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Modified Boussinesq equations and associated parabolic models for water wave propagation

Published online by Cambridge University Press:  26 April 2006

Yongze Chen  and
Philip L.-F. Liu
Yongze Chen
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA Present address: Center For Coastal Studies, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093-0209, USA.
Philip L.-F. Liu
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA
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Abstract

The modified Boussinesq equations given by Nwogu (1993a) are rederived in terms of a velocity potential on an arbitrary elevation and the free surface displacement. The optimal elevation where the velocity potential should be evaluated is determined by comparing the dispersion and shoaling properties of the linearized modified Boussinesq equations with those given by the linear Stokes theory over a range of depths from zero to one half of the equivalent deep-water wavelength. For regular waves consisting of a finite number of harmonics and propagating over a slowly varying topography, the governing equations for velocity potentials of each harmonic are a set of weakly nonlinear coupled fourth-order elliptic equations with variable coefficients. The parabolic approximation is applied to these coupled fourth-order elliptic equations for the first time. A small-angle parabolic model is developed for waves propagating primarily in a dominant direction. The pseudospectral Fourier method is employed to derive an angular-spectrum parabolic model for multi-directional wave propagation. The small-angle model is examined by comparing numerical results with Whalin's (1971) experimental data. The angular-spectrum model is tested by comparing numerical results with the refraction theory of cnoidal waves (Skovgaard & Petersen 1977) and is used to study the effect of the directed wave angle on the oblique interaction of two identical cnoidal wavetrains in shallow water.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 288 , 10 April 1995 , pp. 351 - 381
Copyright
© 1995 Cambridge University Press

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