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Water waves in shallow channels of rapidly varying depth

  • A. Nachbin (a1) (a2) and G. C. Papanicolaou (a1)


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.



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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, 427450.
Asch M., Kohler W., Papanicolaou G., Postel, M. & White B. 1991 Frequency content of randomly scattered signals. SIAM Rev. 33, 519625.
Belzons M., Guazzelli, E. & Parodi O. 1988 Gravity waves on a rough bottom: experimental evidence of one-dimensional localization. J. Fluid Mech. 186, 539558.
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, 582607.
Carrier G. F. 1966 Gravity waves on water of variable depth. J. Fluid Mech. 24, 641659.
Devillard P., Dunlop, F. & Souillard B. 1988 Localization of gravity waves on a channel with a random bottom. J. Fluid Mech. 186, 521538.
Dias, F. & Vanden-Broeck J.-M. 1989 Open channel flows with submerged obstructions. J. Fluid Mech. 206, 155170.
Hamilton J. 1977 Differential equations for long-period gravity waves on a fluid of rapidly varying depth. J. Fluid Mech. 83, 289310.
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, 607614.
Kohler W. 1977 Power reflection at the input of a randomly perturbed rectangular waveguide. SIAM J. Appl. Maths 32 (3), 521533.
Kohler, W. & Papanicolaou G. C. 1973 Power statistics for waves in one dimension and comparison with radiative transport theory I. J. Math. Phys. 14, 17331745.
Kohler, W. & Papanicolaou G. C. 1974 Power statistics for waves in one dimension and comparison with radiative transport theory II. J. Math. Phys. 15, 21862197.
Kreisel G. 1949 Surface waves. Q. Appl. Maths 7, 2144.
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, 499511.
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).
Papanicolaou G. C. 1978 Asymptotic analysis of stochastic equations. In Studies in Probability (ed. M. Rosenblatt), pp. 111179. 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, 217232.
Rosales, R. R. & Papanicolaou G. C. 1983 Gravity waves in a channel with a rough bottom. Stud. Appl. Maths 68, 89102.
Salmon J. R., Liu, P. L.-F. & Liggett J. A. 1980 Integral equation method for linear water waves. J. Hydraul. Div. ASCE 106 (HY12), 19952010.
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, 10001003.
White B., Sheng P. S., Zang, Z.-Q. & Papanicolaou G. 1987 Wave localization characteristics in the time domain. Phys. Rev. Lett. 59, 19181921.
Whitham G. B. 1974 Linear and Nonlinear Waves. John Wiley.
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Water waves in shallow channels of rapidly varying depth

  • A. Nachbin (a1) (a2) and G. C. Papanicolaou (a1)


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