THE DIAGNOSTICS OF A STRUCTURED MEDIUM BY LONG NONLINEAR WAVES: THEORETICAL JACTIFICATION

Abstract

The asymptotic averaged model is suggested for the description of the wave processes in structured heterogeneous media. The obtained integral differential system of equations cannot be reduced to the average terms (pressure, mass velocity, specific volume) and contains the terms with characteristic sizes of individual components. On the microstructure level of the medium, the dynamical behavior is governed only by the laws of thermodynamics. On the macrolevel, the motion of the medium can be described by the wave-dynamical laws for the averaged variables with the integro-differential equation of state containing the characteristics of the medium microstructure. A rigorous mathematical proof is given to show that finite amplitude long waves respond to the structure of the medium in such a way that the homogeneous medium model is insufficient for the description of the behavior of the structured medium. An important result that follows from this model is that, for a finite-amplitude wave, the medium structure (in particular, existence of microcracks) produces nonlinear effects even if the individual components of the medium are described by a linear law. Finding the wave fields in the structured medium is the direct problem, on the one hand. On the other hand, the system analyzed here is not expressed in the average hydrodynamical terms; hence the dynamical behavior of the medium cannot be modelled by a homogeneous medium even for long waves, if these waves are nonlinear. The heterogeneity of the medium structure always introduces additional nonlinearity that does not arise in a homogeneous medium. This effect enabled one to formulate the theoretical grounds of a new diagnostic method that determines the characteristics of a heterogeneous medium with the use of finite-amplitude long waves (inverse problem). This diagnostic method can also be employed to find the mass contents of individual components.