One-Dimensional Wave Dispersion in Layered Media
H. Ockendon, J. R. Ockendon, Chris L. Farmer, David J. Allwright · SIAM Journal on Applied Mathematics · 2015
Homogenization theory reveals that long one-dimensional waves propagating in a medium with spatially periodic wave speed can behave like dispersive waves in an appropriately homogenized medium. However, the precise time and length scales over which this dispersive behavior is a good approximation are not clear. This paper describes what happens in a specific case when the sound speed and initial conditions allow the problem to be reduced to a finite system of difference equations which can be solved exactly. The long-time asymptotics are compared with the predictions of homogenization theory, and new light is shed on the dispersion phenomenon.