Diagnosis of the properties of a structurized medium by long nonlinear waves

Abstract

It was traditionally believed that long-wave processes in inhomogeneous media can be simulated within the framework of a homogeneous medium. It is known [1-3] that, at the acoustic level, the structure of a medium for long waves can be taken into account by means of disperse-dissipative properties of a homogeneous medium. At the same time, the evolution of finite-amplitude long waves is directly affected by the medium structure, as is shown by a rigorous mathematical analysis using asymptotic averaging [4-6l. This paper proves that the effect of the structure on nonlinear long-wave perturbations is so significant that one can predict the properties of the medium from the features of evolution of the wave field. 1. Asymptotic Averaged Model. Media with regular structures are elementary inhomogeneous media for which the effect of the structure can be analyzed. The mechanism of propagation of long-wave perturbations is studied for a periodic medium with equalization of stresses and mass velocities on the boundaries of adjacent components. It is assumed that the microstructural element of the medium is sufficiently large that the laws of classical continuum mechanics can be applied to it. The medium is barotropic. We consider media in a hydrodynamic approximation ignoring shear stresses. The specific volume V = p-1 and the sound velocity c are considered periodically varying properties of undisturbed media. One method of the averaged description of media of regular structure is that of asymptotic averaging [7, 8]. It is used to simulate long waves in compressible media [4]. Taking into account the well-known results for the case of plane symmetry [4-6], we derive an averaged system of equations for one-dimensional motion of arbitrary symmetry. The assumed equations of one-dimensional unsteady motion are the equations of motion for each individual component in Lagrangian variables: