Trigonometrical Representation of Transfer Matrix for Layered Elastic Material

A harmonic wave of a fixed frequency propagates across the periodic system of elastic layers. The elementary cell consists of three layers. The transfer matrix M may be expressed by two real parameters ϕ, ψ and a set P of 64 further scalar parameters M = M(ϕ, ψ, P). The parameters are uniquely defined for the particular M and may be calculated from a system of trigonometrical equations. It has been proved numerically that, for materials and dimensions given in advance, this function for each integer n satisfies the identity [M (ϕ, ψ, P)]ⁿ = M (nϕ, nψ, P). The derived identity drastically simplifies the calculation of displacements and stresses in the periodically layered medium.