One-dimensional wave dispersion in layered media

Homogenisation theory reveals that long one-dimensional waves propagating in a medium with spatially periodic wave speed can behave like dispersive waves in an appropriately homogenised medium. However the precise time and length scales over which this dispersive behaviour 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 homogenisation theory and new light is shed on the dispersion phenomenon