Remarks on acoustic propagation in inhomogeneous fluids: Single-phase shock regularization under the Maxwell model

Abstract

The atmospheric shock problem considered in a recent article by Jordan (2020) is revisited. We show that the amplitude blow-up suffered by upward propagating acoustic shocks in an atmosphere whose ambient temperature is constant can also be eliminated by regarding air as a Maxwell fluid (i.e., a type of viscoelastic fluid), thus obviating the need for the dual-phase model assumed by Jordan (2020). Using an approach based on the Laplace transform, we then establish the existence of a range of values of the Maxwell relaxation time parameter for which the shock amplitude tends to zero and, moreover, that the upper bound of this range can be determined exactly by solving a certain quartic polynomial. And while linear, the equation of motion studied herein is shown to exhibit behaviors remarkably similar to those of certain PDEs that arise in nonlinear acoustics theory.