Long-wave processes in a periodic medium

Abstract

LONG-WAVE PROCESSES IN A PERIODIC MEDIUM V. A. Vakhnenko and V. V. Kulich UDC 532.59:517.19 The development of experimental techniques has shown that the internal structure of a medium affects wave motion and transport processes. Nonlinear effects start playing an in- creasing role with increasing wave amplitude. To the same class belong high-gradient, fast- flowing nonequilibrium processes. All these features must be taken into account in the math- ematical simulation of waves in real media. There exists a class of effects, in which the characteristic size considered L is much larger than the size of medium inhomogeneity E (L ~ ~). The drawback of assigning microprop- erties of an inhomogeneous medium creates major difficulties for direct solution of these problems. Numerical solutions are difficult due to high cost of computer time. One way of investigating an inhomogeneous medium is the spatial averaging method. The averaged description has an asymptotic nature. There exist different mathematical methods of asymptotic averaging of long-wave processes with detailed account of the structure. In the present study, for wave modeling in a barotropic periodic medium we use the asymptotic averaging method developed for a composite regular structure [I, 2]. Processes in an inhomo- geneous medium can be described by equations with fast-oscillating coefficients. For media of regular structure the fast-oscillating coefficients are periodic functions. The essence of asymptotic averaging consists of applying the multiple scale method [3] in conjuction with spatial averaging [4]. The method provides an asymptotically correct approximation to the solution. In the general case one obtains a system of integrodifferential equations. Some- times the problem can be reduced to the average characteristics of wave fields. At the same time, in solving the integrodifferential system one can find a numerical method in which one succeeds in selecting a step in the spatial coordinate substantially exceeding the period of the structure. i. >ystem of Averaged Equations. The asymptotic averaging method can be applied to de- scribe processes in a compressible inhomogeneous medium [5]. The equations of motion are conveniently written in Lagrangian coordinates. In these variables the structure of the periodic compressible medium is nonvarying, making it possible to use the averaging procedure. In the general case the coefficients and solutions of the equations, describing pro- cesses in inhomogeneous media, have discontinuous values. The equations must then be repre- sented in integral form. Under the assumption of smooth coefficients and solutions they are equivalent to the differential equation of motion. It was shown in [i, 6] that for insignif- icant differences in physical properties of medium inhomogeneity the approximate continuous solutions of the system of differential equations formally constructed by means of the asymp- totic method satisfy the integral conservation laws with a certain accuracy. This fact indi- cates the correctness of using a system of differential equations of motion in an asymptotic averaging method. The original equations of motion are the equations of a barotropic periodic medium, written down for each structural element [7]: