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The localization length of randomly scattered water waves

Published online by Cambridge University Press:  26 April 2006

André Nachbin
Affiliation:
Department of Mathematics, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA

Abstract

A theory for the reflection-transmission problem of linear water waves in shallow channels with large-amplitude, rapidly varying topographies is given in Nachbin & Papanicolaou (1992b). However, it is very difficult to extract quantitative information from the theory in the large-amplitude regime. In this work, theoretical parameters are evaluated numerically, through the use of a numerical Schwarz–Christoffel transformation and of a Monte Carlo simulation. This enables the theory to be applied to its full extent. As a result we calculate the localization length for any given type of random bottom topography. Additionally, the numerical conformal mapping provides further insight into depth effects arising from potential theory. Statistical results, for numerically generated reflected waves, are in very good agreement with the theory for both piecewise-linear and piecewise-constant topographies of large amplitude.

Type
Research Article
Copyright
© 1995 Cambridge University Press

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References

Ahlfors, L. V. 1979 Complex Analysis, 3rd Edn. McGraw Hill.
Asch, M., Kohler, W., Papanicolaou, G., Postel, M. & White, B. 1991 Frequency content of randomly scattered signals SIAM. Rev. 33, 519625.Google Scholar
Belzons, M., Guazzelli, E. & Parodi, O. 1988 Gravity waves on a rough bottom: experimental evidence of one-dimensional localization, J. Fluid Mech. 186, 539558.Google Scholar
Dalrymple, R. A. & Kirby, J. T. 1986 Water waves over ripples. J. Waterway Port Coastal Ocean Engng 112, 309319.Google Scholar
Devillard, P., Dunlop, F. & Souillard, B. 1988 Localization of gravity waves on a channel with a random bottom. J. Fluid Mech. 186, 521538.Google Scholar
Floryan, J. M. 1985 Conformal-mapping based coordinate generation method for channel flows. J. Comput. Phys. 58, 229245.Google Scholar
Gredeskul, S. A. & Kivshar, Y. S. 1992 Propagation and scattering of nonlinear waves in disordered systems. Phys. Rep. 216, 161.Google Scholar
Guazzelli, E., Rey, V. & Belzons, M. 1992 Higher-order Bragg reflection of gravity surface waves by periodic beds. J. Fluid Mech. 245, 301317.Google Scholar
Hamilton, J. 1977 Differential equations for long-period gravity waves on a fluid of rapidly varying depth. J. Fluid Mech. 83, 289310.Google Scholar
Hasselmann, K. 1966 Feynman diagrams and interaction rules of wave-wave scattering processes. Rev. Geophys. Space Phys. 4, 132.Google Scholar
Henrici, P. 1986 Applied and Computational Complex Analysis, Vol. 3. John Wiley.
Howell, L. H. & Trefethen, L. N. 1990 A modified Schwarz-Christoffel transformation for elongated regions. SIAM J. Sci. Statist. Comput. 11, 928949.Google Scholar
Kohler, W. 1977 Power reflection at the input of a randomly perturbed rectangular waveguide. SIAM J. Appl. Maths 32, 521533.Google Scholar
Long, R. B. 1973 Scattering of surface waves by an irregular bottom. J. Geophys. Res. 78, 78617870.Google Scholar
Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. John Wiley.
Mysak, L. A. 1978 Wave propagation in random media, with ocean applications. Rev. Geophys. Space Phys. 16, 233261.Google Scholar
Nachbin, A. 1993 Modelling of Water Waves in Shallow Channels. Computational Mechanics Publications, Southampton, UK.
Nachbin, A. & Papanicolaou, G. C. 1992a Boundary element method for the long-time water wave propagation over rapidly varying bottom topography. Intl J. Num. Meth. Fluids 14, 13471365.Google Scholar
Nachbin, A. & Papanicolaou, G. C. 1992b Water waves in shallow channels of rapidly varying depth. J. Fluid Mech. 241, 311332.Google Scholar
O'Donnel, S. & Rohklin, V. 1989 A fast algorithm for the numerical evaluation of conformal mappings. SIAM J. Sci. Statist. Comput. 10, 475487.Google Scholar
O'Hare, T. J. & Davies, A. G. 1990 A laboratory study of sandbar evolution. J. Coastal Res. 6, 531544.Google Scholar
Papanicolaou, G. C. & Keller, J. B. 1971 Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media. SIAM J. Appl. Maths 21, 287305.Google Scholar
Rey, V., Belzons, M. & Guazzelli, E. 1992 Propagation of surface gravity waves over a rectangular submerged bar. J. Fluid Mech. 235, 453479.Google Scholar
Rosales, R. R. & Papanicolaou, G. C. 1983 Gravity waves in a channel with a rough bottom. Stud. Appl. Maths 68, 89102.Google Scholar
Sridhar, K. P. & Davies, R. T. 1985 A Schwarz-Christoffel method for generating two-dimensional flow grids. Trans. ASME I: J. Fluids Engng 107, 330337.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley.