series but under a different basis such that $$\psi_t(z,0) = \sum_{n=0}^{\infty} B_n \cos{\frac{n \pi z}{z_o}}$$
I can somehow get to the expression of $A_n$ in terms of $B_n$ which was what I intended if I exploit inner products well. But I have a feeling that it's going to be something that's rather complicated. I am wondering if it is just what it is, or the solution I got so far through separation of variables is wrong, or etc.