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

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.

On the impossibility of localization in linear thermoelasticity

In the middle of the 1990s, Green & Naghdi proposed three theories of thermoelasticity that they labelled as types I, II and II. The type II theory, which is also called thermoelasticity without energy dissipation, is conservative and the solutions cannot decay with respect to time. It is well known that, in general, in the linear theories of thermoelasticity of types I and III, the solutions decay with respect to time. In many situations this decay is at least exponential. In this paper we study whether this decay can be fast enough to guarantee the solutions to be zero in a finite time. We investigate the impossibility of the localization in time of the solutions of linear thermoelasticity for the theories of Green & Naghdi. This means that the only solution that vanishes after a finite time is the null solution. The main idea is to show the uniqueness of solutions for the backward in time problem. To be precise, for type III thermoelasticity we will prove the impossibility of localization of solutions in the case of bounded domains, and for the type I thermoelasticity in the case of exterior domains, even when the solutions can be unbounded, whether the spatial variable goes to infinity.