The scattering matrix approach: A study of elastic waves propagation in one-dimensional disordered phononic crystals

Abstract

A straightforward scattering matrix method derived from the Hybrid matrix method is proposed to study band gaps of elastic waves propagating along an arbitrary direction in one-dimensional ordered and disordered phononic crystals. We show that this is a suitable alternative methodology to overcome the numerical degradation manifested by the standard transfer matrix method even in calculations where nonlossy elastic medium and/or relatively low angles of incidence are involved. Using the wave equation in the matrix Sturm-Liouville form, we show analytically that we can use the value of the determinant of the associated transfer matrix T, to check the numerical accuracy of our calculations. The localization factor concept and transmittance spectra are used to describe the band gaps. In contrast to the matrix T, the numerical stability of the proposed scattering matrix allows to obtain true transmittance spectra whose band gaps correspond to those predicted by the localization factor values for both ordered and disordered phononic crystals. Furthermore, for the numerical examples provided, the proposed method requires fewer iterations to obtain the same value of the Lyapunov exponent compared with the standard transfer matrix method.

Vibration Isolation and Noise Reduction Method Based on Phononic Crystal

“…Phononic crystals are periodic coincidence materials or structures with elastic wave band gaps and can be regarded as an extension of the crystal concept in solid state physics. In essence, the study of phononic crystals is to study elastic wave propagation in periodic inhomogeneous media [ 9 , 10 ]. Therefore, the lattice and band theory in elastic dynamics and solid state physics is the theoretical basis of phononic crystals.…”

Surface corrugated laminates as elastic grating couplers: Splitting of SV- and P-waves by selective diffraction

“…Due to its relative simplicity, TMMs are often utilised to obtain transmission and reflection coefficients from finite structures, periodic or otherwise, (in any number of dimensions); for N identical layers the resultant transfer matrix is simply raised to the N th power. As such this method has a long history, with much success in its applications to layered structures in electromagnetism 17 , solid-fluid interactions 18 , disordered systems 19 and notably elasticity [20][21][22][23] . By additionally invoking the quasi-periodic Floquet-Bloch conditions, the resultant matrix equation can be re-written as an eigenvalue problem which allows the dispersion relation to be obtained, with the Bloch wavenumber the eigenvalue for a given (nondimensional) frequency Ω, incidence angle and wave type.…”

Ubiquity of the Sturm-Liouville problem in multilayer systems

“…On the other hand, in our practice dealing with multilayered systems we have often encountered that many elementary excitations obey equations of motion having the form of a matrix Sturm-Liouville problem [11,12]. Some of these equations belong to the elasticity theory (see for example [13]), electromagnetism [14] and several other areas of classical physics. The Sturm-Liouville pattern appears also in quantum mechanics and solid state physics.…”