The far field of a rayleigh wave in an elastic half-strip under end loads

The paper [1] gives a classification of nonstationary stress-strain states of plates and shells under impact end loads. An application of asymptotic methods permitted one [2-4] to split the nonstationary stress-strain state due to longitudinal loads of tangential and bending types into components with various variability and dynamic characteristics: one has used tangential and transverse low-frequency approximations, parabolic boundary layers in a neighborhood of the quasifront (the two-dimensional tensile wave front), high-frequency short-wave approximations, and also hyperbolic boundary layers in a small neighborhood of the extension wave front. The proof of the existence of matching domains showed that the approximations used are complete and the splitting scheme is well posed. The nonstationary stress-strain state under normal end loads has a distinguished position in the classification [1]. In this case, there is quasifront in a neighborhood of the conventional front of the surface Rayleigh wave, and the hyperbolic equations of a Timoshenko-type theory mistake this quasifront for the front of a new special shear wave. This property distinguishes this type of stress-strain states and prevents one from using the splitting scheme described above in this case. The paper [5] gives an asymptotic model aimed at the extraction of the solution in a neighborhood of the conventional front of the surface Rayleigh wave in the classical Lamb problem for an elastic half-space. In this model, one first analyzes the hyperbolic equation describing one-dimensional propagation of Rayleigh waves along the boundary of the half-plane. At the second stage, one finds the signal attenuation in the interior of the half-plane by successively solving two Neumann problems of similar type. In [6], the same asymptotic method was used to construct the far field equations for a Rayleigh wave in an infinite elastic layer. The present paper is the first to construct equations determining the solution in a neighborhood of the conventional front of surface Rayleigh waves under normal end loads by an asymptotic method. The approach developed has a general character and can be used when studying wave propagation in shells.