OneDWaveEquation
This module solves the one-dimensional wave equation:
\[\xi_{tt} - c^2 \xi_{xx} = - \beta \xi_t\]
where $\xi$ is the displacement associated with the wave field, $c$ is the wave speed, and $\beta$ is a damping rate.
Numerical formulation
Before solving the wave equation numerically, we 'diagonalize' the dispersion term by defining
\[\chi = u + c \xi_x \\[1ex]
\psi = u - c \xi_x\]
To derive equations for $\chi$ and $\psi$, we start with a first-order formulation of the wave equation,
\[\xi_t = u \\[1ex]
u_t = c^2 \xi_{xx} - \beta u\]
where $u$ is the velocity, or the rate of change of the displacement. We find
\[\chi_t - c \chi_x = - \beta u \\[1ex]
\psi_t + c \psi_x = - \beta u\]
Taking the Fourier transform, we have
\[\hat \chi = \hat u + \mathrm{i} c k \hat \xi \\[1ex]
\hat \psi = \hat u - \mathrm{i} c k \hat \xi \]
We define
\[\sigma \equiv c k\]
and note that
\[ \hat u = \frac{1}{2} \left ( \hat \chi + \hat \psi \right ) \\[1ex]
\hat \xi = \frac{\mathrm{i}}{2 \sigma} \left ( \hat \psi - \hat \chi \right )\]
to obtain
\[\hat \chi_t - \mathrm{i} \sigma \hat \chi = - \frac{\beta}{2} \left ( \hat \chi + \hat \psi \right ) \\[1ex]
\hat \psi_t + \mathrm{i} \sigma \hat \psi = - \frac{\beta}{2} \left ( \hat \chi + \hat \psi \right )\]
Using this form allows us to use the ETDRK4 time-stepper to solve the oscillatory part of this system exactly.