Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1901306 | Wave Motion | 2010 | 13 Pages |
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 Ï in a unidirectionally periodic medium, for which case the monodromy matrix M(Ï) implies a propagator of the wave field over a period. The main interest to the matrix logarithm lnM(Ï) is owing to the fact that it yields the 'effective' matrix Qeff(Ï) of the dynamic-homogenization method. For the typical case of a unimodular matrix M(Ï)(detM=1), it is established that the components of lnM(Ï) diverge as (Ï-Ï0)-1/2 with ÏâÏ0, where Ï0 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(Ï) and of its diverging asymptotics at ÏâÏ0 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(Ï) 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(Ï) near the point Ï=0 is also elucidated.
Related Topics
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Authors
A.L. Shuvalov, A.A. Kutsenko, A.N. Norris,