Plane short-wave oscillations in the vicinity of the convex boundary of an elastic body: PMM vol. 40, n≗ 4, 1976, pp. 706–714

Abstract

We investigate short-wave oscillations of a plane elastic body, concentrated in the vicinity of a smooth convex boundary. We develop an asymptotic process of integrating the dynamic equations of the plane theory of elasticity. We obtain the expressions for the eigenfunctions and natural frequencies of the short-wave oscillations for free and clamped boundaries. The short-wave (high frequency) oscillations can be studied with the help of various asymptotic methods based, in particular, on the method of rays of geometrical optics. A systematic presentation of the method of rays and its development in the boundary value problems of mathematical physics are given in [1, 2]. The method is used to investigate the asymptotic behavior of the eigenfunctions and eigenvalues of the Laplace operator for the case of large eigenvalues. Use of the ray representations to describe the elastic high frequency oscillations was apparently first made in [3] in connection with the problem of reflection of a cylindrical wave from the boundary of a half-space. Authors of the later papers used the method of rays to solve various types of external problems of high frequency oscillations in elastic media. An extensive bibliography related to this problem is given in the survey [4]. There is, however, still no solution available to the problem of free high frequency oscillations of an elastic medium which fills a bounded region, when the oscillations penetrate the region to a certain depth. The general theoretical studies carried out in [5 – 7] also failed to supply sufficiently simple final results. Below we present a generalization of the asymptotic method given in [2], to the solution of the fundamental internal dynamic problems of the plane theory of elasticity for the regions with smooth convex boundaries, in the case of free steady-state oscillations. The solution is developed for short-wave oscillations concentrated in a narrow strip in the vicinity of the boundary. The strip is contained between the boundary of the body and the caustic-curve behind which (in the inward direction) the oscillations decay exponentially. It was shown in [1] that the convexity of the boundary is a necessary condition for the appearance of such oscillations.