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, a, 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.

Poroacoustic acceleration waves in a Jordan–Darcy–Cattaneo material

“…In particular the papers of Jordan [9][10][11][12], Jordan and Fulford [13], Ciarletta and Straughan [14], Ciarletta et al [15] and Ciarletta et al [16] all address this topic paying particular attention to models of Jordan-Darcy type which originate with some form of Darcy's law. Related analyses on acoustic wave propagation in non-linear media are those of Jordan and Saccomandi [17] and Christov and Jordan [18]. The approach proposed herein is to generalise in an appropriate way the Jordan-Darcy theory.…”

Thermal effects on nonlinear acceleration waves in the Biot theory of porous media

“…We suppose the wave is moving into a region in which u X , p and θ are constants, so that u + X , p + and θ + are constants, where the jump notation [ü] =ü − −ü + is used. The idea is to differentiate equation (19), and take the jumps, and then employ the one-dimensional equivalents of equations (14), (15) and (16) together with the Hadamard relation and the equation for the jump of a product. Since the calculations are now well known we simply state the final result.…”

“…The topic of wave propagation in porous and acoustic media is one of great interest in the current research literature, see e.g. Biot [1], Brunnhuber and Jordan [2], Christov [3], Christov and Jordan [4], Christov et al [5], Ciarletta and Straughan [6][7][8], Jordan [9][10][11][12][13][14], Jordan and Puri [15], Jordan and Saccomandi [16], Jordan et al [17,18], Paoletti [19], Rossmanith and Puri [20,21], Wei and Jordan [22].…”

Acoustic traveling waves in thermoviscous perfect gases: Kinks, acceleration waves, and shocks under the Taylor–Lighthill balance

“…It should be stressed that simply because a TWS is piecewise-defined does not necessarily mean that it exhibits an acceleration or shock wave; see, e.g., Refs. [19,27,38,62].…”