Modeling weakly nonlinear acoustic wave propagation

Three weakly nonlinear models of lossless, compressible fluidflow—a straightforward weakly nonlinear equation (WNE),the inviscid Kuznetsov equation (IKE) and the Lighthill–Westerveltequation (LWE)—are derived from first principles and theirrelationship to each other is established. Through a numericalstudy of the blow-up of acceleration waves, the weakly nonlinearequations are compared to the ‘exact’ Euler equations,and the ranges of applicability of the approximate models areassessed. By reformulating these equations as hyperbolic systemsof conservation laws, we are able to employ a Godunov-type finite-differencescheme to obtain numerical solutions of the approximate modelsfor times beyond the instant of blow-up (that is, shock formation),for both density and velocity boundary conditions. Our studyreveals that the straightforward WNE gives the best results,followed by the IKE, with the LWE's performance being the poorestoverall.