×

zbMATH — the first resource for mathematics

Self-channelling of surface water waves in the presence of an additional surface pressure. (English) Zbl 0942.76020
Summary: We examine the nonlinear instability of long periodic waves of a small amplitude on a surface of a water layer of finite depth either subjected to surface tension or in the presence of an elastic ice-sheet floating on the water surface. Wave processes in both cases are described by a model equation which generalizes the Kadomtsev-Petviashvili equation to the presence of higher-order dispersive effects. The treatment is based on the analysis of the Benjamin-Feir type instability, governed by the Davey-Stewartson equations for slowly varying in time and space complex amplitudes of periodic waves. The homogeneous periodic wave is shown to be unstable under perturbations transversal to the direction of wave propagation. Such kind of instability leads to a formation of a lattice of essential wave-guides, i.e. waves periodic in the direction of propagation and localized in the transversal direction. We discuss some natural effects of ice damage, which can be explained with the help of such an instability.

MSC:
76E17 Interfacial stability and instability in hydrodynamic stability
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76E30 Nonlinear effects in hydrodynamic stability
86A40 Glaciology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Benjamin, T.B., The solitary wave with surface tension, Q. appl. maths, 40, 231-234, (1982) · Zbl 0489.76029
[2] Grimshaw, R.; Malomed, B.; Benilov, E., Solitary wave with damped oscillatory tails: an analysis of the fifth order Korteweg-de Vries equation, Physica D, 77, 473-485, (1994) · Zbl 0824.35113
[3] Forbes, L.K., Surface waves of large amplitude beneath an elastic sheet. part 1. high order series solution, J. fluid mech., 188, 409-428, (1986) · Zbl 0607.76015
[4] Hǎrǎgu¢, M.; Kirchgässner, K., Breaking the dimension of a steady wave: some examples, (), 119-129 · Zbl 0837.35131
[5] Hǎrǎgu¢-Courcelle, M.; Il’ichev, A., Three-dimensional solitary waves in the presence of additional surface effects, Eur. J. mech. B/fluids, 17, 739-768, (1998) · Zbl 0933.76012
[6] Il’ichev, A.; Kirchgässner, K., Nonlinear water waves beneath an elastic ice sheet, ()
[7] Iooss, G.; Kirchgässner, K., Bifurcation d’ondes solitaires en présence d’une faible tension superficielle, C.R. acad. sci. Paris, 311, 265-268, (1990) · Zbl 0705.76020
[8] Iooss, G.; Kirchgässner, K., Water waves for small surface tension: an approach via normal form, (), 267-299 · Zbl 0767.76004
[9] Karpman, V.I., Nonlinear waves in dispersive media, (1975), Pergamon Press
[10] Müller, A.; Ettema, B., Dynamic response of an icebreaker hull to ice breaking, (), 287-296
[11] Squire, V.A., On the critical angle for Ocean waves entering Shore fast ice, Cold. reg. sci. tech., 10, 59-68, (1984)
[12] Vanderbauwhede, A.; Iooss, G., Center manifold theory in infinite dimensions, Dynamics reported, 1, 125-163, (1992), New Series · Zbl 0751.58025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.