The theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations that exhibit nonlinearity and dispersion. In this paper we derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary conditions), and of a free surface and a free interface, this latter situation occurring more commonly in experiment and in nature.From the formulation, we develop a Hamiltonian perturbation theory for the long-wave limits, and we carry out a systematic analysis of the principal longwave scaling regimes. This analysis provides a uniform treatment of the classical works of Peters and Stoker [28], Benjamin [3,4], Ono [26], and many others. Our considerations include the Boussinesq and Korteweg-de Vries (KdV) regimes over finite-depth fluids, the Benjamin-Ono regimes in the situation in which one fluid layer is infinitely deep, and the intermediate long-wave regimes. In addition, we describe a novel class of scaling regimes of the problem, in which the amplitude of the interface disturbance is of the same order as the mean fluid depth, and the characteristic small parameter corresponds to the slope of the interface.Our principal results are that we highlight the discrepancies between the case of rigid lid and of free surface upper boundary conditions, which in some circumstances can be significant. Motivated by the recent results of Choi and Camassa [6,7], we also derive novel systems of nonlinear dispersive long-wave equations in the large-amplitude, small-slope regime. Our formulation of the dynamical free-surface, free-interface problem is shown to be very effective for perturbation calculations; in addition, it holds promise as a basis for numerical simulations.

The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation.In this work, we identify a scaling regime in which the Whitham equation can be derived from the Hamiltonian theory of surface water waves. The Whitham equation is integrated numerically, and it is shown that the equation gives a close approximation of inviscid free surface dynamics as described by the Euler equations. The performance of the Whitham equation as a model for free surface dynamics is also compared to two standard free surface models: the KdV and the BBM equation. It is found that in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than both the KdV and BBM equations.

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Abstract. We prove the existence of a global bifurcation branch of 2π-periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of Hölder class. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning point and a 'highest', cusped wave. Both analytic and numerical results are compared to traveling-wave solutions of the KdV equation.