Consider small amplitude oscillations of a string fixed between $x=0$ and $x=L$. I know how to analyze this problem both in terms of eigenfunction expansion and travelling wave solutions for a homogeneous string.
But suppose the string is in-homogeneous, so that its density $\rho(x)$ obeys $\rho=\rho_1$ for $0<x<a$ and $\rho = \rho_2$ for $a<x<L$, where $a \in (0,L)$ and, in general, $\rho_1 \neq \rho_2$. We could state the problem as
$$\frac{\partial^2 u}{\partial t^2} - \frac{S}{\rho_1}\frac{\partial ^2 u}{\partial x^2} = 0, \ 0 < x < a$$ $$\frac{\partial^2 v}{\partial t^2} - \frac{S}{\rho_2}\frac{\partial ^2 v}{\partial x^2} = 0, \ a < x < L$$ $$u(0,t) = 0, v(L,t) = 0$$ $$u(a,t)=v(a,t)$$ $$u'_x(a,t) = v'_x(a,t)$$
if we assume the tension $S$ to be the same in both strings.
We could also state the problem as
$$\frac{\partial^2 u}{\partial t^2} - \frac{S}{\rho}\frac{\partial ^2 u}{\partial x^2} = 0, \ 0 < x < L$$ $$u(0,t)=u(L,t)$$ and $\rho=\rho_1$ for $0<x<a$ and $\rho = \rho_2$ for $a<x<L$.
Either way, I'm not sure how to approach this problem. For the second formulation, I thought that one could maybe consider $-\frac{1}{\rho}\frac{\partial^2}{\partial x^2}$ as a Sturm-Liouville operator on $[0,L]$ with weight function $\rho$, and expand $u$ in the corresponding eigenfunctions. However, since $\rho$ is not continous I'm not sure whether this is allowed.
Any input would be appreciated!
If you write the equation for the eigenfunctions of $$ Lf=\frac{1}{w}\left[-\frac{d}{dx}\left(p\frac{df}{dx}\right)+qf\right] $$ as $Lf=\lambda f$ and form an equivalent integral equation, you get $$ -(pf')'+qf = \lambda w f \\ -pf'= p(a)f(a)+\int_{a}^{x}(\lambda w-q)f \\ f(x)=f(a)-\int_{a}^{x}\left(p(a)f(a)+\int_{a}^{x}(\lambda w-q)fdt\right)\frac{1}{p}dx. $$ So there's really nothing preventing you from getting eigenfunctions once you have the equation in this integral form, provided $p$ does not vanish, even if $p$ has a discontinuity. What will happen is that $pf'$ will be differentiable. So I guess that's the condition you'll want to impose when dealing with the problem piecewise.