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Comptes Rendus. Mécanique
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On the sub-harmonic instabilities of three-dimensional interfacial gravity–capillary waves in infinite depths
Comptes Rendus. Mécanique, Volume 350 (2022), pp. 191-203.

In order to analyze the stability of gravity–capillary short-crested waves propagating at the interface of two fluids of infinite depths, a numerical procedure has been developed using a collocation method. The three-dimensional wave is generated by an oblique reflection of a uniform wave train arriving at a vertical wall at some angle of incidence θ measured from the perpendicular to the wall. Fixing the reflection at angle θ=45°, and wave steepness at h=0.25, we studied the influence of the density ratio μ and the inverse Bond number δ on the unstable area and the growth rate. It was observed that the unstable regions and the maximum growth rate decrease when μ and δ increase. Also, we found that the dominant instability belongs to the Class Ib as in the case of free-surface gravity waves in deep water.

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Revised:
Accepted:
Published online:
DOI: 10.5802/crmeca.111
Keywords: Linear stability, Sub-harmonic perturbation, Three-dimensional waves, Gravity–capillary waves, Interfacial waves, Short-crested water waves
Sara Chikhi 1; Mohammed Debiane 1; Nabil Allalou 1

1 Université des Sciences et de la Technologie Houari Boumediene, Faculté de Physique, BP 32 El Alia, Alger 1611, Algérie
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     title = {On the sub-harmonic instabilities of three-dimensional interfacial gravity{\textendash}capillary waves in infinite depths},
     journal = {Comptes Rendus. M\'ecanique},
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     year = {2022},
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Sara Chikhi; Mohammed Debiane; Nabil Allalou. On the sub-harmonic instabilities of three-dimensional interfacial gravity–capillary waves in infinite depths. Comptes Rendus. Mécanique, Volume 350 (2022), pp. 191-203. doi : 10.5802/crmeca.111. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.111/

[1] M. Iouallalen; C. Kharif On the subharmonic instabilities of steady three-dimensional deep water waves, J. Fluid Mech., Volume 262 (1994), pp. 265-291 | Article

[2] M. Iouallalen; C. Kharif Fourth order approximation of short-crested waves, C. R. Acad. Sci. Paris, Volume 316 (1993), pp. 1193-1200

[3] S. Badulin; I. Shrira; C. Kharif; M. Iouallalen On two approaches to the problem of instability of short-crested water waves, J. Fluid Mech., Volume 303 (1995), pp. 297-326 | Article | MR: 1363798 | Zbl: 0945.76025

[4] V. E. Zakharov Stability of periodic waves of finite amplitude on the surface of a deep, fluid, J. Appl. Mech. Tech. Phys. (USSR), Volume 2 (1968), pp. 190-194 | Article

[5] M. Iouallalen; C. Kharif; A. J. Roberts Stability regimes of finite depth short-crested water waves, J. Phys. Oceanogr., Volume 29 (1999), pp. 2318-2331 | Article

[6] O. Kimmoun; M. Iouallalen; C. Kharif Instabilities of steep short-crested surface waves in deep water, Phys. Fluids, Volume 11 (1999), pp. 1679-1681 | Article | MR: 1690525 | Zbl: 1147.76432

[7] M. Iouallalen; M. Okamura Structure of the Instability associated with harmonic resonance of short-crested waves, J. Phys. Oceanogr., Volume 32 (2002), pp. 1331-1337 | Article | MR: 1901461

[8] N. Allalou; M. Debiane; C. Kharif On the superharmonic instability of nonlinear three-dimensional interfacial waves of two infinite layers, Eur. J. Mech. B Fluids, Volume 59 (2016), pp. 135-139 | Article | MR: 3532781 | Zbl: 1408.76061

[9] O. M. Phillips On the dynamics of unsteady gravity waves of finite amplitude, J. Fluid Mech., Volume 9 (1960), pp. 193-217 | Article | MR: 115465 | Zbl: 0094.41101

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