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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on September 18, 2007
The Quarterly Journal of Mechanics and Applied Mathematics 2007 60(4):473-495; doi:10.1093/qjmam/hbm017
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Q. Jl Mech. Appl. Math, Vol. 60. No. 4 © The author 2007. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Modeling weakly nonlinear acoustic wave propagation

Ivan Christov {dagger}

( Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA )

C. I. Christov

( Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, USA )

P. M. Jordan{ddagger}

( Code 7181, Naval Research Laboratory, Stennis Space Center, MS 39529-5004, USA )

{ddagger} Corresponding author < pjordan{at}nrlssc.navy.mil>

Received 16 May 2007. Revise 20 June 2007. Accepted 23 June 2007.


  Abstract

Three weakly nonlinear models of lossless, compressible fluid flow—a straightforward weakly nonlinear equation (WNE), the inviscid Kuznetsov equation (IKE) and the Lighthill–Westervelt equation (LWE)—are derived from first principles and their relationship to each other is established. Through a numerical study of the blow-up of acceleration waves, the weakly nonlinear equations are compared to the ‘exact’ Euler equations, and the ranges of applicability of the approximate models are assessed. By reformulating these equations as hyperbolic systems of conservation laws, we are able to employ a Godunov-type finite-difference scheme to obtain numerical solutions of the approximate models for times beyond the instant of blow-up (that is, shock formation), for both density and velocity boundary conditions. Our study reveals that the straightforward WNE gives the best results, followed by the IKE, with the LWE's performance being the poorest overall.

{dagger} Present address: Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208-3125, USA.


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