ZAMM · Z. Angew. Math. Mech. 84, No. 7, 499 – 504 (2004) / DOI 10.1002/zamm.200310128 Short Communication A one-dimensional asymptotic model for higher order acoustic perturbations Mehdi Foroughi ∗ and Jens Struckmeier ∗∗ Fachbereich Mathematik, Universit¨ at Hamburg, Bundesstr.55, 20146 Hamburg, Germany Received 26 March 2003, revised 5 September 2003, accepted 27 October 2003 Published online 30 April 2004 Key words nonlinear acoustics, Euler’s equations, asymptotic expansions MSC (2000) 34E05, 35L60, 74J30 We consider a simplified acoustic model to describe nonlinear phenomena occurring in loudspeakers. The first simplification is that we restrict to the one-dimensional isentropic Euler equations in a slab, where on the right end a membrane is moving periodically with frequency ω and maximal displacement ε ≪ 1. Moreover we apply a perturbation method to the nonlinear model based on the small parameter ε, which yields linear hyperbolic first order systems coupled by nonlinear source terms of lower order. The asymptotic model is investigated numerically for two different frequencies ω. c  2004 WILEY-VCH Verlag GmbH & Co. KGaA,Weinheim 1 Introduction Modern bass-reflex loudspeakers should combine a small bass-reflex enclosure together with a high power and an excellent sound quality. The design and optimization of such moving iron loudspeakers requires the accurate prediction of the generated sound fields. During the operation the change in the enclosure volume due to the moving membrane is often no longer negligible. In such cases, the linear wave equation is no longer an appropriate mathematical model. To take into account nonlinear phenomena in the sound field one should go back to the fundamental equations from gas dynamics, like the Euler equations for the density, pressure and velocity of the enclosed gas. Nevertheless, numerical simulations for the nonlinear equations from gas dynamics, which include the moving membrane as a moving boundary, are quite difficult, because the maximal displacement of the membrane is still small compared to the typical dimension of the enclosure. In the present work we consider a simplified model from gas dynamics, namely the one-dimensional isentropic Euler equations in a slab. On the right boundary of the slab there is a moving membrane with frequency ω and a maximal displacement ε ≪ 1. Transforming this time-dependent domain to a fixed interval results in a modified nonlinear system for the density and velocity, which contains the small parameter ε. An asymptotic expansion of this system yields a hierarchy of hyperbolic systems, which are coupled via nonlinear right hand sides. We show that the second order system exactly describes the first harmonic in the sound field and is therefore appropriate to include nonlinear phenomena. Our work is connected with the description of oscillations of a gas contained in a tube closed at one end and driven by a piston located at the other end, see for example the recently published work by Cox, Mortell, and Reck [1] and the references given there; in particular, Klein and Peters [2]. 2 The acoustic model The starting point for our acoustic model are the isentropic Euler equations in one-space dimensions given by ∂ ∂t ρ + ∂ ∂x (ρu)=0 , (1) ∂ ∂t (ρu)+ ∂ ∂x ( ρu 2 ) + ∂ ∂x p =0 (2) ∗ Grant sponsor: Fraunhofer-Institute for Industrial Mathematics, Kaiserslautern, Germany. ∗∗ Corresponding author, e-mail: struckmeier@math.uni-hamburg.de c  2004 WILEY-VCH Verlag GmbH & Co. KGaA,Weinheim