Compact acoustic travelling waves in a class of fluids with nonlinear material dispersion

Abstract

We apply a phenomenological theory of continua put forth by Rubin, Rosenau and Gottlieb in 1995 to an important class of compressible media. Regarding the material characteristic length coefficient, α , not as constant, but instead as a quadratic function of the velocity gradient, we carry out an in-depth analysis of one-dimensional acoustic travelling waves in inviscid, non-thermally conducting fluids. Analytical and numerical methods are employed to study the resulting waveforms, a special case of which exhibits compact support. In particular, a phase plane analysis is performed; simplified approximate/asymptotic expressions are presented; and a weakly nonlinear, KdV-like model that admits compact travelling wave solutions (TWSs), but which is not of the class K ( m , n ), is derived and analysed. Most significantly, our formulation allows for compact, pulse-type, acoustic waveforms in both gases and liquids.