Divergence of logarithm of a unimodular monodromy matrix near the edges of the Brillouin zone

Abstract

A first-order ordinary differential system with a matrix of periodic coefficients Q ( y ) = Q ( y + T ) is studied in the context of time-harmonic elastic waves travelling with frequency ω in a unidirectionally periodic medium, for which case the monodromy matrix M ( ω ) implies a propagator of the wave field over a period. The main interest to the matrix logarithm ln M ( ω ) is owing to the fact that it yields the ‘effective’ matrix Q eff ( ω ) of the dynamic-homogenization method. For the typical case of a unimodular matrix M ( ω ) ( det M = 1 ) , it is established that the components of ln M ( ω ) diverge as ( ω - ω 0 ) - 1 / 2 with ω → ω 0 , where ω 0 is the set of frequencies of the passband/stopband crossovers at the edges of the first Brillouin zone. The divergence disappears for a homogeneous medium. Mathematical and physical aspects of this observation are discussed. Explicit analytical examples of Q eff ( ω ) and of its diverging asymptotics at ω → ω 0 are provided for a simple model of scalar waves in a two-component periodic structure consisting of identical bilayers or layers in spring–mass–spring contact. The case of high contrast due to stiff/soft layers or soft springs is elaborated. Special attention in this case is given to the asymptotics of Q eff ( ω ) near the first stopband that occurs at the Brillouin-zone edge at arbitrary low frequency. The link to the quasi-static asymptotics of the same Q eff ( ω ) near the point ω = 0 is also elucidated.