Waves without the wave equation: Examples from nonlinear acoustics

In a nonlinear system of first order partial differential equations the criterion for hyperbolicity, necessary for the description of wave phenomena, involves the solution. It is therefore possible that solutions may evolve in such a way that hyperbolicity is challenged in the sense that the system comes close to not being hyperbolic. We use the recently introduced formulation for nonlinear acoustic disturbances to illustrate. When hyperbolicity deteriorates, standard numerical methods and the heuristics surrounding wave motion may be compromised. To overcome such difficulties we introduce the notion of inverse characteristic which, at least in the examples, reduces numerical calculations to elementary techniques and clarifies intuition. Analysis of inverse characteristics leads to two systems of ordinary differential equations that have time-like trajectories and space-varying associated curves. Time-like trajectories give rise to an alternative measure of time in terms of which space-like trajectories are easier to analyze. Space-varying curves enable the analysis of shock phenomena in a direct way. We give conditions under which an initially mild challenge of hyperbolicity, represented by pressure, develops into a severe challenge. Under these conditions violent velocity shocks develop from an initially undisturbed state.