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The Green’s function of the mild-slope equation: The case of a monotonic bed profile. (English) Zbl 1074.76598
Summary: In the present work the Green’s function of the mild-slope and the modified mild-slope equations is studied. An effective numerical Fourier inversion scheme has been developed and applied to the construction and study of the source-generated water-wave potential over an uneven bottom profile with different depths at infinity. In this sense, the present work is a prerequisite to the study of the diffraction of water waves by localized bed irregularities superimposed over an uneven bottom. In the case of a monotonic bed profile, the main characteristics of the far-field are: (i) the formation of a shadow zone with an ever expanding width, which is located along the bottom irregularity, and (ii) in each of the two sectors not including the bottom irregularity the asymptotic behavior of the wave field approaches the form of an outgoing cylindrical wave, propagating with an amplitude of order \(O(R^{-1/2})\), where \(R\) is the distance from the source, and wavelength corresponding to the sector-depth at infinity. Moreover, the weak wave system propagating in the shadow zone is of order \(O(R^{-3/2})\), and along the bottom irregularity consists of the superposition of two outgoing waves with wavelengths corresponding to the two depths at infinity.

76Q05 Hydro- and aero-acoustics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
86A05 Hydrology, hydrography, oceanography
Full Text: DOI
[1] L.B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, Prentice-Hall, Englewood Cliffs, NJ, 1973. · Zbl 0050.41802
[2] L.M. Brekhovskikh, O.A. Godin, Acoustics of Layered Media II. Point Sources and Bounded Beams, Springer, Berlin, 1992. · Zbl 0753.76003
[3] C.C. Mei, The Applied Dynamics of Ocean Surface Waves, World Scientific, Singapore, 1989 (2nd reprint, 1994). · Zbl 0991.76003
[4] Massel, S., Extended refraction – diffraction equations for surface waves, Coastal eng., 19, 97-126, (1993)
[5] Chamberlain, P.G.; Porter, D., The modified mild-slope equation, J. fluid mech., 291, 393-407, (1995) · Zbl 0843.76006
[6] Miles, J.W.; Chamberlain, P.G., Topographical scattering of gravity waves, J. fluid mech., 361, 175-188, (1998) · Zbl 0924.76014
[7] Booij, N., A note on the accuracy of the mild-slope equation, Coastal eng., 7, 191-203, (1983)
[8] Porter, D.; Staziker, D.J., Extensions of the mild-slope equation, J. fluid mech., 300, 367-382, (1995) · Zbl 0848.76010
[9] Athanassoulis, G.A.; Belibassakis, K.A., A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions, J. fluid mech., 389, 275-301, (1999) · Zbl 0959.76009
[10] Givoli, D., Non-reflecting boundary conditions, J. comput. phys., 94, 1-29, (1991) · Zbl 0731.65109
[11] D. Givoli, Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992. · Zbl 0788.76001
[12] Tsynkov, S.V., Numerical solution of problems in unbounded domains, Appl. numer. math., 27, 465-532, (1998) · Zbl 0939.76077
[13] J.C.W. Berkhoff, Linear wave propagation problems and the finite element method, in: R.H. Gallagher, J.T. Oden, C. Taylor, O.C. Zienkiewicz (Eds.), Finite Elements in Fluids, Vol. 1, Wiley, New York, 1975, pp. 251-264.
[14] Xu, Y., Radiation condition and scattering problem for time-harmonic acoustic waves in a stratified medium with a nonstratified inhomogeneity, IMA J. appl. math., 54, 9-29, (1995) · Zbl 0833.76070
[15] Holford, R., Elementary source-type solution of the reduced wave equation, J. acoust. soc. am., 70, 1427-1436, (1981) · Zbl 0478.35030
[16] Li, Y.L.; Liu, C.H.; Franke, S.J., Three-dimensional green’s function for wave propagation in a linearly inhomogeneous medium — the exact analytic solution, J. acoust. soc. am., 87, 2285-2291, (1990)
[17] Manolis, G.D.; Shaw, R.P., Green’s function for the vector wave equation in a mildy heterogeneous continuum, Wave motion, 24, 59-83, (1996) · Zbl 0954.74522
[18] Raspet, R.; Lee, S.W.; Kuester, E.; Chang, D.C.; Richards, W.F.; Gilbert, R.; Bong, N., A fast-field program for sound propagation in a layered atmosphere above an impedance ground, J. acoust. soc. am., 77, 345-352, (1985)
[19] Richards, T.L.; Attenborough, K., Accurate FFT-based Hankel transforms for predictions of outdoor sound propagation, J. sound vibr., 109, 157-167, (1986) · Zbl 1235.76116
[20] Di Napoli, F.R.; Deavenport, R.L., Theoretical and numerical green’s function solution in a plane layered medium, J. acoust. soc. am., 67, 92-105, (1980) · Zbl 0421.73030
[21] F.B. Jensen, W.A. Kuperman, M.B. Porter, H. Schmidt, Computational Ocean Acoustics, AIP Press, New York, 1994. · Zbl 1234.76003
[22] A.G. Ramm, Multidimensional Inverse Scattering Problems, Longman, New York, 1992. · Zbl 0746.35056
[23] B. Noble, Methods Based on the Wiener-Hopf Technique, Chelsea, New York, 1988. · Zbl 0657.35001
[24] N. Bleinstein, Mathematical Methods for Wave Phenomena, Academic Press, New York, 1984.
[25] Y.A. Brychkov, A.P. Prudnikov, Integral Transforms of Generalized Functions, Gordon and Breach, London, 1989. · Zbl 0729.46016
[26] V.S. Vladimirov, Generalized Functions in Mathematical Physics, MIR, Moscow, 1979. · Zbl 0515.46033
[27] D.C. Champeney, A Handbook of Fourier Theorems, Cambridge University Press, Cambridge, 1987. · Zbl 0632.42001
[28] Y.V. Sidorov, M.V, Fedoryuk, M.I. Shabunin, Lectures on the Theory of Complex Functions, MIR, Moscow, 1985. · Zbl 0519.30001
[29] I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals Series and Products, Academic Press, New York, 1965. · Zbl 0918.65002
[30] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1964. · Zbl 0171.38503
[31] A.V. Oppenheim, R.W. Schafer, Discrete-Time Signal-Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989. · Zbl 0676.42001
[32] N.C. Geckinkli, D. Yavuz, Discrete Fourier Transformation and its Applications to Power Spectra Estimation, Elsevier, Amsterdam, 1983.
[33] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. · Zbl 0064.33002
[34] G.A. Athanassoulis, K.A. Belibassakis, Water-wave Green’s function for a 3D uneven-bottom problem with different depths at x→+∞ and x→−∞, in: Proceedings of the IUTAM Symposium on Computational Methods for Unbounded Domains, Fluid Mechanics and its Applications, Vol. 39, Kluwer Academic Publishers, Dordrecht, 1998, pp. 21-32. · Zbl 0969.76010
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