The problem is to solve the PDE: $$ \frac{\partial^2 u}{\partial t^{2}} - \frac{\partial}{\partial x}\left(K(x)\frac{\partial u}{\partial x}\right) $$ subject to the following boundary conditions: $$ \begin{split} u(0,t) &= 1, \\ u(1,t) &= 0, \\ u(x,0) &= u_t(x,0) &=0 \end{split} $$ with $$ K(x)= \begin{cases} 1,& 0 \lt x \lt 1/2 \\ 2,& 1/2 \lt x \lt 1 \end{cases} $$
So far I have solved the steady state solution and got: $$ u_{ss}(x)= \begin{cases} -\frac{4x}{3} + 1, & 0 < x < 1/2 \\ -\frac{2x}{3} + 2/3, & 1/2 < x < 1 \end{cases} $$ But this is where I get stuck, I know that I need to solve for $$v(x,t) = u(x,t) - u_{ss}(x)$$The piece wise is throwing me off here, and I know that once I have $v(x,t)$ I will likely sovle it through separation of variables. Any help is much appreciated!