Low Frequency Bloch Waves for Wave Equations Whose Speed Is a Deterministic, or Randomlike, Periodic Function

This paper examines the low frequency behavior of the Bloch wave solution to wave equations whose sound speed is a three-dimensional periodic or almost-periodic function of position. The case where the sound speed is a randomlike function of position is also considered. The low frequency dependence of the Bloch wave phase velocity is obtained as a power series in the frequency. The frequency independent term is related to the average value of the inhomogeneities and is independent of the direction of propagation. The first dispersive term, which is also independent of the direction of propagation, is quadratic in frequency. Its coefficient is related to the spatial correlation of the inhomogeneities.