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On the evolution of packets of long surface waves

Three dimensional inviscid nonlinear waves on the surface of water of finite depth are examined in the limit of long waves. It is shown that small amplitude waves having a suitably slow variation in the direction transverse to that of propagation satisfy a two dimensional analogue of the well known Korteweg-de Vries equation when the parameter Δ =ε /h2k2 is finite; where ε is an amplitude parameter, h is the depth and k is the wavenumber. When Δ is small this analogue is reduced, to first approximation, to a scaled form of the nonlinear Schrödinger-Poisson type equations adumbrated by Davey & Stewartson (1974).