Simplified higher-order Boussinesq equations I. Linear simplifications

In this paper, we derive and test simplified higher-order Boussinesq equations, i.e., higher-order Boussinesq equations which only show lower-order terms. Simplifications are performed linearly for flat beds and slopes of O(rh). With proper coefficient choice, dispersion and shoaling properties are found to be good, while interior fluid velocities show relatively greater error at high wavenumbers. The resulting sets of equations are found to be variants of already-existing equations, which may be easily modified to improve performance. The new equations have dispersion identical to previous results but significantly improved shoaling. D 2002 Elsevier Science B.V. All rights reserved.

[1]  Philip L.-F. Liu,et al.  Modified Boussinesq equations and associated parabolic models for water wave propagation , 1995 .

[2]  Ge Wei,et al.  A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)4 , 2000, Journal of Fluid Mechanics.

[3]  O. Nwogu.  Alternative form of Boussinesq equations for nearshore wave propagation , 1993 .

[4]  James T. Kirby,et al.  Wave evolution over submerged sills: tests of a high-order Boussinesq model , 1999 .

[5]  H. Schäffer,et al.  Higher–order Boussinesq–type equations for surface gravity waves: derivation and analysis , 1998, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[6]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[7]  J. Kirby,et al.  Boussinesq-type equations with improved nonlinear performance , 2001 .

[8]  David R. Basco,et al.  Water Wave Mechanics for Engineers and Scientists , 1985 .

[9]  H. Schäffer,et al.  Further enhancements of Boussinesq-type equations , 1995 .

[10]  G. Wei,et al.  A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves , 1995, Journal of Fluid Mechanics.

[11]  J. Kirby,et al.  Improved Performance in Boussinesq-Type Equations , 2001 .

[12]  M. Nicholas,et al.  Coastal engineering. , 1969, Science.

[13]  P. A. Madsen,et al.  A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry , 1992 .

[14]  Z. Zou.  A new form of higher order Boussinesq equations , 2000 .

[15]  T. Y. Wu.  Modeling Nonlinear Dispersive Water Waves , 1999 .

[16]  P. A. Madsen,et al.  A new form of the Boussinesq equations with improved linear dispersion characteristics , 1991 .