Abstract: Waves propagating in media with periodic structures have attracted a lot of attention in recent years. These media show interesting macroscopic properties, which may be quite different from those of the individual materials constituting the stratified system. More recently, waves on fluids and gases with an underlying periodic structure have been studied. These waves present a peculiar, somehow unexpected, behaviour. For example, there is evidence that in spite of the fact that the waves are governed by a genuinely quasi-linear hyperbolic system, they do not break into a shock. Here we present several systems in which hyperbolic waves show a dispersive behaviour, and deduce approximate effective equations which explain the peculiar phenomena. In all cases, the effective equations are obtained by asymptotic expansion of the solution in a small parameter representing the ratio between the period of the structure and a typical wavelength.

We start by considering 1D shallow water system with periodic bathymetry. The detailed numerical solutions of the system are compared with the ones satisfied by the effective equations at various orders in the small parameter. Traveling wave solutions of the dispersive models are also computed and compared with the traveling waves emerging from the shallow water system. As a second example we consider 2D shallow water, with waves propagating in the x-direction, while the bathymetry is periodic in the y-direction. The last example concerns the Euler equations in gas dynamics. The stationary background is a state with constant pressure, zero velocity, and a periodic variation in the density. Open problems and related work in progress will be mentioned.

Contact: Peter Miller