How to solve wave equation with discontinuous wave speed?

Question

I would like to solve the following equation $$\frac{\partial^2 y}{\partial t^2} - c^2(x,t) \frac{\partial^2 y}{\partial x^2}=0,$$ for $y=y(x,t)$ and $c(x,t)$ is of the form $$c(x,t)=\begin{cases} c_1(t), & x < 0 \\ c_2(t), & x \ge 0. \end{cases}$$ The initial conditions are $$y(x,0)=f(x),$$ $$\left.\frac{\partial y}{\partial t}\right|_{t=0}=g(x).$$ The boundary conditions in $x$ are $$y(-l, t)=y(l,t)=0.$$ If $c^2(x,t)$ were constant in $x$ I could convert this into a set of ODEs by assuming $y$ is of the form $$y(x,t) = \sum_{n=-\infty}^\infty Y_n(t)\exp(ik_nx),$$ where $$k_n = \frac{n \pi}{2l}.$$ Is there a way to convert this into a set of ODEs? Perhaps by using the Laplace transform? Any advice for approaching this problem is appreciated.