Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-25T11:57:35.325Z Has data issue: false hasContentIssue false
  • English
  • Français

Third-order theory for multi-directional irregular waves

Published online by Cambridge University Press:  02 April 2012

Per A. Madsen  and
David R. Fuhrman
Per A. Madsen*
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs Lyngby, Denmark
David R. Fuhrman
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs Lyngby, Denmark
*
†Email address for correspondence: prm@mek.dtu.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

A new third-order solution for multi-directional irregular water waves in finite water depth is presented. The solution includes explicit expressions for the surface elevation, the amplitude dispersion and the vertical variation of the velocity potential. Expressions for the velocity potential at the free surface are also provided, and the formulation incorporates the effect of an ambient current with the option of specifying zero net volume flux. Harmonic resonance may occur at third order for certain combinations of frequencies and wavenumber vectors, and in this situation the perturbation theory breaks down due to singularities in the transfer functions. We analyse harmonic resonance for the case of a monochromatic short-crested wave interacting with a plane wave having a different frequency, and make long-term simulations with a high-order Boussinesq formulation in order to study the evolution of wave trains exposed to harmonic resonance.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 698 , 10 May 2012 , pp. 304 - 334
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Badulin, S. I., Shrira, V. I., Kharif, C. & Ioualalen, M. 1995 On two approaches to the problem of instability of short-crested waves. J. Fluid Mech. 303, 297–325.CrossRefGoogle Scholar
2. Bryant, P. J. 1985 Doubly periodic progressive permanent waves in deep water. J. Fluid Mech. 161, 27–42.CrossRefGoogle Scholar
3. Chappelear, J. E. 1961 On the description of short-crested waves. Tech. Memo no. 125, Beach Erosion Board, U.S. Army Corps of Engineers.Google Scholar
4. Craig, W. & Nicholls, D. P. 2002 Travelling gravity water waves in two and three dimensions. Eur. J. Mech. (B/Fluids) 21, 615–641.CrossRefGoogle Scholar
5. Fuhrman, D. R. & Bingham, H. B. 2004 Numerical solutions of fully nonlinear and highly dispersive Boussinesq equations in two horizontal dimensions. Intl J. Numer. Meth. Fluids 44, 231–255.CrossRefGoogle Scholar
6. Fuhrman, D. R. & Madsen, P. A. 2006 Short-crested waves in deep water: a numerical investigation of recent laboratory experiments. J. Fluid Mech. 559, 391–411.CrossRefGoogle Scholar
7. Fuhrman, D. R., Madsen, P. A. & Bingham, H. B. 2006 Numerical simulation of lowest-order short-crested wave instabilities. J. Fluid Mech. 563, 415–441.CrossRefGoogle Scholar
8. Hammack, J. L., Henderson, D. M. & Segur, H. 2005 Progressive waves with persistent, two-dimensional surface patterns in deep water. J. Fluid Mech. 532, 1–51.CrossRefGoogle Scholar
9. Hammack, J. L., Scheffner, N. & Segur, H. 1989 Two-dimensional periodic waves in shallow water. J. Fluid Mech. 209, 567–589.CrossRefGoogle Scholar
10. Hasselmann, K. 1962 On the nonlinear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech. 12, 481.CrossRefGoogle Scholar
11. Henderson, D. M., Patterson, M. S. & Segur, H. 2006 On the laboratory generation of two-dimensional, progressive, surface waves of nearly permanent form in deep water. J. Fluid Mech. 559, 413–427.CrossRefGoogle Scholar
12. Hsu, J. R. C., Tsuchiya, Y. & Silvester, R. 1979 Third-order approximation to short-crested waves. J. Fluid Mech. 90 (1), 179–196.CrossRefGoogle Scholar
13. Ioualalen, M. 1993 Fourth order approximation of short-crested waves. C. R. Acad. Sci. Paris 316 (II), 1193–1200.Google Scholar
14. Ioualalen, M. & Kharif, C. 1993 Stability of three-dimensional progressive gravity waves on deep water to superharmonic disturbances. Eur. J. Mech. B 12 (3), 401–414.Google Scholar
15. Ioualalen, M. & Kharif, C. 1994 On the subharmonic instabilities of steady three-dimensional deep water waves. J. Fluid Mech. 262, 265–291.CrossRefGoogle Scholar
16. Ioualalen, M., Kharif, C. & Roberts, A. J. 1999 Stability regimes of finite depth short-crested water waves. J. Phys. Oceanogr. 29, 2318–2331.2.0.CO;2>CrossRefGoogle Scholar
17. Ioualalen, M., Okamura, M., Cornier, S., Kharif, C. & Roberts, A. J. 2006 Computation of short-crested deepwater waves. ASCE J. Waterway Port Coastal Ocean Engng 132 (3), 157–165.CrossRefGoogle Scholar
18. Ioualalen, M., Roberts, A. J. & Kharif, C. 1996 On the observability of finite-depth short-crested water waves. J. Fluid Mech. 322, 1–19.CrossRefGoogle Scholar
19. Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33, 863–884.2.0.CO;2>CrossRefGoogle Scholar
20. Janssen, P. A. E. M. & Onorato, M. 2007 The intermediate water depth limit of the Zakharov equation and consequences for wave prediction. J. Phys. Oceanogr. 37, 2389–2400.CrossRefGoogle Scholar
21. Kimmoun, O., Ioualalen, M. & Kharif, C. 1999 Instabilities of steep short-crested surface waves in deep water. Phys. Fluids 11 (6), 1679–1681.CrossRefGoogle Scholar
22. Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1–20.CrossRefGoogle Scholar
23. Longuet-Higgins, M. S. 1962 Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12, 321–332.CrossRefGoogle Scholar
24. Madsen, P. A., Bingham, H. B. & Liu, H. 2002 A new Boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech. 462, 1–30.CrossRefGoogle Scholar
25. Madsen, P. A., Bingham, H. B. & Schäffer, H. A. 2003 Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves – derivation and analysis. Proc. R. Soc. Lond. A 459, 1075–1104.CrossRefGoogle Scholar
26. Madsen, P. A. & Fuhrman, D. R. 2006 Third-order theory for bichromatic bi-directional water waves. J. Fluid Mech. 557, 369–397.CrossRefGoogle Scholar
27. Madsen, P. A., Fuhrman, D. R. & Wang, B. H. 2006 A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Engng. 53, 487–504.CrossRefGoogle Scholar
28. Marchant, T. R. & Roberts, A. J. 1987 Properties of short-crested waves in water of finite depth. J. Austral. Math. Soc. B 29, 103–125.CrossRefGoogle Scholar
29. McLean, J. W. 1982 Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331–341.CrossRefGoogle Scholar
30. Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part I. J. Fluid Mech. 9, 193–217.CrossRefGoogle Scholar
31. Roberts, A. J. 1981 The behaviour of harmonic resonant steady solutions to a model differential equation. Q. J. Mech. Appl. Maths 34, 287–310.CrossRefGoogle Scholar
32. Roberts, A. J. 1983 Highly nonlinear short-crested water waves. J. Fluid Mech. 135, 301–321.CrossRefGoogle Scholar
33. Roberts, A. J. & Peregrine, D. H. 1983 Notes on long-crested water waves. J. Fluid Mech. 135, 323–335.CrossRefGoogle Scholar
34. Roberts, A. J. & Schwartz, L. W. 1983 The calculation of nonlinear short-crested gravity waves. Phys. Fluids 26 (9), 2388–2392.CrossRefGoogle Scholar
35. Sharma, J. & Dean, R. 1981 Second-order directional seas and associated wave forces. Soc. Petrol. Engng J. 21, 129–140.CrossRefGoogle Scholar
36. Smith, D. H. & Roberts, A. J. 1999 Branching behaviour of standing waves – the signatures of resonance. Phys. Fluids 11 (5), 1051–1065.CrossRefGoogle Scholar
37. Stiassnie, M. & Gramstad, O. 2009 On Zakharov’s kernel and the interaction of non-collinear wavetrains in finite water depth. J. Fluid Mech. 639, 433–442.CrossRefGoogle Scholar
38. Stokes, G. G. 1847 On the theory of of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
39. Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar
40. Zakharov, V. 1999 Statistical theory of gravity and capillary waves on the surface of a finite depth fluid. Eur. J. Mech. (B/Fluids) 18, 327–344.CrossRefGoogle Scholar
41. Zhang, J. & Chen, L. 1999 General third-order solutions for irregular waves in deep water. J. Engng Mech. 125 (7), 768–779.Google Scholar