Hostname: page-component-7dd5485656-2pp2p Total loading time: 0 Render date: 2025-10-30T23:36:09.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Localization of gravity waves on a channel with a random bottom

Published online by Cambridge University Press:  21 April 2006

Pierre Devillard ,
François Dunlop  and
Bernard Souillard
Pierre Devillard
Affiliation:
Centre de Physique Théorique, Ecole Polytechnique, F-91128 Palaiseau, France
François Dunlop
Affiliation:
Centre de Physique Théorique, Ecole Polytechnique, F-91128 Palaiseau, France
Bernard Souillard
Affiliation:
Centre de Physique Théorique, Ecole Polytechnique, F-91128 Palaiseau, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a theoretical study of the localization phenomenon of gravity waves by a rough bottom in a one-dimensional channel. After recalling localization theory and applying it to the shallow-water case, we give the first study of the localization problem in the framework of the full potential theory; in particular we develop a renormalized-transfer-matrix approach to this problem. Our results also yield numerical estimates of the localization length, which we compare with the viscous dissipation length. This allows the prediction of which cases localization should be observable in and in which cases it could be hidden by dissipative mechanisms.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 186 , January 1988 , pp. 521 - 538
Copyright
© 1988 Cambridge University Press

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

Akkermans, E. & Maynard, R. 1984 Chains of random impedances. J. Phys. Paris 45, 1549.Google Scholar
Anderson, P. W. 1958 Absence of diffusion in some random lattices. Phys. Rev. 109, 1492.Google Scholar
Belzons, M., Devillard, P., Dunlop, F., Guazzelli, E., Parodi, O. & Souillard, B. 1988a Localization of gravity waves on a rough bottom, theory and experiments. Europhys. lett. (to appear).Google Scholar
Belzons, M., Guazzelli, E. & Parodi, O. 1988b Gravity waves on a rough bottom: experimental evidence of one-dimensional location. J. Fluid Mech. 186, 539–558.Google Scholar
Davies, A. G. & Heathershaw, A. D. 1984 Surface wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419.Google Scholar
Delyon, F., Simon, B. & Souillard, B. 1987 Exponential localization for a class of discrete and continuous generalized Schrödinger equations. Commun. Math. Phys. 109, 157.Google Scholar
Devillard, P. 1986 Thèse de Troisième Cycle, Université de Paris VI.
Devillard, P. & Souillard, B. 1986 Power decaying transmission for the non-linear Schrödinger equation in a disordered medium. J. Statist. Phys. 43, 423.Google Scholar
Fürstenberg, H. 1963 Trans. Am. Math. Soc. 108, 377.
Guazzelli, E. 1986 Thèse d'Etat, Marseille, Université de Saint Jérôme.
Guazzelli, E., Guyon, E. & Souillard, B. 1983 On the localization of shallow water waves by a random bottom. J. Phys. Lett. 44, L-837.Google Scholar
Hodges, C. H. 1982 Confinement of vibration by structural irregularity. J. Sound Vib. 82, 411.Google Scholar
Kotani, S. 1982 In Proc. Taniguchi Symp., Katata, 1982, p. 225.
Landau, L. & Lifschitz, E. 1971 Mécanique des Fluides. Moscow: Editions MIR.
Lee, P. A. & Ramakrishnan, T. V. 1985 Disordered electronic systems. Rev. Mod. Phys. 57, 287.Google Scholar
Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. Wiley-Interscience.
Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315.Google Scholar
Miles, J. W. 1967 Surface wave scattering matrix for a shelf. J. Fluid Mech. 28, 755.Google Scholar
Minami, N. 1986 An extension of Kotani's theorem for random generalized Sturm-Liouville operators. Commun. Math. Phys. 103, 387.Google Scholar
Mysak, L. A. 1978 Wave propagation in random media with oceanic applications. Rev. Geophys. Space Phys. 16, 233.Google Scholar
Newman, J. N. 1965 Propagation of water waves past long two-dimensional obstacles. J. Fluid Mech. 23, 23.Google Scholar
Papanicolaou, G. C. 1978 In CIME 1978 (ed. J. P. Cecconi), p. 193. Napoli: Liguori Editore.
Simon, B. & Souillard, B. 1984 Franco-American meeting on the mathematics of random and almost periodic potentials. J. Stat. Phys. 36, 273.Google Scholar
Souillard, B. 1987 Waves and electrons in inhomogeneous media. In Chance and Matter, Proc. Les Houches (ed. J. Souletie, J. Vannimenus & R. Stora). North-Holland.
Srokosz, M. A. & Evans, D. V. 1979 A theory for wave-power absorption by two independently oscillating bodies. J. Fluid Mech. 90, 337.Google Scholar
Thouless, D. J. 1979 Percolation and localization. In Ill-Condensed Matter, Proc. Les Houches. North-Holland.