Available online at www.sciencedirect.com International Journal of Non-Linear Mechanics 38 (2003) 1545–1552 Linear interactions aecting the propogation of waves in a linear elastic uid E. Momoniat * School of Computational and Applied Mathematics, Centre for Dierential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa Received 18 April 2001; accepted 26 June 2002 Abstract Small linear interactions aecting the propogation of waves in a linear elastic uid are investigated. These linear interactions may occur as a result of impurities on the surface of a linear elastic uid. These interactions are imposed on the linear wave equations which were investigated in Momoniat (Propogation of waves in a linear elastic uid, submitted for publication) using the non-classical contact symmetry method. The occurrence of a small parameter in the wave equations under consideration in this paper makes the problem ideal for analysis using an approximate non-classical contact symmetry method. Approximate contact symmetries and approximate solutions are determined and discussed for the problems under consideration. Comparisons are made with the case of no interaction. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Linear interaction; Wave propogation; Approximate non-classical contact symmetry 1. Introduction Wave propogation in linear and non-linear uids and solids are discussed in [1]. The propogation of waves in a linear elastic uid in particular, were in- vestigated in [2] using the non-classical contact sym- metry method. In this paper we investigate the eect of a simple linear interaction on the propogation of waves in a linear elastic uid. This interaction may be due to impurities on the surface of the uid. Alterna- tively, if an experiment were being conducted, interac- tionswithanon-smoothsurfacemaycausetheselinear ∗ Tel.: +27-11-716-3905; fax: +27-11-403-9317. E-mail address: ebrahim@cam.wits.ac.za (E. Momoniat). interactions. We consider these interactions on two of the wave equations derived in [2]. The rst linear wave equation under consideration is given by u tt - E 3 (t ) 2 f(x) f ′′ (x) u xx = E 4 (t )f(x)+ K (t; x; u; u t ;u x ) (1.1) (subscripts denote dierentiation unless otherwise indicated) where 1 and K is an arbitrary func- tion of t; x; u; u t and u x and f ′ (x)=df= d x; f ′′ (x)= d 2 f= d x 2 ;::: : The interaction term K is assumed to depend on up to rst derivatives of the dependent variable u. The solution of the unperturbed equation from (1.1) is given by (see [2]) u(x; t )= E 5 (t )f(x); (1.2) 0020-7462/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII:S0020-7462(02)00118-X