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

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 lnM(ω) is owing to the fact that it yields the 'effective' matrix Qeff(ω) of the dynamic-homogenization method. For the typical case of a unimodular matrix M(ω)(detM=1), it is established that the components of lnM(ω) diverge as (ω-ω0)-1/2 with ω→ω0, where ω0 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 Qeff(ω) and of its diverging asymptotics at ω→ω0 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 Qeff(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 Qeff(ω) near the point ω=0 is also elucidated.