Harmonic decomposition of the trace of 1D transfer matrices in layered media

The Transfer Matrix formalism is ubiquitous when it comes to study wave propagation in various stratified media, applications ranging from Seismology to Quantum Mechanics. A relation between variables at two points in two different layers can be established via a matrix, termed (global) transfer matrix (product of “atomic” single-layer matrices). As a matter of convenience, we focus on 1D propagation of axial waves in rods, but our results extend to other fields where the formalism applies. When the layering corresponds to a representative repeated cell in an otherwise infinitely periodic medium, the trace of the cumulative multi-layer matrix is known to control the dispersion relation. It is proved that this trace has a discrete spectrum made up of distinct 2N−1 harmonics (not necessarily orthogonal to each other in any sense) which are exactly characterized in terms of both the period of their oscillations (each period being defined by N constituents) and their amplitudes; the phase shift among harmonics is either zero or π. This definite appraisal of the discrete spectrum of the trace opens new strategies for rational design of bandgaps, going beyond parametric or sensitivity studies.