Trace spectrum of 1D Transfer Matrices for wave propagation in layered media

The Transfer Matrix formalism is ubiquitous when considering wave propagation in various stratified media, applications ranging from Seismology to Quantum Mechanics. The relation between variables at two points in the laminate 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 phononic structures, but our derivation can be extended to other fields where the formalism applies. We present exact expressions for entries of the global propagator for N layers. 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. We show how this trace has a discrete spectrum made up of distinct 2**(N-1) harmonics (not necessarily orthogonal to each other in any sense) which we characterize exactly both in terms of the periods that they contain and their amplitudes; we also show that the phase shift among harmonics is either zero or pi. This definite appraisal of the spectrum of the trace opens the path for rational design of band gaps, going beyond parametric or sensitivity studies.