I am looking for a solution to a wave equation
$\frac{\partial^2 u}{\partial \tau^2} = \frac{\partial^2 u}{\partial \xi^2}$
in which $t_c\tau = t$, $L\xi = x$,
and $t_c = L/v_c$ is the characteristic time,
$L$ is the sample thickness,
and $v_c$ is the characteristic wave speed,
with an IC of
$\left [\frac{\partial u}{\partial \tau} \right]_{x,t=0} = \theta \left (x, t=0 \right)$
and a BC of
$\left [\frac{\partial u}{\partial \xi} \right]_{x=0,t} = \phi \left (x=0, t \right)$
I have tried the D' Alembert solution, but I get a function $u\left(\xi, \tau \right)$ that is a function of the integral of phi which I don't know since it is not analytic, and it also introduces two new unknowns, $f\left (\tau_0 \right)$ and $g\left (\tau_0 \right)$ and I'm actually trying to find $\frac{\partial u}{\partial \tau}$ and $\frac{\partial u}{\partial \xi}$ not u.
I haven't tried separation of variables, Sturm-Liouville or Fourier transform yet.
This system is similar to Cauchy-Riemann equations.