QUARTERLY OF APPLIED MATHEMATICS VOLUME LV, NUMBER 1 MARCH 1997, PAGES 23-39 DISSIPATION OF ENERGY FOR MAGNETOELASTIC WAVES IN A CONDUCTIVE MEDIUM By ELIAS ANDREOU (Department of Mathematics, University of Patras, Greece) GEORGE DASSIOS (Univ. of Patras and Institute of Chem. Eng. and High Temperature Chemical Processes, Patras, Greece) Abstract. We consider the propagation of magnetoelastic waves within a homoge- neous and isotropic elastic medium exhibiting finite electric conductivity. An appropriate physical analysis leads to a decoupling of the governing system of equations which in turn effects an irreducible factorization of the ninth-degree characteristic polynomial into a product of first, third, and fifth-degree polynomials. Regular and singular perturba- tion methods are then used to deduce asymptotic expansions of the characteristic roots which reflect the low and the high frequency dependence of the frequency on the wave number. Dyadic analysis of the spacial spectral equations brings the general solution into its canonical dyadic form. Extensive asymptotic analysis of the quadratic forms that define the kinetic, the strain, the magnetic and the dissipation energy provides the rate of dissipation of these energies as the time variable approaches infinity. The rate of dissipation obtained coincides with the corresponding rate for thermoelastic waves. Therefore, a similarity between the dissipative effects of thermal coupling and that of finite conductivity upon the propagation of elastic waves is established. 1. Introduction. Elastic wave propagation in a medium exhibiting electric conduc- tivity is by now an extended branch of continuum mechanics and the reader can consult the fundamental two-volume book by Eringen and Maugin [12] for an almost complete account of the subject. References [16, 20, 21, 22] are also very helpful. Knopoff [18] and Chadwick [6] stated the equations governing the propagation of elastic waves in a solid conductive medium where a homogeneous external magnetic field has been applied and they studied the propagation of plane waves within such a medium. See also the relative work of Banos [1]. The discussion of magnetoelastic waves for parameters that correspond to real earth data can be found in the work of Keilis-Borok and Munin [17]. Dunkin and Eringen [11] published an extensive investigation of the propagation of elas- tic waves under the influence of a magnetic field and under the influence of an electric field. If thermal effects are also taken into consideration then a much more complicated Received December 3, 1993. 1991 Mathematics Subject Classification. Primary 35B40, 35Q60, 35Q72, 73D99, 73R05. ©1997 Brown University 23