Method of the transfer matrix is used to describe the six-wave diffraction in layered media. The system of differential equations that determine the 6-order transfer matrix is solved by the scaling method with the aid of 6-th order symmetric polynomials. This method is applied to the elastic waves in anisotropic media. Influence of layer thickness and frequency of the wave on the scaling parameter is found. Analytic solutions describing the transfer of thermoelastic waves in isotropic media and elastic stresses in the crystalline layers of the cubic, tetragonal and orthorhombic systems are obtained.
The method of symmetric polynomials (MSP) was developed for computation analytical functions of matrices, in particular, integer powers of matrix. MSP does not require for its realization finding eigenvalues of the matrix. A new type of recurrence relations for symmetric polynomials of order n is found. Algorithm for the numerical calculation of high powers of the matrix is proposed.This computational procedure is more accurate in comparison with ordinary matrix multiplication.
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