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

Abstract

A first-order differential system with matrix of periodic coefficients $Q(y)=Q(y+T) $ is studied for time-harmonic elastic waves in a unidirectionally periodic medium, for which the monodromy matrix $M(\omega) $ implies a propagator of the wave field over a period. The main interest in the matrix logarithm $\ln M(\omega) $ is due to the fact that it yields the 'effective' matrix $Q_{eff}(\omega) $ of the dynamic-homogenization method. For the typical case of a unimodular matrix $M(\omega)$ ($\det M=1$), it is established that the components of $\ln M(\omega) $ diverge as $(\omega -\omega_0)^{-1/2}$ with $\omega \to \omega_0,$ where $\omega_0$ 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}(\omega) $ and of its diverging asymptotics at $\omega \to \omega_0$ are provided for a model of scalar waves in a two-component periodic structure. 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 $Q_{eff}(\omega)$ 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 $Q_{eff}(\omega)$ near the point $\omega=0$ is also elucidated.