Stability of bichromatic gravity waves on deep water

European Journal of Mechanics, B/Fluids

Volumn 28, Issue 5, 2009, Pages 605-612

  Leblanc, Stéphane ^a  

^a UNIVERSITÉ DE TOULON   (France)

Author keywords

Gravity waves; Modulational instability; Nonlinear interactions

Indexed keywords

BI-CHROMATIC WAVES; DEEP WATER; FINITE AMPLITUDE; GENERIC MECHANISM; LINEAR GROWTH; MODULATIONAL INSTABILITY; NONLINEAR INTERACTIONS; NUMERICAL INTEGRATIONS; PHILLIPS; RESONANT INTERACTION; STANDING WAVE; TWO-COMPONENT; WAVE TRAIN; WAVE VECTOR; ZAKHAROV EQUATIONS;

BATHYMETRY; GRAVITATIONAL EFFECTS; GRAVITY WAVES; HYDRODYNAMICS; LOCOMOTIVES; RAILROAD CARS; WAVES;

STABILITY CRITERIA;

EID: 67651108761     PISSN: 09977546     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.euromechflu.2009.05.001     Document Type: Article

References (43)

1. Benjamin T.B., and Feir F.E. The disintegration of wave trains on deep water. Part 1. J. Fluid Mech. 27 (1967) 417-430
2. Benney D.J., and Newell A.C. The propagation of nonlinear wave envelopes. J. Math. Phys. 46 (1967) 133-145
3. Phillips O.M. Theoretical and experimental studies of gravity wave interactions. Proc. R. Soc. Lond. A 299 (1967) 104-119
4. Zakharov V.E. Stability of periodic waves of finite amplitude on the surface of deep fluid. J. Appl. Mech. Tech. Phys. 2 (1968) 190-194
5. Crawford D.R., Lake B.M., Saffman P.G., and Yuen H.C. Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech. 14 (1981) 577-584
6. Longuet-Higgins M.S. The instabilities of gravity waves of finite amplitude in deep water. Part 1. Proc. R. Soc. Lond. A 360 (1978) 471-488
7. Phillips O.M. Wave interactions-the evolution of an idea. J. Fluid Mech. 106 (1981) 215-227
8. Yuen H.C., and Lake B.M. Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22 (1982) 67-229
9. Craik A.D.D. Wave interactions and fluid flows (1985), Cambridge University Press
10. Shemer L., and Stiassnie M. The Zakharov and modified Zakharov equations and their applications. In: Osborne A.R. (Ed). Nonlinear topics in ocean physics (1988), Italian Physical Society 581-620
11. Dias F., and Kharif C. Nonlinear gravity and capillary-gravity waves. Annu. Rev. Fluid Mech. 31 (1999) 301-346
12. Janssen P. The Interaction of ocean waves and wind (2004), Cambridge University Press
13. Zakharov V.E., and Ostrovsky L.A. Modulation instability: the beginning. Physica D 238 (2009) 540-548
14. Bridges T.J., and Mielke A. A proof of the Benjamin-Feir instability. Arch. Rat. Mech. Anal. 133 (1995) 145-198
15. Roskes G.J. Nonlinear multiphase deep-water wavetrains. Phys. Fluids 19 (1976) 1253-1254
16. Dhar A.K., and Das K.P. Fourh-order nonlinear evolution equation for two Stokes wave trains in deep water. Phys. Fluids A 3 (1991) 3021-3026
17. Pierce R.D., and Knobloch E. On the modulational stability of traveling an standing water waves. Phys. Fluids 6 (1994) 1177-1190
18. Onorato M., Osborne A.R., and Serio M. Modulational instability in crossing sea states: a possible mechanism for the formation of freak waves. Phys. Rev. Lett. 96 (2006) 014503 (4 pages)
19. Craig W., Henderson D.M., Oscamou M., and Segur H. Stable three-dimensional waves of nearly permanent form on deep water. Math. Comput. Simul. 74 (2007) 135-144
20. Okamura M. Instabilities of weakly nonlinear standing gravity waves. J. Phys. Soc. Jap. 53 (1984) 3788-3796
21. Ioualalen M., and Kharif C. Stability of three-dimensional progressive gravity waves on deep water to superharmonic disturbances. Eur. J. Mech. B/Fluids 12 (1993) 401-414
22. Ioualalen M., and Kharif C. On the subharmonic instabilities of steady three-dimensional deep water waves. J. Fluid Mech. 262 (1994) 265-291
23. Badulin S.I., Shrira V.I., Kharif C., and Ioualalen M. On two approaches to the problem of instability of short-crested water waves. J. Fluid Mech. 303 (1995) 297-326
24. Kimmoun O., Ioualalen M., and Kharif C. Instabilities of steep short-crested surface waves in deep water. Phys. Fluids 11 (1999) 1679-1681
25. Bridges T.J., and Laine-Pearson F.E. Nonlinear counter propagating waves, multisympletic geometry, and the instability of standing waves. SIAM J. Appl. Math. 64 (2004) 2096-2120
26. Bridges T.J., and Laine-Pearson F.E. The long-wave instability of short-crested waves, via embedding in a oblique two-wave interaction. J. Fluid Mech. 543 (2005) 147-182
27. Phillips O.M. On the dynamics of unsteady gravity waves of finite amplitude. Part 1. J. Fluid Mech. 9 (1960) 193-217
28. Longuet-Higgins M.S. Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12 (1962) 321-332
29. Longuet-Higgins M.S., and Phillips O.M. Phase velocity effects in tertiary wave interactions. J. Fluid Mech. 12 (1962) 333-336
30. Longuet-Higgins M.S., and Smith N.D. An experiment on third-order resonant wave interactions. J. Fluid Mech. 25 (1966) 417-435
31. McGoldrick L.F., Phillips O.M., Huang N.E., and Hodgson T.H. Measurements of third-order resonant wave interactions. J. Fluid Mech. 25 (1966) 437-456
32. Krasitskii V.P. On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272 (1994) 1-20
33. Hogan S.J., Gruman I., and Stiassnie M. On the changes in phase speed of one train of water waves in the presence of another. J. Fluid Mech. 192 (1988) 97-114
34. Roberts A.J., and Peregrine D.H. Notes on long-crested water waves. J. Fluid Mech. 135 (1983) 323-335
35. Benney D.J. Non-linear gravity wave interactions. J. Fluid Mech. 14 (1962) 577-584
36. Hasselmann K. On the non-linear energy transfer in a gravity-wave spectrum. Part 1. J. Fluid Mech. 12 (1962) 481-500
37. Hasselmann K. On the non-linear energy transfer in a gravity-wave spectrum. Part 2. J. Fluid Mech. 15 (1963) 273-281
38. Hasselmann K. On the non-linear energy transfer in a gravity-wave spectrum. Part 3. J. Fluid Mech. 15 (1963) 385-398
39. Bretherton F.P. Resonant interactions between waves. The case of discrete oscillations. J. Fluid Mech. 20 (1964) 457-480
40. Stiassnie M., and Shemer L. On the interaction of four water waves. Wave Motion 41 (2005) 307-328
41. Hammack J.L., Henderson D.M., and Segur H. Progressive waves with persistent two-dimensional surface patterns in deep water. J. Fluid Mech. 532 (2005) 1-52
42. Fuhrman D.R., Madsen P.A., and Bingham H.B. Numerical simulation of lowest-order short-crested wave instabilities. J. Fluid Mech. 563 (2006) 415-441
43. D. Harvar, B.C.P. Mercy, Personnal communication, 2009.