Periodic solutions and multiharmonic expansions for the Westervelt equation

Abstract

<p style='text-indent:20px;'>In this paper we consider nonlinear time periodic sound propagation according to the Westervelt equation, which is a classical model of nonlinear acoustics and a second order quasilinear strongly damped wave equation exhibiting potential degeneracy. We prove existence, uniqueness and regularity of solutions with time periodic forcing and time periodic initial-end conditions, on a bounded domain with absorbing boundary conditions. In order to mathematically recover the physical phenomenon of higher harmonics, we expand the solution as a superposition of contributions at frequencies that are multiples of a fundamental excitation frequency. This multiharmonic expansion is proven to converge, in appropriate function spaces, to the periodic solution in time domain.