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In this work, we obtain closed-form expressions for the transfer matrix of free oscillations in finite periodic Timoshenko–Ehrenfest beams with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for free oscillations of a beam composed of N cells, we obtain a fourth-order recursive relation for the matrix coefficients, which defines the so-called Tetranacci Polynomials. Such recursive relation provides an algorithm to compute the Nth power of the transfer matrix, avoiding the matrix product of the N matrices. Furthermore, in the symmetric case of free oscillations in finite periodic Timoshenko-Ehrenfest beams, closed-form expressions for the solutions to the recursive relation have recently been derived, which we use to write the transfer matrix of a finite beam composed of N cells in a closed form. We find a good agreement between the natural frequencies calculated with the obtained expressions and finite element simulations. These expressions are very useful for studying the interaction of evanescent oscillations and find application, for instance, in the development of phononic topological insulators. Our formalism can be applied to waves propagating in finite periodic layers described by a 4 × 4-transfer matrix, such as electromagnetic waves in anisotropic optical media.

In this work, we obtain closed expressions for the transfer matrix and the transmittance of electromagnetic waves propagating in finite one-dimensional anisotropic periodic stratified media with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for the electromagnetic field in a periodic stratified media formed by N cells, we obtain a fourth-order recursive relation for the matrix coefficients that defines the so-called Tetranacci polynomials (TPs). In the symmetric case, corresponding to a unit-cell transfer matrix with a characteristic polynomial where the coefficients of the linear and cubic terms are equal, closed expressions for the solutions to the recursive relation, known as symmetric TPs, have recently been derived, allowing us to write the transfer matrix and transmittance in a closed form. We show as sufficient conditions that the 4 × 4 differential propagation matrix of each layer in the binary unit cell, Δ , a) has eigenvalues of the form ± p 1 , ± p 2 , with p 1 ≠ p 2 , and b) its off-diagonal 2 × 2 block matrices possess the same symmetric structure in both layers. Otherwise, the recursive relations are still solvable for any 4 × 4 matrix and provide an algorithm to compute the N th power of the transfer matrix without carrying out explicitly the matrix multiplication of N matrices. We obtain analytical expressions for the dispersion relation and transmittance, in closed form, for two finite periodic systems: the first one consists of two birefringent uniaxial media with their optical axis perpendicular to the z -axis, and the second consists of two isotropic media subject to an external magnetic field oriented along the z -axis and exhibiting the Faraday effect. Our formalism applies also to lossy media, magnetic anisotropy or optical activity. [ABSTRACT FROM AUTHOR]