Suppose $0<s<1$ and $\phi,\psi:\mathbb{R}\to\mathbb{R}$ are $C^2$. Find a solution formula for the following boundary value problem for the wave equation in $\mathbb{R}^2$: \begin{align} u_{tt}&=u_{xx}, &t>sx,\\ u(x,t)&=\phi(x), &t=sx,\\ u_t(x,t)&=\psi(x),&t=sx. \end{align}
I’ve recognized this can be solved similarly to D’Alembert’s solution, but I’m not confident how to deal with the $sx$ term. I’ve attempted to solve it following Evans 2.4.1, but have not been successful.