Harmonic oscillations on non-homogeneous media

Abstract

We introduce a model to study the vibrational properties of non-homogeneous materials. In our work these materials, naturally idealized, are composed by one dimensional chains of harmonic oscillators represented by an alternating sequence of particles and springs. Despite the system has explicit solutions since it is linear, the formulas can be very complicated. We use homogeneous chains as building blocks for characterizing the whole system and the global dynamics. In particular, we determine the solution for a chain composed of two distinct homogeneous chains in terms of the original solutions for these two homogeneous chains, when uncoupled. This gives us a general procedure to deal easily with hon-homogeneous chains.