Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-01T09:23:12.777Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral data for travelling water waves: singularities and stability

Published online by Cambridge University Press:  10 April 2009

DAVID P. NICHOLLS
DAVID P. NICHOLLS*
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
*
†Email address for correspondence: nicholls@math.uic.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we take up the question of the spectral stability of travelling water waves from a new point of view, namely that the spectral data of the water-wave operator linearized about fully nonlinear Stokes waves is analytic as a function of a height parameter. This observation was recently made rigorous by the author using a boundary perturbation approach which is amenable to approximation by a stable high-order numerical method. Using this algorithm, we investigate, for both super- and sub-harmonic disturbances, the evolution of the spectrum, in particular the ‘first collision’ of eigenvalues and the ‘smallest singularity’ in the perturbation expansion. The former is studied in response to MacKay & Saffman's (1986) work on the water-wave problem which demonstrated that instability can only arise after the collision of two eigenvalues of opposite Krein signature. However, we present results which show, quite explicitly, that eigenvalue collision (even of opposite Krein signature) is insufficient to conclude instability. With this in mind, we have identified a new criterion for the loss of spectral stability, namely the appearance of a singularity in the expansion of the spectral data (as a function of the height parameter mentioned above). We give some heuristic reasons why this should be so, and then provide complete numerical spectral stability results for four representative depths, two above (h = ∞, 2) and two below (h = 1, 1/2) Benjamin's (1967) critical value, hc ≈ 1.363, above which the Benjamin–Feir instability emerges. We find that the strongest (two-dimensional) instability appears to be among the long waves, but we notice that there is a sharp difference between ‘shallow-water’ and ‘deep-water’ waves in that first eigenvalue collision and smallest expansion singularity are synonymous for shallow water, while this is not so in deep water where ‘windows of stability’ beyond the first eigenvalue collision exist.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 624 , 10 April 2009 , pp. 339 - 360
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Baker, G. A. Jr. & Graves-Morris, P. 1996 Padé Approximants, 2nd ed.Cambridge University Press.CrossRefGoogle Scholar
Benjamin, T. B. 1967 Instability of periodic wavetrains in nonlinear dispersive systems. Proc. R. Soc. Lond. A 299 (1456), 59–76.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
Bridges, T. J. & Mielke, A. 1995 A proof of the Benjamin–Feir instability. Arch. Rational Mech. Anal. 133 (2), 145–198.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.CrossRefGoogle Scholar
Chandezon, J., Maystre, D. & Raoult, G. 1980 A new theoretical method for diffraction gratings and its numerical application. J. Opt. 11 (7), 235–241.CrossRefGoogle Scholar
Craig, W. & Nicholls, D. P. 2000 Traveling two and three dimensional capillary–gravity water waves. SIAM J. Math. Anal. 32 (2), 323–359.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
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary–gravity waves. In Annual Review of Fluid Mech, vol. 31, pp. 301–346. Annual Reviews.Google Scholar
Francius, M. & Kharif, C. 2003 On the disappearance of the lowest-order instability for steep gravity waves in finite depth. Phys. Fluids 15 (8), 2445–2448.CrossRefGoogle Scholar
Francius, M. & Kharif, C. 2006 Three-dimensional instabilities of periodic gravity waves in shallow water. J. Fluid Mech. 561, 417–437.CrossRefGoogle Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26.Google Scholar
Groves, M. D. 2007 Three-dimensional travelling gravity–capillary water waves. GAMM-Mitt. 30 (1), 8–43.CrossRefGoogle Scholar
Kharif, C. 1987 A comparison between 3 and 2 dimensional instabilities of very steep gravity-waves. J. Mech. Theor. Appl. 6 (6), 843–864.Google Scholar
Kharif, C. & Ramamonjiarisoa, A. 1988 Deep-water gravity–wave instabilities at large steepness. Phys. of Fluids 31 (5), 1286–1288.CrossRefGoogle Scholar
Lamb, H. 1993 Hydrodynamics, 6th ed.Cambridge University Press.Google Scholar
Levi-Civita, T. 1925 Détermination rigoureuse des ondes permanentes d'ampleur finie. Math. Ann. 93, 264–314.CrossRefGoogle Scholar
Li, L., Chandezon, J., Granet, G. & Plumey, J. P. 1999 Rigorous and efficient grating-analysis method made easy for optical engineers. Appl. Opt. 38 (2), 304–313.CrossRefGoogle ScholarPubMed
Longuet-Higgins, M. S. 1978 a The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. R. Soc. Lond. Ser. 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. Ser. A 360 (1703), 489–505.Google Scholar
MacKay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406, 115–125.Google Scholar
Marchant, T. R. & Roberts, A. J. 1987 Properties of short-crested waves in water of finite depth. J. Austral. Math. Soc. Ser. B 29 (1), 103–125.CrossRefGoogle Scholar
McLean, J. W. 1982 a Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331–341.CrossRefGoogle Scholar
McLean, J. W. 1982 b Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.CrossRefGoogle Scholar
McLean, J. W., Ma, Y. C., Martin, D. U., Saffman, P. G. & Yuen, H. C. 1981 Three-dimensional instability of finite-amplitude water waves. Phys. Rev. Lett. 46 (13), 817–820.CrossRefGoogle Scholar
Mielke, A. 1997 Instability and stability of rolls in the Swift–Hohenberg equation. Comm. Math. Phys. 189 (3), 829–853.CrossRefGoogle Scholar
Nicholls, D. P. 2007 a Boundary perturbation methods for water waves. GAMM–Mitt. 30 (1), 44–74.CrossRefGoogle Scholar
Nicholls, D. P. 2007 b Spectral stability of traveling water waves: analytic dependence of the spectrum. J. Nonlin. Sci. 17 (4), 369–397.CrossRefGoogle Scholar
Nicholls, D. P. & Reitich, F. 2001 a A new approach to analyticity of Dirichlet–Neumann operators. Proc. R. Soc. Edinburgh Sect. A 131 (6), 1411–1433.CrossRefGoogle Scholar
Nicholls, D. P. & Reitich, F. 2001 b Stability of high-order perturbative methods for the computation of Dirichlet–Neumann operators. J. Comput. Phys. 170 (1), 276–298.CrossRefGoogle Scholar
Nicholls, D. P. & Reitich, F. 2003 Analytic continuation of Dirichlet–Neumann operators. Numer. Math. 94 (1), 107–146.CrossRefGoogle Scholar
Nicholls, D. P. & Reitich, F. 2004 a Shape deformations in rough surface scattering: cancellations, conditioning, and convergence. J. Opt. Soc. Am. A 21 (4), 590–605.CrossRefGoogle ScholarPubMed
Nicholls, D. P. & Reitich, F. 2004 b Shape deformations in rough surface scattering: improved algorithms. J. Opt. Soc. Am. A 21 (4), 606–621.CrossRefGoogle ScholarPubMed
Nicholls, D. P. & Reitich, F. 2005 On analyticity of traveling water waves. Proc. R. Soc. Lond. A 461 (2057), 1283–1309.Google Scholar
Nicholls, D. P. & Reitich, F. 2006 Rapid, stable, high-order computation of traveling water waves in three dimensions. Eur. J. Mech. B Fluids 25 (4), 406–424.CrossRefGoogle Scholar
Phillips, N. A. 1957 A coordinate system having some special advantages for numerical forecasting. J. Atmosph. Sci. 14 (2), 184–185.Google Scholar
Reed, M. & Simon, B. 1978 Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press (Harcourt Brace Jovanovich).Google Scholar
Roberts, A. J. 1983 Highly nonlinear short-crested water waves. J. Fluid Mech. 135, 301–321.CrossRefGoogle Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes' expansion for gravity waves. J. Fluid Mech. 62, 553–578.CrossRefGoogle Scholar
Struik, D. 1926 Détermination rigoureuse des ondes irrotationnelles périodiques dans un canal à profondeur finie. Math. Ann. 95, 595–634.CrossRefGoogle Scholar
Zakharov, V. 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