Dispersion, Attenuation and Spatial Localization of Thermoelastic Waves in a Medium with Point Defects

Abstract

This paper deals with the study of the propagation of plane longitudinal waves in an unbounded point-defected medium located in a nonstationary inhomogeneous temperature field. The problem is considered in the self-consistent formulation making allowance for both the influence of an acoustic wave on the formation and movement of defects and the effect of defects on the propagation specificity of an acoustic wave. It is shown that with no heat diffusion, the system of equations is reduced to a nonlinear evolutionary equation, which is the formal generalization of the Korteweg–de Vries–Burgers equation (KdVB). Using the truncated expansion method, a specific solution of an evolutionary equation has been found in the form of a stationary shock wave with a monotonic decrease. It is noted that dissipative effects induced by available defects prevail over the dispersion associated with the migration of defects in the medium. The influence of the initial temperature and type of defects have been herein studied on the main parameters of a stationary wave such as velocity, amplitude and width. Three limiting cases of the evolutionary equation have been considered, and some generalizations of the known equations have been obtained, namely the equations of Korteweg–de Vries (KdV), Burgers and Riemann. The obtained equations as well as the generalized KdVB equation have solutions in the form of stationary shock waves. The propagation of a harmonic wave in a thermoelastic defective medium is herein analyzed. It is shown that the availability of defects in the medium promotes the occurrence of the frequency-dependent dissipation and dispersion. The influence of diffusion parameters and the type of defects on the harmonic wave propagation are studied.