Analytical Solutions for a Monochromatic Wave Propagating in an Elastic Layer with Gradient Properties

Abstract

The paper studies the regularities of wave propagation in an inhomogeneous elastic layer with gradient properties varying along the layer thickness. The laws of change in elastic parameters are established at which the waves propagating in the direction of parameter change are harmonic. The waves propagating in the layer with a linear law of change in the elastic parameters are noт-harmonic. Analytical solutions obtained for a monochromatic wave defined on the boundary of a plane layer with a rigidly fixed second boundary are represented by special functions. In the case of gradient variation of the layer density, the solution is determined by a superposition of the Airy functions, in the case of equal relative variations of the density and elastic moduli, they are expressed through the modified Bessel functions, and in the case of arbitrary linear laws of variation in the elastic parameters, they are expressed through the confluent hypergeometric functions. Based on the analytical solution obtained for a layer with inhomogeneous density, the conditions for extreme values of displacements and strains are determined, and their distributions over the layer thickness are calculated and analyzed.