Nonlinear Acoustics : what happens when the constraint is challenged?

Abstract

The Lagrangian formulation for the propagation of sonic disturbances consists of a system of rst order partial di erential equations and an inequality constraint under which the system is hyperbolic. We study the behaviour of solutions when the system is initially at rest but strained under an initial pressure eld that challenges the constraint. The result is that the challenge is transferred to the solutions, that the pressure decays, the rest state changes violently, and shock discontinuities appear in velocity as well as pressure. The main tool of investigation is the notion of inverse characteristics in which a chosen point in the time-space plane is associated with points on the initial manifold where characteristics through the given point emanate from. They also lead to the introduction of an alternative measure of time in terms of which explicit expressions for the onset of shocks are derived.