Hamiltonian long-wave expansions for water waves over a rough bottom

This paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom. The object is to describe the character of wave propagation in a long-wave asymptotic regime, extending the results of R. Rosales & G. Papanicolaou. (1983 Stud. Appl. Math. 68, 89-102) on periodic bottoms for two-dimensional flows. We take the point of view of perturbation of a Hamiltonian system dependent on a small scaling parameter, with the starting point being Zakharov's Hamiltonian (V. E. Zakharov 1968 J. Appl. Mech. Tech. Phys. 9, 1990-1994) for the Euler equations for water waves. We consider bottom topography which is periodic in horizontal variables on a short length-scale, with the amplitude of variation being of the same order as the fluid depth. The bottom may also exhibit slow variations at the same length-scale as, or longer than, the order of the wavelength of the surface waves. We do not take up the question of random bottom variations, a topic which is considered in Rosales & Papanicolaou (1983). In the two-dimensional case of waves in a channel, we give an alternate derivation of the effective Korteweg-de Vries (KdV) equation that is obtained in Rosales & Papanicolaou (1983). In addition, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves, in cases in which the bottom possesses both short and long-scale variations. In certain cases we also obtain unidirectional equations that are similar to the KdV equation. In three dimensions we obtain effective three-dimensional long-wave equations in a Boussinesq scaling regime, and again in certain cases an effective Kadomtsev-Petviashvili (KP) system in the appropriate unidirectional limit. The computations for these results are performed in the framework of an asymptotic analysis of multiple-scale operators. In the present case this involves the Dirichlet-Neumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium.