Skip to main content
SpringerLink
Log in

A Boussinesq model with alleviated nonlinearity and dispersion

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The classical Boussinesq equation is a weakly nonlinear and weakly dispersive equation, which has been widely applied to simulate wave propagation in off-coast shallow waters. A new form of the Boussinesq model for an uneven bottoms is derived in this paper. In the new model, nonlinearity is reduced without increasing the order of the highest derivative in the differential equations. Dispersion relationship of the model is improved to the order of Padé (2,2) by adjusting a parameter in the model based on the long wave approximation. Analysis of the linear dispersion, linear shoaling and nonlinearity of the present model shows that the performances in terms of nonlinearity, dispersion and shoaling of this model are improved. Numerical results obtained with the present model are in agreement with experimental data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Boussinesq M J. Théorie des ondes et des ramous qui se propagent le long d’un canal rectangularies horizontal, en communiquant au liquide contenudans ce canal des vitesses sensiblement pareilles de la surface au fond[J]. Journal de Mathématique Pures et Appliquées, Deuxieme Série, 1872, 17(1):55–108.

    Google Scholar 

  2. Peregrine D H. Long waves on a beach[J]. J Fluid Mechanics, 1967, 27(7):815–827.

    Article  MATH  Google Scholar 

  3. Madsen P A, Sorensen O R. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry[J]. Coastal Engineering, 1992, 18(3/4):183–204.

    Article  Google Scholar 

  4. Nwogu O. Alternative form of Boussinesq equations for nearshore wave propagation[J]. Journal of Waterway, Port, Coastal and Ocean Engineering, 1993, 119(6):618–638.

    Article  Google Scholar 

  5. Wei G, Kirby J T, Grilli S T et al. A fully nonlinear Boussinesq model for surface waves. Part 1, hightly nonlinear unsteady waves[J]. J Fluid Mechanics, 1995, 294(13):71–92.

    Article  MathSciNet  MATH  Google Scholar 

  6. Agnon B Y, Madsen P A, Schäffer H A. A new approach to high-order Boussinesq models[J]. J Fluid Mechanics, 1999, 399(22):319–333.

    Article  MATH  Google Scholar 

  7. Gobbi M F, Kirby J T, Wei G. A fully nonlinear Boussinesq model for surface waves. Part 2. extension to O((kh)4)[J]. J Fluid Mechanics, 2000, 405(4):181–210.

    Article  MathSciNet  MATH  Google Scholar 

  8. Madsen P A, Bingham B B, Hua Liu. A new Boussinesq method for fully nonlinear waves from shallow to deep water[J]. J Fluid Mechanics, 2002, 462(13):1–30.

    Article  MATH  Google Scholar 

  9. Madsen P A, Fuhrman D R, Wang Benlong. A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry[J]. Coastal Engineering, 2006, 53(5/6):487–504.

    Article  Google Scholar 

  10. Tao jianhua. Numerical simulation of water waves[J]. Tianjin: Tianjin University Press, 2005, 108 (in Chinese).

    Google Scholar 

  11. Gobbi M F, Kirby J T. A fourth order Boussinesq-type wave model[J]. Coastal Engineering, 1996, 1(1):1116–1129.

    Google Scholar 

  12. Schäffer H A, Madsen P A. Further enhancements of Boussinesq-type equations[J]. Coastal Engineering, 1995, 26(1/2):1–14.

    Article  Google Scholar 

  13. Madsen P A, Schäffer H A. Higher order Boussinesq-type equations for surface gravity waves-derivation and analysis[J]. Phil Trans R Soc Lond, 1998, 356(12):1–59.

    Google Scholar 

  14. Abbott M B, McCowan A D, Warren I R. Accuracy of short wave numerical models[J]. J Hydraul Eng, 1984, 110(10):1287–1301.

    Article  Google Scholar 

  15. Drago M. Mathematical models for propagation of sea waves in the surf zone[D]. Ph D dissertation. University of Bologna, 1991, 135 (in Italian).

  16. Dean R G, Sharma J N. Simulation of wave systems due to nonlinear directional spectra[J]. Proc Intl Symp Hydrodynamics in Ocean Engng, 1981, 2(1):1211–1222.

    Google Scholar 

  17. Beji S, Battjes J A. Experimental investigations of wave propagation over a bar[J]. Coastal Engineering, 1993, 19(1/2):151–162.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin, 300072, P. R. China

    Dian-xin Zhang  (张殿新) & Jian-hua Tao  (陶建华)

Authors
  1. Dian-xin Zhang  (张殿新)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian-hua Tao  (陶建华)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dian-xin Zhang  (张殿新).

Additional information

Communicated by ZHOU Heng

Project supported by the National Natural Science Foundation of China (No. 50509018)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Dx., Tao, Jh. A Boussinesq model with alleviated nonlinearity and dispersion. Appl. Math. Mech.-Engl. Ed. 29, 897–908 (2008). https://doi.org/10.1007/s10483-008-0708-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-008-0708-6

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation