Hostname: page-component-f554764f5-rvxtl Total loading time: 0 Render date: 2025-04-21T11:52:41.928Z Has data issue: false hasContentIssue false
  • English
  • Français

HARMONIC RESONANCE OF SHORT-CRESTED GRAVITY WAVES ON DEEP WATER: ON THEIR PERSISTENCY

Part of: Incompressible inviscid fluids Hydrodynamic stability Special Collection in honour of Professor Anthony Roberts

Published online by Cambridge University Press:  03 December 2024

SIREL C. COLÓN USECHE Open the ORCID record for SIREL C. COLÓN USECHE [Opens in a new window] ,
SYLVERT PAUL Open the ORCID record for SYLVERT PAUL [Opens in a new window]  and
MANSOUR IOUALALEN Open the ORCID record for MANSOUR IOUALALEN [Opens in a new window]
SIREL C. COLÓN USECHE
Affiliation:
Fundación Venezolana de Investigaciones Sismológicas (Funvisis), Prolongación Calle Mara, El Llanito, Caracas 1073, Venezuela; e-mail: [email protected]
SYLVERT PAUL
Affiliation:
Université d’Etat d’Haïti, Faculté des Sciences, Laboratoire URGéo, 270, rue Monseigneur Guilloux, Port-au-Prince, Haiti; e-mail: [email protected]
MANSOUR IOUALALEN*
Affiliation:
IRD, CNRS, OCA, Géoazur, Univ. Côte d’Azur, 250 rue Albert Einstein, Sophia Antipolis, 06560 Valbonne, France
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Three-dimensional short-crested water waves are known to host harmonic resonances (HRs). Their existence depends on their sporadicity versus their persistency. Previous studies, using a unique yet hybrid solution, suggested that HRs exhibit sporadic instability, with the domain of instability exhibiting a bubble-like structure which experiences a loss of stability followed by a re-stabilization. Through the calculation of their complete multiple solution structures and normal forms, we discuss the particular harmonic resonance (2,6). The (2,6) resonance was chosen, not only because it is of lower order, and thus more likely to be significant, but also because it is representative of a fully developed three-dimensional water wave field. Its appearance, growth rate and persistency are discussed. On our converged solutions, we show that, at an incidence angle for which HR (2,6) occurs, the associated superharmonic instability is no longer sporadic. It was also found that the multiple solution operates a subcritical pitchfork bifurcation, so regardless of the value of the control parameter, the wave steepness, a stable branch of the solution always exists. As a result, the analysis reveals two competing processes that either provoke and enhance HRs, or inhibit their appearance and development.

Keywords

short-crested wavesharmonic resonancesstability

MSC classification

Primary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Secondary: 76E17: Interfacial stability and instability
Type
Research Article
Information
The ANZIAM Journal , Volume 67 , 2025 , e1
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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.)

Article purchase

Temporarily unavailable

References

Chappelear, J. E., On the description of the short-crested waves, Volume 125 of Techn. Memorandum (Beach Erosion Board, U. S. Army Corps. Engrs., New England, 1961) 1–26; https://erdc-library.erdc.dren.mil/server/api/core/bitstreams/81b728f8-7432-4ef8-e053-411ac80adeb3/content.Google Scholar
d’Alembert, J., Opuscules, Volume 5 of 35ième Mémoire, Réflexion sur les suites & les Racines imaginaire (Chez Briasson, Libraire, rue Saint-Jacques, à la Science, 1768), 171–183; http://gallica.bnf.fr/ark:/12148/bpt6k62424s.image.f192.Google Scholar
Gilewicz, J., Approximants de Padé, Volume 67 of Lect. Notes in Math. (Springer Verlag, Berlin–Heidelberg, 1978); doi:10.1007/BFb0061327.CrossRefGoogle Scholar
Hsu, J. R. C., Tsuchiya, Y. and Silvester, R., “Third approximation to short-crested waves”, J. Fluid Mech. 90 (1979) 179–196; doi:10.1017/S0022112079002135.CrossRefGoogle Scholar
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; https://api.semanticscholar.org/CorpusID:125338722.Google Scholar
Ioualalen, M. and Kharif, C., “On the subharmonic instabilities of steady three-dimensional deep water waves”, J. Fluid Mech. 262 (1994) 265–291; doi:10.1017/S0022112094000509.CrossRefGoogle Scholar
Ioualalen, M. and Okamura, M., “Structure of the instability associated with harmonic resonance of short-crested waves”, J. Phys. Oceanogr. 32 (2002) 1331–1337; doi:10.1175/1520-0485(2002)032<1331:SOTIAW>2.0.CO;2.2.0.CO;2>CrossRefGoogle Scholar
Moler, C. B. and Stewart, G. W., “An algorithm for generalized matrix eigenvalue problems”, SIAM J. Numer. Anal. 10 (1973) 241–256; doi:10.1137/0710024.CrossRefGoogle Scholar
Okamura, M., “Notes on short-crested waves in deep water”, J. Phys. Soc. Jpn. 65 (1996) 2841–2845; doi:10.1143/JPSJ.65.2841.CrossRefGoogle Scholar
Roberts, A. J., “Highly nonlinear short-crested water waves”, J. Fluid Mech. 135 (1983) 301–321; doi:10.1017/S0022112083003092.CrossRefGoogle Scholar
Roberts, A. J. and Schwartz, L. W., “The calculation of nonlinear short-crested wave”, Phys. Fluids 26 (1983) 2388–2392; doi:10.1063/1.864422.CrossRefGoogle Scholar
Wynn, P., “On a device for computing the em(Sn) transformation”, Math. Tables Other Aids Comput. 10 (1956) 91–96; doi:10.2307/2002183.CrossRefGoogle Scholar