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Wave Motion 31 (2000) 7176 Hamiltonian formulation for surface waves in a layered fluid

Summary: Wave Motion 31 (2000) 7176
Hamiltonian formulation for surface waves in a layered fluid
D. Ambrosi
Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
Received 21 July 1998; received in revised form 30 March 1999
An Hamiltonian formulation for irrotational isoentropic flow of two fluids of different density subject to the gravity force
is given. Canonical variables are the elevation of the free surface and the jump of the momentum potential density evaluated
at the interface between fluids and at the upper free boundary. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
A large class of water waves at the free surface of a fluid of constant density can be studied assuming irrotational
isoentropic flow [1,2]. The Laplace equation, holding in the interior of the fluid, must be supplemented by two
nonlinear coupled equations at the free surface: one describing the evolution in time of the free surface itself, the
other ensuring the boundary condition for the elliptic problem. In this context, the dynamics of surface waves of a
non-stratified fluid can be described in any regime of wave height or length until they do not break down.
Because of the assumption of potential velocity, it is not surprising that the dynamics of the waves is completely
determined by quantities evaluated at the boundary of the fluid, namely the elevation of the free-surface and the
velocity potential evaluated at the free surface itself. The former fixes the fluid domain, the latter provides boundary
conditions for a well posed elliptic boundary value problem of mixed type. Much less obviously, Zakharov [3]
showed that the water elevation and the potential at the free surface are canonical variables when formulating the


Source: Ambrosi, Davide - Dipartimento di Matematica, Politecnico di Torino


Collections: Mathematics