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.