Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T17:45:23.563Z Has data issue: false hasContentIssue false
  • English
  • Français

Water waves in shallow channels of rapidly varying depth

Published online by Cambridge University Press:  26 April 2006

A. Nachbin  and
G. C. Papanicolaou
A. Nachbin
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA Present address: Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210-1174, USA.
G. C. Papanicolaou
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse the linear water-wave equations for shallow channels with arbitrary rapidly varying bottoms. We develop a theory for reflected waves based on an asymptotic analysis for stochastic differential equations when both the horizontal and vertical scales of the bottom variations are comparable to the depth but small compared to a typical wavelength so the shallow water equations cannot be used. We use the full, linear potential theory and study the reflection–transmission problem for time-harmonic (monochromatic) and pulse-shaped disturbances. For the monochromatic waves we give a formula for the expected value of the transmission coefficient which depends on depth and on the spectral density of the O(1) random depth perturbations. For the pulse problem we give an explicit formula for the correlation function of the reflection process. We compare our theory with numerical results produced using the boundary-element method. We consider several realizations of the bottom profile, let a Gaussian-shaped disturbance propagate over each topography sampled and record the reflected signal for each realization. Our numerical experiments produced reflected waves whose statistics are in good agreement with the theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 311 - 332
Copyright
© 1992 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.)

References

Abramowitz, M. & Stegun I. A. 1968 Handbook of Mathematical Functions. Dover.
Arnold L., Papanicolaou, G. & Wihstutz V. 1986 Asymptotic analysis of the Lyapounov exponent and rotation number of the random oscillator and applications. SIAM J. Appl. Maths 46, 427–450.Google Scholar
Asch M., Kohler W., Papanicolaou G., Postel, M. & White B. 1991 Frequency content of randomly scattered signals. SIAM Rev. 33, 519–625.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, 539–558.Google Scholar
Brebbia C. A., Telles, J. C. F. & Wrobel L. C. 1984 Boundary Element Technique. Springer.
Burridge R., Papanicolaou G. C., Sheng, P. & White B. 1989 Probing a random medium with a pulse. SIAM J. Appl. Maths 49, 582–607.Google Scholar
Carrier G. F. 1966 Gravity waves on water of variable depth. J. Fluid Mech. 24, 641–659.Google Scholar
Devillard P., Dunlop, F. & Souillard B. 1988 Localization of gravity waves on a channel with a random bottom. J. Fluid Mech. 186, 521–538.Google Scholar
Dias, F. & Vanden-Broeck J.-M. 1989 Open channel flows with submerged obstructions. J. Fluid Mech. 206, 155–170.Google Scholar
Hamilton J. 1977 Differential equations for long-period gravity waves on a fluid of rapidly varying depth. J. Fluid Mech. 83, 289–310.Google Scholar
Jaswon, M. A. & Symm G. T. 1977 Integral Equation Methods in Potential Theory and Elastostatics. Academic.
Keller J. B. 1958 Surface waves on water on non-uniform depth. J. Fluid Mech. 4, 607–614.Google Scholar
Kohler W. 1977 Power reflection at the input of a randomly perturbed rectangular waveguide. SIAM J. Appl. Maths 32 (3), 521–533.Google Scholar
Kohler, W. & Papanicolaou G. C. 1973 Power statistics for waves in one dimension and comparison with radiative transport theory I. J. Math. Phys. 14, 1733–1745.Google Scholar
Kohler, W. & Papanicolaou G. C. 1974 Power statistics for waves in one dimension and comparison with radiative transport theory II. J. Math. Phys. 15, 2186–2197.Google Scholar
Kreisel G. 1949 Surface waves. Q. Appl. Maths 7, 21–44.Google Scholar
Mei C. C. 1983 The Applied Dynamics of Ocean Surface Waves. John Wiley.
Mei, C. C. & Black J. L. 1969 Scattering of surface waves by rectangular obstacles in waters of finite depth. J. Fluid Mech. 38, 499–511.Google Scholar
Nachbin A. 1989 Reflection and transmission of water waves in shallow channels with rough bottoms. PhD thesis, New York University.
Nachbin, A. & Papanicolaou G. C. 1992 Boundary element method for the long-time water wave propagation over rapidly varying bottom topography. Intl J. Numer. Meth. Fluids (to appear).Google Scholar
Papanicolaou G. C. 1978 Asymptotic analysis of stochastic equations. In Studies in Probability (ed. M. Rosenblatt), pp. 111–179. MAA Studies in Mathematics, vol. 18.
Papanicolaou, G. C. & Kohler W. 1975 Asymptotic analysis of deterministic and stochastic equations with rapidly varying components. Commun. Math. Phys. 45, 217–232.Google Scholar
Rosales, R. R. & Papanicolaou G. C. 1983 Gravity waves in a channel with a rough bottom. Stud. Appl. Maths 68, 89–102.Google Scholar
Salmon J. R., Liu, P. L.-F. & Liggett J. A. 1980 Integral equation method for linear water waves. J. Hydraul. Div. ASCE 106 (HY12), 1995–2010.Google Scholar
Sheng P., Zang Z.-Q., White, B. & Papanicolaou G. 1986 Multiple-scattering noise in one dimension: universality through localization-length scaling. Phys. Rev. Lett. 57, 1000–1003.Google Scholar
White B., Sheng P. S., Zang, Z.-Q. & Papanicolaou G. 1987 Wave localization characteristics in the time domain. Phys. Rev. Lett. 59, 1918–1921.Google Scholar
Whitham G. B. 1974 Linear and Nonlinear Waves. John Wiley.