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High-frequency instabilities of Stokes waves

Published online by Cambridge University Press:  28 February 2022

Ryan P. Creedon Open the ORCID record for Ryan P. Creedon [Opens in a new window] ,
Bernard Deconinck  and
Olga Trichtchenko
Ryan P. Creedon*
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA98195, USA
Bernard Deconinck
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA98195, USA
Olga Trichtchenko
Affiliation:
Department of Physics and Astronomy, The University of Western Ontario, London, ONN6A 3K7, Canada
*
†Email address for correspondence: [email protected]
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Abstract

Euler's equations govern the behaviour of gravity waves on the surface of an incompressible, inviscid and irrotational fluid of arbitrary depth. We investigate the spectral stability of sufficiently small-amplitude, one-dimensional Stokes waves, i.e. periodic gravity waves of permanent form and constant velocity, in both finite and infinite depth. We develop a perturbation method to describe the first few high-frequency instabilities away from the origin, present in the spectrum of the linearization about the small-amplitude Stokes waves. Asymptotic and numerical computations of these instabilities are compared for the first time, with excellent agreement.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 937 , 25 April 2022 , A24
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Ablowitz, M.J., Fokas, A.S. & Musslimani, Z.H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–343.CrossRefGoogle Scholar
Ablowitz, M.J. & Haut, T.S. 2008 Spectral formulation of the two fluid Euler equations with a free interface and long wave reduction. Anal. Applics. 6 (4), 323–348.CrossRefGoogle Scholar
Akers, B. & Nicholls, D.P. 2012 Spectral stability of deep two-dimensional gravity water waves: repeated eigenvalues. SIAM J. Appl. Maths 130 (2), 81–107.Google Scholar
Akers, B. & Nicholls, D.P. 2014 The spectrum of finite depth water waves. Eur. J. Mech. (B/Fluids) 46, 181–189.CrossRefGoogle Scholar
Akers, B. 2015 Modulational instabilities of periodic traveling waves in deep water. Physica D 300, 26–33.CrossRefGoogle Scholar
Benjamin, T.B. 1967 Instability of periodic wave trains in nonlinear dispersive systems. Proc. R. Soc. Lond. A 299, 59–79.Google Scholar
Benjamin, T.B. & Feir, J.E. 1967 The disintegration of wave trains on deep water. Part I. Theory. J. Fluid Mech. 27, 417–430.CrossRefGoogle Scholar
Bohr, H. 1947 Almost Periodic Functions. Chelsea Publishing Company.Google Scholar
Bridges, T.H. & Mielke, A. 1995 A proof of the Benjamin–Feir instability. Arch. Rat. Mech. Anal. 133, 145–198.CrossRefGoogle Scholar
Bryant, P.J. 1974 Stability of periodic waves in shallow water. J. Fluid Mech. 66, 81–96.CrossRefGoogle Scholar
Bryant, P.J. 1978 Oblique instability of periodic waves in shallow water. J. Fluid Mech. 86, 783–792.CrossRefGoogle Scholar
Constantin, A. & Strauss, W.A. 2010 Pressure beneath a Stokes wave. Commun. Pure Appl. Maths 63 (4), 533–557.Google Scholar
Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108 (1), 73–83.CrossRefGoogle Scholar
Craik, A.D.D. 2004 The origins of water wave theory. Annu. Rev. Fluid Mech. 36, 1–28.CrossRefGoogle Scholar
Creedon, R.P., Deconinck, B. & Trichtchenko, O. 2021 a High-frequency instabilities of the Kawahara equation: a perturbative approach. SIAM J. Appl. Dyn. Syst. 20, 1571–1595.Google Scholar
Creedon, R.P., Deconinck, B. & Trichtchenko, O. 2021 b High-frequency instabilities of a Boussinesq–Whitham system: a perturbative approach. Fluids 6 (4), 136.CrossRefGoogle Scholar
Curtis, C. & Deconinck, B. 2010 On the convergence of Hill's method. Maths Comput. 79 (269), 169–187.CrossRefGoogle Scholar
Deconinck, B. & Kutz, J.N. 2006 Computing spectra of linear operators using the Floquet–Fourier–Hill method. J. Comput. Phys. 219 (1), 296–321.CrossRefGoogle Scholar
Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141–167.CrossRefGoogle Scholar
Deconinck, B. & Trichtchenko, O. 2017 High frequency instabilities of small-amplitude solutions of Hamiltonian PDE's. Disc. Contin. Dyn. Syst. A 37 (3), 1323–1358.Google Scholar
Francius, M. & Kharif, C. 2006 Three-dimensional instabilities of periodic gravity waves in shallow water. J. Fluid Mech. 561, 417–437.CrossRefGoogle Scholar
Hur, V.M. & Yang, Z. 2020 Unstable Stokes waves. arXiv:2010.10766.Google Scholar
Kato, T. 1966 Perturbation Theory for Linear Operators. Springer.Google Scholar
Kharif, C. & Ramamonjiarisoa, A. 1990 On the stability of gravity waves on deep water. J. Fluid Mech. 218, 163–170.CrossRefGoogle Scholar
Levi-Civita, T. 1925 Determination rigoureuse des ondes permanentes dampleur finie. Math. Ann. 93 (1), 264–314.CrossRefGoogle Scholar
Longuet-Higgins, M.S. 1978 a The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics. Proc. R. Soc. Lond. A 360 (1703), 471–488.Google Scholar
Longuet-Higgins, M.S. 1978 b The instabilities of gravity waves of finite amplitude in deep water II. Subharmonics. Proc. R. Soc. Lond. A 360 (1703), 489–505.Google Scholar
MacKay, R.S. & Saffman, P.G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406 (1830), 115–125.Google Scholar
McLean, J.W. 1982 Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.CrossRefGoogle Scholar
Nekrasov, A.I. 1921 On waves of steady species. Math. Ivanovo Voznesensky Polytechnic. Inst. 3, 52–65.Google Scholar
Nguyen, H.Q. & Strauss, W.A. 2020 Proof of modulational instability of Stokes waves in deep water. arXiv:2007.05018.Google Scholar
Nicholls, D.P. 2007 Spectral stability of traveling water waves: analytic dependence of the spectrum. J. Nonlinear Sci. 17 (4), 369–397.CrossRefGoogle Scholar
Oliveras, K. 2009 Stability of periodic surface gravity water waves. PhD thesis, University of Washington.Google Scholar
Stokes, G.G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Struik, D. 1926 Determination rigoureuse des ondes irrotationnelles periodiques dans un canalá profondeur finie. Math. Ann. 95 (1), 595–634.CrossRefGoogle Scholar
Whitham, G.B. 1967 Non-linear dispersion of water waves. J. Fluid Mech. 27 (2), 399–412.CrossRefGoogle Scholar
Yuen, H.C. & Lake, B.M. 1980 Instabilities of waves in deep water. Annu. Rev. Fluid Mech. 12 (1), 303–334.CrossRefGoogle Scholar
Zakharov, V.E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.CrossRefGoogle Scholar