Sturm-Liouville problem: How to calculate the eigenfunctions of a periodic system?

Question

Consider the following Sturm-Liouville equation $$\frac{d^2 y}{d x^2}+\frac{\omega^2}{c^2(x)}y(x)=0,$$ where $$c(x)=\begin{cases} c_1, & x \le l \\ c_2, & x > l. \end{cases}$$ The system is periodic in $x$ with period $L>l$, i.e. $y(x)=y(x+L)$. The boundary conditions are given by $y(0)=y(L)$, $dy/dx|_{x=0}=dy/dx|_{x=L}$. Assume continuity of $y$ and $dy/dx$. Do you know how to calculate the eigenfunctions, $y_n$, and eigenfrequencies $\omega_n$? If $c_1=c_2=c$, then the eigenfunctions are given by $$y_n = \exp(ik_nx),$$ where $$k_n=\frac{\omega_n}{c},$$ $$\omega_n = n\pi\frac{c}{L},$$ where $n$ is an integer which can be negative.