Traveling waves in deep water with gravity and surface tension

SIAM Journal on Applied Mathematics

Volumn 70, Issue 7, 2010, Pages 2373-2389

  Akers, Benjamin ^a   Nicholls, David P ^a  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

Author keywords

Potential flow; Ripples; Surface tension

Indexed keywords

APPROXIMATE MODEL; BOND NUMBERS; CAPILLARY EFFECTS; DEEPWATER; EXPLICIT SOLUTIONS; FREE SURFACES; NON-LINEAR MODEL; NUMERICAL SCHEME; NUMERICAL SIMULATION; PERTURBATION THEORY; RIPPLES; SMALL AMPLITUDE; THREE DIMENSIONS; TRAVELING WAVE; WATER-WAVE PROBLEM;

HYDRODYNAMICS; NONLINEAR EQUATIONS; PERTURBATION TECHNIQUES; POTENTIAL FLOW; SURFACE PROPERTIES; SURFACE TENSION; WATER WAVES; WAVE EQUATIONS; WAVES;

COMPUTER SIMULATION;

EID: 77956241032     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/090771351     Document Type: Article

References (45)

1. A. D. D. Craik, The origins of water wave theory, Ann. Rev. Fluid Mech., 36 (2004), pp. 1-28.
2. A. D. D. Craik, George Gabriel Stokes on water wave theory, Ann. Rev. Fluid Mech., 37 (2005), pp. 23-42.
3. W. J. Harrison, The influence of viscosity and capillarity on waves of finite amplitude, Proc. Lond. Math. Soc., 7 (1909), pp. 107-121.
4. R. J. C. Kamesvara, On ripples of finite amplitude, Proc. Indian Ass. Cultiv. Sci., 6 (1920), pp. 175-193.
5. H. Lamb, Hydrodynamics, Cambridge University Press, Cambridge, UK, 1879.
6. L. F. McGoldrick, An experiment on second-order capillary gravity resonant wave interactions, J. Fluid Mech., 40 (1970), pp. 251-271.
7. J. Reeder and M. Shinbrot, Three dimensional, nonlinear wave interaction in water of constant depth, Nonlinear Anal., 5 (1981), pp. 303-323.
8. W. J. Pierson and P. Fife, Some nonlinear properties of long-crested periodic waves with lengths near 2.44 centimeters, J. Geophys. Res., 66 (1961), pp. 163-179.
9. J. R. Wilton, On ripples, Philos. Mag., 29 (1915), pp. 689-700.
10. J.-M. Vanden-Broeck, Wilton ripples generated by a moving pressure disturbance, J. Fluid Mech., 451 (2002), pp. 193-201.
11. J.-M. Vanden-Broeck, Nonlinear gravity-capillary standing waves in water of arbitrary uniform depth, J. Fluid Mech., 139 (1984), pp. 97-104.
12. P. Christodoulides and F. Dias, Resonant capillary-gravity interfacial waves, J. Fluid Mech., 265 (1994), pp. 303-343.
13. J. Reeder and M. Shinbrot, On Wilton ripples II: Rigorous results, Arch. Rational Mech. Anal., 77 (1981), pp. 321-347.
14. D. P. Nicholls and F. Reitich, Stable, high-order computation of traveling water waves in three dimensions, Eur. J. Mech. B Fluids, 25 (2006), pp. 406-424.
15. B. Akers and P. A. Milewski, Dynamics of three-dimensional gravity-capillary solitary waves in deep water, SIAM J. Appl. Math., 70 (2010), pp. 2390-2408.
16. F. Dias and C. Kharif, Nonlinear gravity and gravity-capillary waves, Ann. Rev. Fluid Mech., 31 (1999), pp. 301-346.
17. A. J. Roberts and L. W. Schwartz, The calculation of nonlinear short-crested gravity waves, Phys. Fluid, 26 (1983), pp. 2388-2392.
18. D. P. Nicholls, Traveling water waves: Spectral continuation methods with parallel implementation, J. Comput. Phys., 143 (1998), pp. 224-240.
19. L. W. Schwartz and J.-M. Vanden-Broeck, Numerical solution of the exact equations for capillary-gravity waves, J. Fluid Mech., 95 (1979), pp. 119-139.
20. O. Kimmoun, H. Branger, and C. Kharif, On short-crested waves: Experimental and analytical investigations, Eur. J. Mech. B Fluids, 18 (1999), pp. 889-930.
21. J. L. Hammack and D. M. Henderson, Experiments on deep-water waves with two dimensional surface patterns, J. Offshore Mech. Arct. Eng., 125 (2003), pp. 48-53.
22. P. G. Saffman, The superharmonic instability of finite amplitude water waves, J. Fluid Mech., 159 (1985), pp. 169-174.
23. T. R. Akylas and Y. Cho, On the stability of lumps and wave collapse in water waves, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), pp. 2761-2774.
24. B. Akers and P. A. Milewski, A stability result for travelling waves in dispersive nonlinear equations, Comm. Math. Sci., 6 (2008), pp. 791-797.
25. A. D. D. Craik, Wave Interactions and Fluid Flows, Cambridge University Press, Cambridge, UK, 1985.
26. D. P. Nicholls, Boundary perturbation methods for water waves, GAMM-Mitt., 30 (2007), pp. 44-74.
27. A. J. Roberts, Highly nonlinear short-crested water waves, J. Fluid Mech., 135 (1983), pp. 301-321.
28. T. R. Marchant and A. J. Roberts, Properties of short-crested waves in water of finite depth, J. Austral. Math. Soc. Ser. B, 29 (1987), pp. 103-125.
29. D. P. Nicholls and J. Shen, A rigorous numerical analysis of the transformed field expansion method, SIAM J. Numer. Anal., 47 (2009), pp. 2708-2734.
30. D. J. Korteweg and G. de Vries, On the change in form of long waves advancing in a rectangular canal and new type of long stationary waves, Philos. Mag., 39 (1895), pp. 422-443.
31. B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Soviet. Phys. Dokl., 15 (1970), pp. 539-541.
32. D. R. Fuhrman and P. A. Madsen, Short-crested waves in deep water: A numerical investigation of recent laboratory experiments, J. Fluid Mech., 559 (2006), pp. 391-411.
33. B. Akers and P. A. Milewski, Model equations for gravity-capillary waves in deep water, Stud. Appl. Math., 121 (2008), pp. 49-69.
34. B. Akers and P. A. Milewski, A model equation for wavepacket solitary waves arising from capillary-gravity flows, Stud. Appl. Math., 122 (2009), pp. 249-274.
35. N. B. Vargraftik, B. N. Volkov, and L. D. Voljak, International tables of the surface tension of water, J. Phys. Chem. Ref. Data, 12 (1983), pp. 817-820.
36. N. A. Phillips, A coordinate system having some special advantages for numerical forecasting, J. Atmos. Sci., 14 (1957), pp. 184-185.
37. J. Chandezon, D. Maystre, and G. Raoult, A new theoretical method for diffraction gratings and its numerical application, J. Opt., 11 (1980), pp. 235-241.
38. D. P. Nicholls and F. Reitich, On analyticity of traveling water waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), pp. 1283-1309.
39. P. A. Milewski, Three-dimensional localized gravity-capillary waves, Comm. Math. Sci., 3 (2005), pp. 89-99.
40. J.-M. Vanden-Broeck and F. Dias, Gravity-capillary solitary waves in water of infinite depth and related free surface flows, J. Fluid Mech., 240 (1992), pp. 549-557.
41. M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech., 92 (1979), pp. 691-715.
42. V. D. Djordjevic and L. G. Redekopp, On two dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), pp. 703-714.
43. T. Kawahara, Nonlinear self-modulation of capillary-gravity waves on a liquid layer, J. Phys. Soc. Japan, 38 (1975), pp. 265-270.
44. T. R. Akylas, Envelope solitons with stationary crests, Phys. Fluids A, 5 (1993), pp. 789-791.
45. T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water, J. Fluid Mech., 27 (1967), pp. 417-430.