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.