Non-homogeneous Chain of Harmonic Oscillators

We study vibrational properties of non-homogeneous materials. These are idealized materials composed by one dimensional chains of harmonic oscillators represented by an alternating sequence of particles and springs. Although the system is linear, thus possesses exact explicit solutions, the formulas for the solutions can be complicated. Homogeneous chains are used as basic building blocks for the characterization of the system global dynamics. In particular, we determine the solutions for a system composed of two distinct homogeneous chains in terms of the original solutions for these two homogeneous chains, when uncoupled. We explore different kinds of possible couplings, each with a distinct physical meaning.