Journal of Engineering Mathematics

, Volume 114, Issue 1, pp 87–114 | Cite as

Semi-explicit solutions to the water-wave dispersion relation and their role in the non-linear Hamiltonian coupled-mode theory

  • T. K. PapathanasiouEmail author
  • Ch. E. Papoutsellis
  • G. A. Athanassoulis


The Hamiltonian coupled-mode theory (HCMT), recently derived by Athanassoulis and Papoutsellis [Proceeding of 34th International Conference on Ocean Offshore Arctic Engineering, ASME, St. John’s, Newfoundland, Canada, 2015], provides an efficient new approach for solving fully non-linear water-wave problems over arbitrary bathymetry. This theory exactly transforms the free-boundary problem to a fixed-boundary one, with space and time-varying coefficients. In calculating these coefficients, heavy use is made of the roots of a local, water-wave dispersion relation with varying parameters, which have to be calculated at every horizontal position and every time instant. Thus, fast and accurate calculation of these roots, valid for all possible values of the varying parameter, are of fundamental importance for the efficient implementation of HCMT. In this paper, new, semi-explicit and highly accurate root-finding formulae are derived, especially for the roots corresponding to evanescent modes. The derivation is based on the successive application of a Picard-type iteration and the Householders root-finding method. Explicit approximate formulae of very good accuracy are obtained, and machine-accurate determination of the required roots is easily achieved by no more than three iterations, using the explicit forms as initial values. Exploiting this procedure in the HCMT, results in an efficient, dimensionally reduced, numerical solver able to treat fully non-linear water waves over arbitrary bathymetry. Applications to four demanding non-linear problems demonstrate the efficiency and the robustness of the present approach. Specifically, we consider the classical tests of strongly non-linear steady wave propagation and the transformation of regular waves due to trapezoidal and sinusoidal bathymetry. Novel results are also given for the disintegration of a solitary wave due to an abrupt deepening. The derived root-finding formulae can be used with any other multimodal methods as well.


Dispersion relation Hamiltonian coupled-mode theory Multimodal techniques Newton–Raphson iterations Non-linear water waves Root approximation 



This research has not been supported by any funding bodies. The authors would like to thank Mr. A. Charalampopoulos for his support in the numerical simulations.

Supplementary material

10665_2018_9983_MOESM1_ESM.mp4 (9.2 mb)
Supplementary material 1 (mp4 9408 KB)


  1. 1.
    Bingham HB, Agnon Y (2005) A Fourier–Boussinesq method for nonlinear water waves. Eur J Mech B/Fluids 24:255–274MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Madsen P, Fuhrman D, Wang B (2006) A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast Eng 53:487–504CrossRefGoogle Scholar
  3. 3.
    Karambas V, Memos C (2009) Boussinesq model for weakly nonlinear fully dispersive water waves. J Waterw Port Coast Ocean Eng 135(5):187–199CrossRefGoogle Scholar
  4. 4.
    Bonneton P, Barthelemy E, Chazel F, Cienfuegos R, Lannes D, Marche F, Tissier M (2011) Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes. Eur J Mech B/Fluids 30:589–597MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Zhao B, Duan W, Ertekin R (2014) Application of higher-level GN theory to some wave transformation problems. Coast Eng 83:177–189CrossRefGoogle Scholar
  6. 6.
    Clamond D, Dutykh D, Mitsotakis D (2017) Conservative modified Serre–Green–Naghdi equations with improved dispersion characteristics. Commun Nonlinear Sci Numer Simul 45:254–257MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brocchini M (2013) A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics. Proc R Soc Lond A 469:20130496MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Memos C, Klonaris G, Chondros M (2015) On Higher-order boussinesq-type wave models. J Waterw Port Coast Ocean Eng 142(1):1–17Google Scholar
  9. 9.
    Bingham HB, Zhang H (2007) On the accuracy of finite difference solutions for nonlinear water waves. J Eng Math 58:211–228MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gagarina E, Ambati VR, Van Der Vegt JJW, Bokhove O (2013) Variational space-time (dis) continuous Galerkin method for nonlinear free surface water waves. J Comput Phys 275:459–483MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brink F, Izsák F, van der Vegt J (2017) Hamiltonian finite element discretization for nonlinear free surface water waves. J Sci Comput 73(1):366–394MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grilli S, Guyenne P, Dias F (2001) A fully non-linear model for three-dimensional overturning waves over an arbitrary bottom. Int J Numer Methods Fluids 35:829–867CrossRefzbMATHGoogle Scholar
  13. 13.
    Fructus D, Clamond D, Grue J, Kristiansen Ø (2005) An efficient model for three-dimensional surface wave simulations. Part I: free space problems. J Comput Phys 205:665–685MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Milewski P (1998) A formulation for water waves over topography. Stud Appl Math 100:95–106MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Athanassoulis G, Belibassakis K (1999) A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions. J Fluid Mech 389:275–301CrossRefzbMATHGoogle Scholar
  16. 16.
    Belibassakis K, Athanassoulis G (2002) Extension of second-order stokes theory to variable bathymetry. J Fluid Mech 464:35–80MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Belibassakis K, Gerostathis T, Athanassoulis G (2011) A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography. Appl Ocean Res 33(4):384–397CrossRefGoogle Scholar
  18. 18.
    Athanassoulis G, Papoutsellis C (2015) New form of the Hamiltonian equations for the nonlinear water-wave problem, based on a new representation of the DtN operator, and some applications. In: Proc 34th int conf ocean offshore arct eng, ASME, St. John’s, Newfoundland, Canada, 2015, p V007T06A029Google Scholar
  19. 19.
    Papoutsellis C, Athanassoulis G (2017) A new efficient Hamiltonian approach to the nonlinear water-wave problem over arbitrary bathymetry.
  20. 20.
    Athanassoulis G, Papoutsellis C (2017) Exact semi-separation of variables in waveguides with nonplanar boundaries. Proc R Soc Lond A 473:20170017CrossRefzbMATHGoogle Scholar
  21. 21.
    Craig W, Sulem C (1993) Numerical simulation of gravity waves. J Comput Phys 108:73–83MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Belibassakis K, Athanassoulis G (2011) A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions. Coast Eng 58:337–350CrossRefGoogle Scholar
  23. 23.
    Eckart C (1952) Propagation of gravity waves from deep to shallow water. In: Proceedings of the NBS semicentennial symposium on gravity waves. National Bureau of Standards Circular 521, November 28, 1952. Department of Commerce, USA, p 165Google Scholar
  24. 24.
    Olson W (1973) An explicit expression for the wavelength of a gravity wave. J Phys Ocean 3:238–239CrossRefGoogle Scholar
  25. 25.
    Hunt N (1979) Direct solution of wave dispersion equation. J Waterw Port Coastal Ocean Eng 4:457–459Google Scholar
  26. 26.
    Nielsen P (1982) Explicit formulae for practical wave calculations. Coast Eng 6:389–398CrossRefGoogle Scholar
  27. 27.
    Chen H, Thompson E (1985) Iterative and Padé solutions for the water-wave dispersion relation. US Army Corps of Engineers, CERC-85-4, Washington, DCGoogle Scholar
  28. 28.
    Wu S, Thornton B (1986) Wave numbers of linear progressive waves. J Waterw Port Coastal Ocean Eng 112:536–570CrossRefGoogle Scholar
  29. 29.
    McKee D (1988) Calculation of evanescent wave modes. J Waterw Port Coastal Ocean Eng 114:373–378CrossRefGoogle Scholar
  30. 30.
    Fenton J, McKee W (1990) On calculating the lengths of water waves. Coast Eng 14:499–513CrossRefGoogle Scholar
  31. 31.
    Newman J (1990) Numerical solutions of the water-wave dispersion relation. Appl Ocean Res 12:14–18CrossRefGoogle Scholar
  32. 32.
    Chamberlain PG, Porter D (1999) On the solution of the dispersion relation for water waves. Appl Ocean Res 21:161–166CrossRefGoogle Scholar
  33. 33.
    Guo J (2002) Simple and explicit solution of wave dispersion equation. Coast Eng 45:71–74CrossRefGoogle Scholar
  34. 34.
    Beji S (2013) Improved explicit approximation of linear dispersion relationship for gravity waves. Coast Eng 73:11–12CrossRefGoogle Scholar
  35. 35.
    Simarro G, Orfilia A (2013) Improved explicit approximation of linear dispersion relationship for gravity waves: another discussion. Coast Eng 80:38–39Google Scholar
  36. 36.
    Vatankhah R, Aghashariatmadari Z (2013) Improved explicit approximation of linear dispersion relationship for gravity waves: a discussion. Coast Eng 78:21–22CrossRefGoogle Scholar
  37. 37.
    Massel SR (1993) Extended refraction–diffraction equation for surface waves. Coast Eng 19:97–126CrossRefGoogle Scholar
  38. 38.
    Porter D, Staziker DJ (1993) Extensions of the mild-slope equation. J Fluid Mech 300:367–382MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Chamberlain PG, Porter D (2006) Multi-mode approximations to wave scattering by an uneven bed. J Fluid Mech 556:421–441MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Belibassakis K, Athanassoulis G, Gerostathis Th (2001) A coupled-mode model for the refraction-diffraction of linear waves over steep three-dimensional bathymetry. Appl Ocean Res 23:319–336CrossRefGoogle Scholar
  41. 41.
    Papathanasiou Th, Karperaki A, Belibassakis K (2016) An efficient coupled-mode/FEM numerical method for linear wave propagation. In: Proc twenty-sixth int ocean polar eng conf, Rhodes, Greece, 2016, pp 1363–1370Google Scholar
  42. 42.
    Beji S, Battjes A (1993) Experimental investigation of wave propagation over a bar. Coast Eng 19:151–162CrossRefGoogle Scholar
  43. 43.
    Dingemans M (1994) Comparison of computations with Boussinesq-like models and laboratory measurements. Mast-G8M note, H1684, Delft HydraulicsGoogle Scholar
  44. 44.
    Davies AG, Heathershaw AD (1984) Surface-wave propagation over sinusoidally varying topography. J Fluid Mech 144:419–443CrossRefGoogle Scholar
  45. 45.
    Zakharov V (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. Z Prikl Mekhaniki I Tekhnicheskoi Fiz 9:86–94Google Scholar
  46. 46.
    Lannes D (2013) Water waves problem: mathematical analysis and asymptotics. American Mathematical Society, Providence, RICrossRefzbMATHGoogle Scholar
  47. 47.
    Nicholls DP (1998) Traveling water waves: spectral continuation methods with parallel implementation. J Comput Phys 143:224–240MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Craig W, Guyenne P, Sulem C (2009) Water waves over a random bottom. J Fluid Mech 640:79–107MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Xu L, Guyenne P (2009) Numerical simulation of three-dimensional nonlinear water waves. J Comput Phys 228:8446–8466MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Craig W, Guyenne P, Nicholls D, Sulem C (2005) Hamiltonian long-wave expansions for water waves over a rough bottom. Proc R Soc Lond A 461:839–873MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Gouin M, Ducrozet G, Ferrant P (2016) Development and validation of a non-linear spectral model for water waves over variable depth. Eur J Mech B-Fluid 57:115–128MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Nicholls DP, Reitich F (2001) Stability of high-order perturbative methods for the computation of Dirichlet–Neumann operators. J Comput Phys 170:276–298MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Higgins J (1977) Completeness and basis properties of sets of special functions. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  54. 54.
    Hazard C, Luneville E (2008) An improved multimodal approach for non-uniform acoustic waveguides. IMA J Appl Math 73:668–690MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Mercier JF, Maurel A (2013) Acoustic propagation in non-uniform waveguides: revisiting Webster equation using evanescent boundary modes. Proc R Soc Lond A 469:20130186MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Maurel A, Mercier JF, Pagneux V (2014) Improved multimodal admittance method in varying cross section waveguides. Proc R Soc Lond A 470(2164):20130448MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Luke J (1967) A variational principle for a fluid with a free surface. J Fluid Mech 27:395–397MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Papoutsellis Ch, Charalambopoulos A, Athanassoulis G (2018) Implementation of a fully nonlinear Hamiltonian coupled-mode theory, and application to solitary wave problems over bathymetry. Eur J Mech B/Fluids 72:199–224MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Nicholls D (2007) Boundary perturbation methods for water waves. GAMM Mitt 30(1):44–74MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Butcher J (2003) Numerical methods for ordinary differential equations. Wiley, West SussexCrossRefzbMATHGoogle Scholar
  61. 61.
    Fenton J, McKee D (1990) On calculating the lengths of water waves. Coast Eng 14:499–513CrossRefGoogle Scholar
  62. 62.
    Householder A (1970) The numerical treatment of a single nonlinear equation. McGraw Hill, New YorkzbMATHGoogle Scholar
  63. 63.
    Bender C, Orszag S (1999) Advanced mathematical methods for scientists and engineers. McGraw Hill, New YorkCrossRefzbMATHGoogle Scholar
  64. 64.
    Isaacson E, Keller B (1994) Analysis of numerical methods. Dover Publications Inc, New YorkzbMATHGoogle Scholar
  65. 65.
    Mező I, Baricz A (2017) On the generalization of the Lambert W function. Trans Am Math Soc 369(11):7917–7934MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Mező I, Keady G (2016) Some physical applications of generalized Lambert functions. Eur J Phys 37:065802CrossRefzbMATHGoogle Scholar
  67. 67.
    Scott TC (2014) Numerics of the generalized Lambert W function. ACM Commun Comput Algebra 48(1–2):42–56MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Maignan A, Scott TC (2016) Fleshing out the generalized Lambert W function. ACM Commun Comput Algebra 50(2):45–60MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Castle P (2018) Taylor series for generalized Lambert W functionsGoogle Scholar
  70. 70.
    Rienecker MM, Fenton JD (1981) A Fourier approximation method for steady water waves. J Fluid Mech 104:119–137CrossRefzbMATHGoogle Scholar
  71. 71.
    Clamond D, Dutykh D (2017) Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth. J Fluid Mech 844:491–518MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Papoutsellis C (2016) Nonlinear water waves over varying bathymetry: theoretical and numerical study using variational methods. National Technical University of AthensGoogle Scholar
  73. 73.
    Williams J (1981) Limiting gravity waves in water of finite depth. Phil Trans R Soc Lond A 302:139–188MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Zhang Y, Kennedy A, Panda N, Dawson C, Westerink J (2014) Generating-absorbing sponge layers for phase-resolving wave models. Coast Eng 84:1–9CrossRefGoogle Scholar
  75. 75.
    Goda Y, Suzuki Y (1976) Estimation of incident and Reflected waves in random wave experiments. In: Proc 15th Conf Coast Eng, Honoloulou, HawaiGoogle Scholar
  76. 76.
    Liu Y, Yue D (1998) On generalized Bragg scattering of surface waves by bottom ripples. J Fluid Mech 356:297–326MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Garnier J, Munoz Grajales JC, Nachbin A (2007) Effective behavior of solitary waves over random topography. Multiscale Model Simul 6:995–1025MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Andrade D, Nachbin A (2018) A three-dimensional Dirichlet-to-Neumann operator for water waves over topography 845:321–345Google Scholar
  79. 79.
    Nachbin A, Papanicolaou GC (1992) Water waves in shallow channels of rapidly varying depth. J Fluid Mech 241:311–332MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Fructus D, Grue J (2007) An explicit method for the nonlinear interaction between water waves and variable and moving bottom topography. J Comput Phys 222:720–739MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Guyenne P, Nicholls D (2007) A high-order spectral method for nonlinear water waves over moving bottom topography. SIAM J Sci Comput 30(1):81–101MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    Clamond D, Dutykh D (2013) Fast accurate computation of the fully nonlinear solitary surface gravity waves. Comput Fluids 84:35–38MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical, Aerospace and Civil EngineeringBrunel University LondonUxbridgeUK
  2. 2.School of Naval Architecture and Marine EngineeringAthensGreece
  3. 3.École Centrale Marseille and Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE)MarseilleFrance
  4. 4.National Technical University of AthensZografosGreece
  5. 5.Research Center for High Performance ComputingITMO UniversitySt. PetersburgRussian Federation

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