I am solving the following wave problem
$$\begin{cases} u_{tt} - u_{xx} = 0, & \text{for } x \in \big( 0, \frac{1}{2} \big) \\ u_{tt} - 4u_{xx} = 0, & \text{for } x \in \big( \frac{1}{2}, 1 \big) \end{cases}$$
with initial conditions $u_0 = \sin(\pi x)$ and $v_0 = 0$ and boundary conditions $u(0,t) = 0$ and $u(1,t) = 0$.
Additional conditions are $[u] \lvert_{x=\frac{1}{2}}=0$ and $u_x \big(\frac{1}{2},t \big) = 4u_x \big( \frac{1}{2},t \big)$, where $[]$ means the jump at specified point.
I tried the separation of variable method but I couldn't find the eigenvalues explicitly and hence the constants terms are difficult to find. Is there any way one can solve the problem? Can one merge the two separate wave using the condition at $x = \frac{1}{2}$?