Abstract
We investigate linear and nonlinear poroacoustic waveforms under the Rubin–Rosenau– Gottlieb (RRG) theory of generalized continua. Working in the context of the Cauchy problem, on both the real line and the case with periodic boundary conditions, exact and asymptotic expressions are obtained. Numerical simulations are also presented, von Neumann–Richtmyer “artificial” viscosity is used to derive an exact kink-type solution to the poroacoustic piston problem, and possible experimental tests of our findings are noted. The presentation concludes with a discussion of possible follow-on investigations.
Highlights
What is known today as the “RRG theory” was put forth by Rubin et al [1] in 1995.This phenomenological-based theory of generalized continua is thought capable of modeling dispersive effects caused by the introduction of a medium’s characteristic length, which Rubin et al denote as α
This theory exhibits a number of appealing features, the two most important of which are the following: (i) it is only the pressure stress part of the Cauchy stress tensor and the specific Helmholtz free energy that are modified, but these modifications are achieved by adding perburtative terms, which must satisfy certain constraint equations, to the constitutive relations of the former and latter; and (ii), no additional boundary nor initial conditions, beyond those required to solve classically formulated problems, are needed ([1], p. 4063)
In Equation (5), which we note reduces to the corresponding equation of motion (EoM) for the (1D) BPM on setting a0 := 0, e = Up /c0 is the Mach number, where e 1 is assumed; δ = νχL/(c0 K ) is the dimensionless Darcy coefficient, where ν =√μ/$0 is the kinematic viscosity of the gas; a0, the dimensionless version of α, is given by a0 = α 2/L; we have set σ := χ/ReB, where ReB = c0 L/νis a Reynolds number, and where ν = μ/$0 ; and β(> 1) denotes the coefficient of nonlinearity [9], which in the case of a perfect gas is given by β = (γ + 1)/2
Summary
What is known today as the “RRG theory” was put forth by Rubin et al [1] in 1995. This phenomenological-based theory of generalized continua is thought capable of modeling dispersive effects caused by the introduction of a medium’s characteristic length, which Rubin et al denote as α. There is an obvious need to investigate the nature of the solutions, e.g., those of the traveling wave type, predicted by this theory in the case of multi-phase media. The aim of this communication is to carry out a preliminary investigation of RRG theory in the context of acoustic problems involving propagation in dual-phase Solid) media—dual-phase media being, the simplest case of multi-phase media Employing both analytical and numerical methodologies, we consider linear and finite-amplitude poroacoustic propagation under the RRG-based generalization of what some refer to as the Brinkman poroacoustic model (BPM) ( he does not refer to it as such, the general, multi-D, version of the BPM follows on setting C = 0 in Burmeister [3].). All traveling wave profiles shall be taken to be propagating to the right along the axis corresponding to the wave variable under consideration
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