Journal of Fluid Mechanics



A new approach to high-order Boussinesq models


Y. AGNON a1, P. A. MADSEN a2 and H. A. SCHÄFFER a3
a1 Civil Engineering, Technion, Haifa 32000, Israel
a2 Department of Mathematical, Modelling Technical University of Denmark, 2800 Lyngby, Denmark
a3 Danish Hydraulic Institute, Agern Alle 5, 2970 Hørsholm, Denmark

Abstract

An infinite-order, Boussinesq-type differential equation for wave shoaling over variable bathymetry is derived. Defining three scaling parameters – nonlinearity, the dispersion parameter, and the bottom slope – the system is truncated to a finite order. Using Padé approximants the order in the dispersion parameter is effectively doubled. A derivation is made systematic by separately solving the Laplace equation in the undisturbed fluid domain and then addressing the nonlinear free-surface conditions. We show that the nonlinear interactions are faithfully captured. The shoaling and dispersion components are time independent.

(Received July 10 1998)
(Revised July 2 1999)



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