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

We study one-dimensional traveling wave phenomena in thermoviscous perfect gases with constant material properties. First, we summarize the known forms of the solution of a classic acoustic signaling problem based on the linearized theory. Next, we review several weakly-nonlinear models, all of which admit traveling wave solutions (TWSs) in the form of classical Taylor shocks, and note some of their features. We then consider traveling waves under the fully-nonlinear theory of gas dynamics, and derive a new third-order equation of motion valid for arbitrary Mach number values. Focusing on the special cases of (a) inviscid, thermally-conducting, and (b) viscous, non-thermally-conducting (i.e., strictly adiabatic) perfect gases, exact and/or numerical solutions of this nonlinear ordinary differential equation (ODE) are determined, asymptotic expressions presented, and critical values of the physical parameters identified. It is shown that, in addition to kinks, the fully-nonlinear theory allows for discontinuous solutions (i.e., shocks) and solutions that exhibit acceleration waves.