Deviation of the solution of a quasi-linear wave equation from the solution of a linear equation in the domain of continuous first derivatives: PMM vol. 37. n≗3, 1973, pp. 434–447

Abstract

We consider one-dimensional transient wave processes which under zero initial conditions are excited by a boundary force and are described by a quasi-linear wave equation of general form. Conditions are imposed on the boundary force such that at the initial stage of the process a domain of continuous first derivatives exists. For the successive approximation of the solution of the quasi-linear equation in this domain we propose a procedure in which the solution of a linear homogeneous wave equation serves as the zeroth approximation, while the succeeding approximations are computed by integrating the inhomogeneous wave equations obtained from the original quasi-linear equation by approximating the nonlinear terms by means of the preceding approximation. We consider the application of this procedure for constructing the asymptotic approximations and we analyze the deviation of the nonlinear solution from the linear solution (the zeroth approximation) as a function of the coefficients of the quasi-linear equation and of the nature of the boundary force. As an illustration we examine geometrically and physically the transient wave processes of deformation of an elastic halfspace. We show that in the special case of an abruptly applied force, which subsequently varies sinusoidally with time, the nonlinear effects lead not only to a variation in the amplitude of the linear solution, but also to the appearance of qualitatively different high-frequency components of the solution. The approximation procedure which in the present paper has been proposed, by example of a second-order quasi-linear equation, for the construction of a solution of the travelling wave type, is related in concept, to a certain extent, to the method of perturbations [1]. We remark that the procedure of successive approximation was applied in [2] for constructing the solution of a second-order quasilinear equation in the form of an expansion in standing waves. To some extent, closely related to the present paper are the investigations in [3–5] in which the dynamic process, modelled by a quasi-linear system of equations, is described approximately as the sum of two components of which one is determined as the solution of the linear wave equation, while the other is constructed in a non-wave form by the method of perturbations.