Distributed Example: 1-D wave equation, solution by FDA approach

Suppose we want to simulate one direction in an acoustic space in which the air is described by the second-order wave equation

$$\zbox{\frac{\partial^{2} u}{\partial^{2} t}=c^{2}\frac{\partial^{2} u}{\partial^{2} x}}$$

where $u(x,t)$ is particle velocity of the air relative to equilibrium.

• This is the familiar 1-D wave equation, with wave speed given by
  $$c=\sqrt {\frac{\gamma P_{0}}{\rho_{0}}}$$
  where
  • $\gamma=1.4$ for air (``adiabatic gas constant''),
  • $P_0$ is ambient pressure, and
  • $\rho_0$ is mass density.
• The same equation holds also for pressure $p(x,t)$ and density $\rho(x,t)$, all with the same wave speed $c$.

Let's ``digitize'' this wave equation to create a finite difference scheme (FDS).