Method for calculating acoustic stresses in six-beam diffraction in layered media

The stresses arising in a layered medium as a result of an acoustic wave are investigated theoretically. In the general case, under the action of an incident elastic wave in an anisotropic layer, six waves are formed, three of which are directed to the reflection region and three of them are directed to the region of transmission. The stress-strain state of the layer is caused by the combined effect of these waves and is described by the equations of motion of a continuous medium and the generalized Hooke's law. This system of differential equations is solved with respect to the components of the displacement vector and the stress tensor in the Cartesian coordinate system in the matrix form. The components of the displacement vector and the stress tensor at two opposite boundaries of a layer of thickness di are expressed through each other by means of a sixth-order transfer matrix Ti = exp(Wi di). The calculation of this exponential is carried out using polynomials of the principal minors of the matrix Wi and does not require finding the eigenvalues of the matrix Wi. This method provides a more accurate and reliable calculation of the transfer matrix of the N-layer medium T = TNTN-1…T1 in comparison with other known algorithms. The amplitudes of the waves scattered by the anisotropic layer are expressed in terms of the elements of the transfer matrix. The distribution of acoustic stresses along the thickness of an anisotropic layer is determined by the amplitudes of the scattered waves and the elements of the corresponding transfer matrices. This method of calculating acoustic stresses is demonstrated for the incident SH-, SV- and P-type waves on the three-layer model: isotropic layer-crystal layer-isotropic layer. We present the comparison of the scattering spectra of elastic waves and the dependence of the stresses on the scattering angles for the crystalline layers of silicon and lead molybdate. The interpretation of the resonances of acoustic stresses arising in the crystalline layer due to the action of shear waves is given.