October  2012, 17(7): 2329-2341. doi: 10.3934/dcdsb.2012.17.2329

Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability

1. 

School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ

2. 

Ship Science, University of Southampton, Southampton SO17 1BJ, United Kingdom

Received  November 2010 Revised  March 2012 Published  July 2012

The Benjamin-Feir instability describes the instability of a uniform oscillatory wave train in an irrotational flow subject to small perturbation of wave number, amplitude and frequency. Their instability analysis is based on the perturbation around the second order Stokes wave which satisfies the dynamic and kinematic free-surface boundary conditions up to the second order. In the same irrotational flow and perturbation framework of the Benjamin-Feir analysis, the perturbation in the present paper is around a nonlinear oscillatory wave train which solves exactly the dynamic free-surface boundary condition and satisfies the kinematic free-surface boundary condition up to the third order. It is shown that the nonlinear oscillatory wave train is stable with respect to the perturbation when the irrotational flow involves small Rayleigh energy dissipation.
Citation: Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329
References:
[1]

Izv. Atmos. Oceanic Phys., 15 (1979), 711-716. Google Scholar

[2]

Math. Comp. Simul., 74 (2007), 159-167. doi: 10.1016/j.matcom.2006.10.010.  Google Scholar

[3]

J. Atmos. Sci., 36 (1979), 1205-1216. doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2.  Google Scholar

[4]

Stud. Appl. Math., 62 (1980), 1-21.  Google Scholar

[5]

J. Fluids Stuct., 28 (2012), 378-391. doi: 10.1016/j.jfluidstructs.2011.10.003.  Google Scholar

[6]

Proc. R. Soc. A, 299 (1967), 59-75. doi: 10.1098/rspa.1967.0123.  Google Scholar

[7]

J. Fluid Mech., 27 (1967), 417-430. doi: 10.1017/S002211206700045X.  Google Scholar

[8]

Arch. Ratianal Mech. Anal., 133 (1995), 145-198. doi: 10.1007/BF00376815.  Google Scholar

[9]

Cambridge University Press, Cambridge, 1932. Google Scholar

[10]

Proc. R. Soc. Lond. Ser. A, 396 (1984), 269-280. doi: 10.1098/rspa.1984.0122.  Google Scholar

[11]

J. Fluid Mech., 27 (1967), 395-397. doi: 10.1017/S0022112067000412.  Google Scholar

[12]

Springer-Verlag, New York, 1979. Google Scholar

[13]

J. Fluid Mech., 539 (2005), 229-271.  Google Scholar

[14]

Trans. Cambridge Philos. Soc., 8 (1847), 441-473. Google Scholar

[15]

J. Phys. Soc. Japan, 52 (1983), 3047-3055. doi: 10.1143/JPSJ.52.3047.  Google Scholar

[16]

J. Fluid Mech., 156 (1985), 281-289. doi: 10.1017/S0022112085002099.  Google Scholar

[17]

Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar

[18]

Comm. Pure Appl. Math., 41 (1988), 19-46. doi: 10.1002/cpa.3160410104.  Google Scholar

[19]

J. Fluid Mech., 556 (2006), 45-54. doi: 10.1017/S0022112005008293.  Google Scholar

[20]

J. Appl. Mech. Tech. Phys., 2 (1968), 190-194. Google Scholar

show all references

References:
[1]

Izv. Atmos. Oceanic Phys., 15 (1979), 711-716. Google Scholar

[2]

Math. Comp. Simul., 74 (2007), 159-167. doi: 10.1016/j.matcom.2006.10.010.  Google Scholar

[3]

J. Atmos. Sci., 36 (1979), 1205-1216. doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2.  Google Scholar

[4]

Stud. Appl. Math., 62 (1980), 1-21.  Google Scholar

[5]

J. Fluids Stuct., 28 (2012), 378-391. doi: 10.1016/j.jfluidstructs.2011.10.003.  Google Scholar

[6]

Proc. R. Soc. A, 299 (1967), 59-75. doi: 10.1098/rspa.1967.0123.  Google Scholar

[7]

J. Fluid Mech., 27 (1967), 417-430. doi: 10.1017/S002211206700045X.  Google Scholar

[8]

Arch. Ratianal Mech. Anal., 133 (1995), 145-198. doi: 10.1007/BF00376815.  Google Scholar

[9]

Cambridge University Press, Cambridge, 1932. Google Scholar

[10]

Proc. R. Soc. Lond. Ser. A, 396 (1984), 269-280. doi: 10.1098/rspa.1984.0122.  Google Scholar

[11]

J. Fluid Mech., 27 (1967), 395-397. doi: 10.1017/S0022112067000412.  Google Scholar

[12]

Springer-Verlag, New York, 1979. Google Scholar

[13]

J. Fluid Mech., 539 (2005), 229-271.  Google Scholar

[14]

Trans. Cambridge Philos. Soc., 8 (1847), 441-473. Google Scholar

[15]

J. Phys. Soc. Japan, 52 (1983), 3047-3055. doi: 10.1143/JPSJ.52.3047.  Google Scholar

[16]

J. Fluid Mech., 156 (1985), 281-289. doi: 10.1017/S0022112085002099.  Google Scholar

[17]

Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar

[18]

Comm. Pure Appl. Math., 41 (1988), 19-46. doi: 10.1002/cpa.3160410104.  Google Scholar

[19]

J. Fluid Mech., 556 (2006), 45-54. doi: 10.1017/S0022112005008293.  Google Scholar

[20]

J. Appl. Mech. Tech. Phys., 2 (1968), 190-194. Google Scholar

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