A ninth-order solution for the solitary wave

Abstract

Several solutions for the solitary wave have been attempted since the work of Boussinesq in 1871. Of the approximate solutions, most have obtained series expansions in terms of wave amplitude, these being taken as far as the third order by Grimshaw (1971). Exact integral equations for the surface profile have been obtained by Milne-Thomson (1964,1968) and Byatt-Smith (1970), and these have been solved numerically. In the present work an exact operator equation is developed for the surface profile of steady water waves. For the case of a solitary wave, a form of solution is assumed and coefficients are obtained numerically by computer to give a ninth-order solution. This gives results which agree closely with exact numerical results for the surface profile, where these are available. The ninth-order solution, together with convergence improvement techniques, is used to obtain an amplitude of 0.85for the solitary wave of greatest height and to obtain refined approximations to physical quantities associated with the solitary wave, including the surface profile, speed of the wave and the drift of fluid particles.

Simulating run-up on steep slopes with operational Boussinesq models; capabilities, spurious effects and instabilities

“…Only some of the formulations described herein possess known closed form solitary wave solutions, namely the non-linear Lagrangian forms, without optimized dispersion (see Appendix B). For full potential theory we have the numerical solution of Tanaka (1986), while a perturbation solution, namely the fourth-order solution of Fenton (1972), may be applied for small amplitudes (A/d < 0.1). A numerical solution is also employed for the standard Boussinesq equation, employed in model 2c (see Pedersen, 1988).…”

A depth semi-averaged model for coastal dynamics

“…Specifically, at the first iteration M is approximated as Q = U d and it is used to compute the forcing term and the boundary conditions of the Poisson equation (17). This provides a first approximate solution for Υ, which is used to obtain a new value of M through the definition in equation (23). Then, the procedure is repeated until the relative error between two subsequent iterations in the L ∞ -norm is below 0.01, that is:…”

Simulations numériques et expérimentales d’un soliton se propageant sur une plage inclinée

“…Des simulations ont été réalisées pour une onde solitaire se propageant dans un domaine rectangulaire ayant tout d'abord un fond plat, puis à partir de x = 0 m, une plage à pente constante de 1 : 10. Le niveau de l'eau au repos est d = 0,5 m. La condition initiale est une onde solitaire approximée au neuvième ordre (Fenton, 1972) d'amplitude a = 0,25 m, initialement centrée en x = -2,5 m. Le calcul est réalisé sur un maillage curviligne qui épouse la forme du fond. Les simulations montrent que la vague se propage sans se déformer jusqu'à la plage, où sa face avant se déforme.…”