Method of averaging and the generalized variational principle for nonsinusoidal waves: PMM vol. 36, n≗1, 1972. pp. 71–78

Abstract

A method of asymptotic expansion is proposed for wave processes which are close to stationary periodic waves of arbitrary form. It is shown that equations of the first approximation can be written in the form of Lagrange equations of the second kind for averaged Lagrange and Rayleigh functions. At the present time most analytical results in the theory of nonlinear wave processes are obtained with the aid of approximate methods based on the smallness of one or another parameter in the initial equations or in the boundary (initial) conditions. Wave processes in media with small nonlinearity and strong dispersion, where the solution is close to one or the superposition of several quasiharmonic waves [1–4] have been studied with relatively great completeness. Furthermore, it is well known that in many problems related to waves on the surface of a liquid, in the plasma [5], in transmission lines for electromagnetic waves [6], and also in problems of nonlinear field theory [7] the necessity arises to examine essentially nonsinusoidal waves with arbitrary relationship of nonlinearity and dispersion parameters. Here also some methods exist for obtaining approximate solutions based on local similarity of the process to a stationary traveling wave [5, 8, 9]. From the point of view of generality and physical clearness, apparently, the Hamilton's variational principle in the averaged form as proposed by Whitham [8] is of greatest interest among these methods. In this connection the equations for envelopes (of amplitude, frequency, etc.) of the quasistationary wave are obtained in the form of differential equations of Euler in the corresponding variational problem with averaged Lagrangian. Such an approach, however, is directly applicable only to strictly conservative systems for which the Lagrangian is known (the determination of the latter frequently represents quite a complicated problem [10]). Furthermore, the independent value has the structure of the scheme for asymptotic expansion which permits to obtain averaged equations in any approximation with respect to the small parameter, as it was already done for one nonlinear second order equation [11] and an arbitrary system of first order equations [12]. This paper examines processes which can be described by partial differential equations of the Lagrangian type, including in particular the Rayleigh dissipation function. The asymptotic method permits to examine processes which are locally similar to a plane stationary wave. We succeed in showing that the equations of the first approximation can be derived from the generalized Hamilton's variational principle in the averaged form.