Phase space evolution of waves in heterogeneous media

For any compact initial condition, the solution of the acoustics equation rapidly evolves in phase space onto two Riemann invariants, corresponding to the two counter-propagating waves.

The same happens for, say, the \(p\)-system if we start with some compact initial pulse. Generically, it also breaks up into two counter-propagating pulses. Each pulse corresponds to states along a single integral curve. Eventually shocks form, but if they do not interact then there is no generation of states on other integral curves.

In a heterogeneous medium, things are much more complicated. For simplicity, let’s think of the variable-coefficient \(p\)-system and piecewise constant media. At every material interface, reflections lead to generation of waves from the other characteristic family.

All plots below use the \(\sigma-u\) phase plane. Here is what the solution looks like after some time in a homogeneous medium:

This is what two stegoton wave trains look like in the phase plane:

The black lines are the homogenized integral curves: \[u = \pm \frac{2}{\bar{Z}} (1-\sqrt{\sigma+1})\] where \[\bar{Z} = \sqrt{\bar{K}\bar{\rho}}\] The blue dots are the solution values; the red dots result from averaging over each layer. Each stegoton oscillates about the corresponding homogenized integral curve in a braid-like fashion.

In a random medium, the picture is not so nice. Here we take a piecewise constant medium with \(K_A=\rho_A=1\) and \(K_{B,i}=\rho_{B_i}\) but the latter values are randomly distributed with a mean of 4. The solution at \(t=300\) looks like this:

The phase plane looks like this:

Notice that the large amplitude values are along the integral curves, but all the small amplitude incoherence is not. Here is a closeup of the area around the origin:

For a linear medium, the picture looks like this:

The red dots are averages over each period. Clearly, they lie very close to the homogenized Riemann invariant.