Method for calculating resonances of acoustic stresses at the boundaries of an anisotropic layer

The conditions of occurrence of acoustic stress resonances at the boundaries of an anisotropic layer are investigated. In general, under the action of an incident elastic wave, six elastic waves are formed in an anisotropic layer. The total effect of these waves determines the stress-strain state of the layer and is displayed in the spectra of waves scattered by the layer into the environment. The scattering spectra and acoustic stresses were modeled by solving 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. The Peano-Becker method of solving a system of differential equations by means of a matrix exponential is used. The components of the displacement vector and the stress tensor at two opposite boundaries of the layer with thickness d are expressed through each other using a sixth-order transfer matrix T = exp(W d ), where matrix W is determined by the parameters of the layer under study. The method of scaling and multiple squaring is used. According to this approach, T = (exp(W d / m )) m . A method for selecting the scaling parameter m is proposed to estimate the errors of truncation and rounding when calculating exp(W d / m ). A guaranteed accuracy and the best efficiency of calculations of all elements of the matrix exponential of the sixth order, in comparison with other known methods, is provided by the use of the method of polynomials of the principal minors of matrix W. The modeling of elastic wave scattering spectra (conversion coefficients) and stress dependences on the angles of incidence for cubic crystal layers is given using the example of indium. The interpretation of resonances of acoustic stresses arising in the crystal layer under the action of a shear wave incident on the crystal is given.