On the transient solutions of three acoustic wave equations: van Wijngaarden’s equation, Stokes’ equation and the time-dependent diffusion equation

Abstract

Acoustic wave propagation in a dispersive medium may be described by a wave equation containing one or more dissipation terms. Three such equations are examined in this article: van Wijngaarden's equation (VWE) for sound propagating through a bubbly liquid; Stokes' equation for acoustic waves in a viscous fluid; and the time-dependent diffusion equation (TDDE) for waves in the interstitial gas in a porous solid. The impulse-response solution for each of the three equations is developed and all are shown to be strictly causal, with no arrivals prior to the activation of the source. However, the VWE is nonphysical in that it predicts instantaneous arrivals, which are associated with infinitely fast, propagating Fourier components in the Green's function. Stokes' equation and the TDDE are well behaved in that they do not predict instantaneous arrivals. Two of the equations, the VWE and Stokes' equation, satisfy the Kramers-Kronig dispersion relations, while the third, the TDDE, does not satisfy Kramers-Kronig, even though its impulse-response solution is causal and physically realizable. The Kramers-Kronig relations are predicated upon the (mathematical) existence of the complex compressibility, a condition which is not satisfied by the TDDE because the Fourier transform of the complex compressibility is not square-integrable.