A Bloch analysis extended to weakly disordered periodic media

This paper develops an asymptotic method to predict the dispersion curves of a \rYL{weakly disordered} medium. The \rYL{disordered} coefficients of the original acoustic equation are related to periodic coefficients through a change of variables, assumed asymptotically close to the identity. At leading order in the amplitude of the disorder, the medium is driven by an acoustic equation with periodic coefficients, which can be analyzed through a classical Floquet-Bloch approach. At first order, with simple eigenvalues, a simple post-processing of the periodic eigenvalues and eigenfunctions of the leading order allows to account for the \rYL{weakly disordered} character of the original coefficients. In the case of repeated eigenvalues (for instance, at Dirac points), a residual is introduced, whose minimization allows to recover the opening of the band gaps with the loss of periodicity. Examples in 1D and 2D illustrate the validity of our asymptotic approach, by comparing its results with eigenmodes and dispersion curves computed over much larger periods for reference.