Use Fourier's method of separation of variables to solve the problem

Question

\begin{align} & u_{tt}=c^2u_{xx}-\alpha u \\[6pt] & u(0,t)=u(L,t)=0, \ \ t>0 \\[6pt] & u(x,0)=f(x) \\[6pt] & u_t(x,0)=g(x), \ \ 0<x<L \\[6pt] & \alpha>0. \end{align}