Causality, Stokes’ wave equation, and acoustic pulse propagation in a viscous fluid

Abstract

Stokes' acoustic wave equation is solved for the impulse response of an isotropic viscous fluid. Two exact integral forms of solution are derived, both of which are causal, predicting a zero response before the source is activated at time t = 0. Moreover, both integral solutions satisfy a stronger causality condition: the pressure pulse is maximally flat, with all its time derivatives identically zero at t = 0, signifying that there is no instantaneous response to the source anywhere in the fluid. A closed-form approximation for each of the two integrals is derived, with distinctly different properties in the two cases, even though the original integrals are equivalent in that they predict identical pulse shapes. One of these approximations, reminiscent of transient solutions that have appeared previously in the literature, is noncausal due to the incorrect representation of high-frequency components in the propagating pulse. In the second approximation, all frequency components are treated correctly, leading to an impulse response that satisfies the strong causality condition, also satisfied by the original integrals, whereby the predicted pressure pulse is zero when t < 0 and maximally flat everywhere in the fluid immediately after t = 0.