Revisiting terrain-following Boussinesq equations on a highly variable periodic bed

Research Article


On re-deriving the Boussinesq-type equations in a terrain-following coordinate system based on the conformal map for an arbitrarily (submerged) periodic bed, it is pointed out that some aspects of the existing approach need to be clarified, specifically the use of velocity potential gradient in the mapped plane in connection with the transformation of velocity vector, and the appropriate frequency–wavenumber relation that is from the solution to the linearized Boussinesq equations. It is shown that as the bed undulation height increases, this relationship increasingly departs from the flat-bottom dispersion relation that has previously been assumed for the terrain-following Boussinesq systems. Over a highly variable periodic bed, the waveforms of linear time harmonic waves already have features reminiscent of nonlinear waveforms, which should be distinguished from the subsequent nonlinear evolution.


Boussinesq equations Terrain-following Conformal mapping 



This research was funded by the Office of Naval Research under program element 0602235N. The author thanks anonymous reviewers for their helpful comments.


  1. Andrade D, Nachbin A (2018) Two dimensional surface wave propagation over arbitrary ridge-like topographies. SIAM J Appl Math 78(5):2465–2490MathSciNetCrossRefGoogle Scholar
  2. Artiles W, Nachbin A (2004) Nonlinear evolution of surface gravity waves over highly variable depth. Phys Rev Lett 93:234501CrossRefGoogle Scholar
  3. Athanassoulis GA, Papoutsellis C (2017) Exact semi-separation of variables in waveguides with non-planar boundaries. Proc R Soc A 473:20170017MathSciNetCrossRefGoogle Scholar
  4. Belibassakis KA, Athanassoulis GA (2011) A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions. Coast Eng 58(4):337–350CrossRefGoogle Scholar
  5. Fokas AS, Nachbin A (2012) Water waves over a variable bottom: a non-local formulation and conformal mappings. J Fluid Mech 695:288–309MathSciNetCrossRefGoogle Scholar
  6. Grajales JCM, Nachbin A (2006) Improved Boussinesq-type equations for highly variable depth. IMA J Appl Math 71(4):600–633MathSciNetCrossRefGoogle Scholar
  7. Howard LN, Yu J (2007) Normal modes of a rectangular tank with corrugated bottom. J Fluid Mech 593:209–234MathSciNetCrossRefGoogle Scholar
  8. Kim JW, Ertekin RC, Bai KJ (2010) Linear and nonlinear wave models based on Hamilton’s principle and stream-function theory: CMSE and IGN. ASME J Offshore Mech Arctic Eng 132(2):021102CrossRefGoogle Scholar
  9. Luz AM, Nachbin A (2013) Wave packet defocusing due to a highly variable disordered bathymetry. Stud Appl Math 130:393–416MathSciNetCrossRefGoogle Scholar
  10. Madsen OS, Mei CC (1969) The transformation of a solitary wave over an uneven bottom. J Fluid Mech 39:781–791CrossRefGoogle Scholar
  11. Mei CC (1989) The applied dynamics of ocean surface waves. World Scientific, SingaporezbMATHGoogle Scholar
  12. Mei CC, Le Méhauté B (1966) Note on the equations of long waves over an uneven bottom. J Geophys Res 71(2):393–400MathSciNetCrossRefGoogle Scholar
  13. Nachbin A (2003) A terrain-following Boussinesq system. SIAM J Appl Math 63(3):905–922MathSciNetCrossRefGoogle Scholar
  14. Nachbin A, Choi W (2007) Nonlinear waves over highly variable topography. Eur Phys J Spec Top 147:113–132CrossRefGoogle Scholar
  15. Nwogu O (1993) Alternative form of Boussinesq equations for nearshore wave propagation. J Waterw Port Coast Ocean Eng 119(6):618–638CrossRefGoogle Scholar
  16. Papoutsellis C, Charalampopoulos AG, Athanassoulis GA (2018) Implementation of a fully nonlinear Hamiltonian coupled-mode theory, and application to solitary wave problems over bathymetry. Eur J Mech B Fluids 72:199–224MathSciNetCrossRefGoogle Scholar
  17. Peng J, Tao A, Liu Y, Zheng J, Zhang J, Wang R (2019) A laboratory study of class III Bragg resonance of gravity surface waves by periodic beds. Phys Fluids 31:067110CrossRefGoogle Scholar
  18. Peregrine DH (1967) Long waves on a beach. J Fluid Mech 27:815–827CrossRefGoogle Scholar
  19. Yu J (2019) Waveform of gravity and capillary-gravity waves over a bathymetry. Phys Rev Fluids 4:014806CrossRefGoogle Scholar
  20. Yu J, Howard LN (2010) On higher order Bragg resonance of water waves by bottom corrugations. J Fluid Mech 659:484–504MathSciNetCrossRefGoogle Scholar
  21. Yu J, Howard LN (2012) Exact Floquet theory for waves over arbitrary periodic topographies. J Fluid Mech 712:451–470MathSciNetCrossRefGoogle Scholar

Copyright information

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Naval Research LaboratoryStennis Space CenterMississippiUSA

Personalised recommendations