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J Boussinesq and the standing water waves problem

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J.Boussinesq and the standing water waves problem J.Boussinesq et le probleme du clapotis G.Iooss IUF, Labo. J.Dieudonne, Parc Valrose, F-06108 Nice, Cedex02 courriel: October 30 2006 Abstract In this short note we present the original Boussinesq's contribution to the nonlinear theory of the two dimensional standing gravity water wave problem, which he defined as ”le clapotis”. Dans cette courte note on presente, en la situant dans le contexte actuel, la contribution originale de Boussinesq sur la theorie non lineaire du ”clapotis”. The two-dimensional standing gravity water wave problem has only recently been solved (see [7], [5], [6]), this is an opportunity for pointing out the sem- inal contribution of J.Boussinesq to this challenging nonlinear fluid dynamics problem. Let consider the classical 2-dimensional water wave problem where H is the depth at rest of the perfect incompressible fluid layer, the flow is assumed to be potential, and ? is the velocity potential. The free surface is y = ?(x, t), where y and x are respectively the vertical and horizontal coordinates. Then the problem is ruled by the nonlinear system ∆? = 0, x, t ? R, ?H < y < ?(x, t) ∂y? = 0, x, t ? R, y = ?H ∂t? + ∂x?∂x?? ∂y? = 0, x, t ? R, y = ?(x, t

  • wave problem

  • coordinates

  • longueur d'onde

  • standing waves

  • contribution originale de boussinesq

  • nonlinear fluid

  • fluid particles


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J.Boussinesq and the standing water waves problem J.Boussinesqetleprobl`emeduclapotis
G.Iooss IUF,Labo.J.Dieudonne´,ParcValrose,F06108Nice,Cedex02 courriel: gerard.iooss@unice.fr
October 30 2006
Abstract In this short note we present the original Boussinesq’s contribution to the nonlinear theory of the two dimensional standing gravity water wave problem, which he defined as ”le clapotis”. Danscettecourtenoteonpre´sente,enlasituantdanslecontexte actuel,lacontributionoriginaledeBoussinesqsurlathe´orienonline´aire du ”clapotis”. The twodimensional standing gravity water wave problem has only recently been solved (see [7], [5], [6]), this is an opportunity for pointing out the sem inal contribution of J.Boussinesq to this challenging nonlinear fluid dynamics problem. Let consider the classical 2dimensional water wave problem whereHis the depth at rest of the perfect incompressible fluid layer, the flow is assumed to be potential, andφThe free surface isis the velocity potential. y=η(x, t), whereyandxThenare respectively the vertical and horizontal coordinates. the problem is ruled by the nonlinear system
Δφ= 0, x, tR,H < y < η(x, t) yφ= 0t, x, R, y=H tη+xη∂xφyφ= 0t, x, R, y=η(x, t) 1 2 tφ+ (φ) += 0t, x, R, y=η(x, t) 2 wheregis the acceleration of gravity, which is the only external force acting on the system. The condition ony=Hexpresses that the velocity is tangent to the boundary at the bottom, in case of an infinite depth layer one has to replace the second equation by φ0 asy→ −∞.
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