A 2×2 transfer matrix approach is used to study the elastic response of multilayered systems. Superlattices with a period of n layers are considered to calculate the dispersion relations of the normal modes for both longitudinal and transverse waves, and the reflectivity of longitudinal modes for finite and semi-infinite structures. Numerical results of the dispersion relation for a two- and three-layer period superlattice are presented to show the band structure of wave propagation. For transverse waves, it is considered that the single layer may support surface modes and it is found that their interaction with those of the adjacent layers also yield a band structure. The calculated reflectivity of longitudinal elastic waves for the semi-inifinite superlattices resembles the allowed and forbidden regions of the dispersion relations. The theoretical reflectivity curves of sound waves are compared with the experimental results for the three-layer systems. A good agreement between theory and experiment is obtained.
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Esquivel–Sirvent et al. (1994) studied this question.