Analytical Solution for Long-Wave Scattering by a Circular Island Mounted on a General Shoal
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 138, Issue 6
Abstract
In this paper, an analytical solution is given in the form of a Taylor series for long-wave scattering by a cylindrical island mounted on a general shoal, where the water depth in the shoal region can be a real constant plus a power function of the radial distance. In other words, the crest of the shoal is not restricted to be on the still water level. The distribution of singular points becomes complicated; therefore, the solution technique needs to be more sophisticated. Because of the generality of the location of the shoal crest, the analytical solutions find several classical analytical solutions to be its special cases, which include long-wave scattering by a cylindrical island mounted on an idealized paraboloidal shoal, a cylindrical island mounted on an idealized conical shoal, a cylindrical island mounted on an idealized shoal, and a cylindrical island located on a flat bottom. The present analytical solution covers a much wider range of problems, and therefore, is much more useful. Finally, the effect of the shoal size to the wave-scattering pattern is investigated using the present analytical solution.
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Acknowledgments
The first writer is supported by the Natural Science Foundation of China (10962001, 51149007), Guangxi Natural Science Foundation (2010GXNSFA013115, 2011GXNSFD018006) and Scientific Research Foundation of Guangxi Universities (201102ZD014). All the writers would like to gratefully acknowledge some very useful suggestions from three anonymous referees.
References
Berkhoff, J. C. W. (1972). “Computation of combined refraction-diffraction.” Proc., 13th Int. Conf. Coastal Eng., ASCE, Reston, VA, 471–490.
Berkhoff, J. C. W. (1976). “Mathematical models for simple harmonic linear water waves, wave diffraction and refraction.” Publication No. 163, Delft Hydraulic Laboratory, Delft, Netherlands.
Bettess, P., and Zienkiewicz, O. C. (1977). “Diffraction and refraction of surface waves using finite and infinite elements.” Int. J. Numer. Methods Eng., 11(8), 1271–1290.
Chamberlain, P. G., and Porter, D. (1995). “The modified mild-slope equation.” J. Fluid Mech., 291, 393–407.
Cheng, Y.-M. (2007). “Analytic solutions of the mild-slope equation.” Ph.D. thesis, National Cheng Kung Univ., Taiwan.
Cheng, Y.-M. (2011). “A new solution for waves incident to a circular island on an axi-symmetric shoal.” Ocean Eng., 38(17–18), 1916–1924.
Dean, R. G. (1964). “Long wave modification by linear transitions.” J. Wtrwy. and Harbors Div., 90(1), 1–29.
Homma, S. (1950). “On the behavior of seismic sea waves around circular island.” Geophys. Mag., 21, 199–208.
Houston, J. R. (1981). “Combined refraction and diffraction of short waves using the finite element method.” Appl. Ocean Res., 3(4), 163–170.
Hsiao, S. S., Chang, C. M., and Wen, C. C. (2009). “Solution for wave propagation through a circular cylinder mounted on different topography ripple-bed profile shoals using DRBEM.” Eng. Anal. Bound. Elem., 33(11), 1246–1257.
Hsiao, S. S., Chang, C. M., and Wen, C. C. (2010). “An analytical solution to the modified mild-slope equation for waves propagating around a circular conical island.” J. Mar. Sci. Technol., 18(4), 520–529.
Hunt, J. N. (1979). “Direct solution of wave dispersion equation.” J. Waterway, Port, Coastal, Oc. Div., 105(4), 457–459.
Jonsson, I. G., Skovgaard, O., and Brink-Kjaer, O. (1976). “Diffraction and refraction calculations for waves incident on an island.” J. Mar. Res., 34(3), 469–496.
Jung, T. H., and Cho, Y.-S. (2009). “Analytical approach for long wave solution to an arbitrarily varying topography.” J. Coast. Res., 25(1), 216–223.
Jung, T. H., and Lee, C. (2012). “Analytical solutions for long waves on a circular island with combined topographies.” Wave Motion, 49(1), 152–164.
Jung, T. H., Lee, C., and Cho, Y.-S. (2010). “Analytical solutions for long waves over a circular island.” Coast. Eng., 57(4), 440–446.
Jung, T. H., and Suh, K. D. (2007). “An analytic solution to the mild slope equation for waves propagating over an axi-symmetric pit.” Coast. Eng., 54(12), 865–877.
Jung, T. H., and Suh, K. D. (2008). “An analytical solution to the extended mild-slope equation for long waves propagating over an axi-symmetric pit.” Wave Motion, 45(6), 835–845.
Jung, T. H., Suh, K. D., Lee, S. O., and Cho, Y.-S. (2008). “Linear wave reflection by trench with various shapes.” Ocean Eng., 35(11–12), 1226–1234.
Lamb, H. (1932). Hydrodynamics, 6th Ed. Dover, New York.
Lie, V., and Tørum, A. (1991). “Ocean waves over shoals.” Coast. Eng., 15(5–6), 545–562.
Lin, P., and Liu, H.-W. (2005). “Analytical study of linear long-wave reflection by a two-dimensional obstacle of general trapezoidal shape.” J. Eng. Mech., 131(8), 822–830.
Lin, P., and Liu, H.-W. (2007). “Scattering and trapping of wave energy by a submerged truncated paraboloidal shoal.” J. Waterway, Port, Coastal, Ocean Eng., 133(2), 94–103.
Liu, H.-W., and Li, Y. B. (2007). “An analytical solution for long-wave scattering by a submerged circular truncated shoal.” J. Eng. Math., 57(2), 133–144.
Liu, H.-W., and Lin, P. (2005). “Discussion of ‘Wave transformation by two-dimensional bathymetric anomalies with sloped transitions' by Christopher J. Bender and Robert G. Dean.” Coast. Eng., 52(2), 197–200.
Liu, H.-W., and Lin, P. (2007). “An analytic solution for wave scattering by a circular cylinder mounted on a conical shoal.” Coast. Eng. J., 49(4), 393–416.
Liu, H.-W., Lin, P., and Shankar, N. J. (2004). “An analytical solution of the mild-slope equation for waves around a circular island on a paraboloidal shoal.” Coast. Eng., 51(5–6), 421–437.
Liu, H.-W., and Xie, J.-J. (2011). “Discussion of ‘Analytic solution of long wave propagation over a submerged hump' by Niu and Yu.” Coast. Eng., 58(9), 948–952.
MacCamy, R. C., and Fuchs, R. A. (1954). “Wave forces on piles: A diffraction theory.” Tech. Mem. Vol. 69. U.S. Army Corps of Engineering, Beach Erosion Board, Washington, D.C.
Massel, S. R. (1996). Ocean surface waves: Their physics and prediction, World Scientific, Singapore.
Mei, C. C. (1989). The applied dynamics of ocean surface waves, World Scientific, Singapore.
Niu, X., and Yu, X. (2011a). “Analytic solution of long wave propagation over a submerged hump.” Coast. Eng., 58(2), 143–150.
Niu, X., and Yu, X. (2011b). “Long wave scattering by a vertical cylinder with idealized scour pit.” J. Waterway, Port, Coastal, Ocean Eng., 137(6), 279–285.
Niu, X., and Yu, X. (2011c). “Analytical study on long wave refraction over a dredge excavation pit.” Wave Motion, 48(3), 259–267.
Suh, K.-D., Jung, T.-H., and Haller, M. C. (2005). “Long waves propagating over a circular bowl pit.” Wave Motion, 42(2), 143–154.
Tsay, T.-K., and Liu, P. L.-F. (1983). “A finite element model for wave refraction and diffraction.” Appl. Ocean Res., 5(1), 30–37.
Vastano, A. C., and Reid, R. O. (1967). “Tsunami response for islands: Verification of a numerical procedure.” J. Mar. Res., 25(2), 129–139.
Xie, J.-J., Liu, H.-W., and Lin, P. (2011). “Analytical solution for long wave reflection by a rectangular obstacle with two scour trenches.” J. Eng. Mech., 137(12), 919–930.
Yu, X., and Zhang, B. (2003). “An extended analytical solution for combined refraction and diffraction of long waves over circular shoals.” Ocean Eng., 30(10), 1253–1267.
Zhang, Y. L., and Zhu, S.-P. (1994). “New solutions for the propagation of long water waves over variable depth.” J. Fluid Mech., 278, 391–406.
Zhu, S.-P. (1993). “A new DRBEM model for wave refraction and diffraction.” Eng. Anal. Bound. Elem., 12(4), 261–274.
Zhu, S.-P., and Harun, F.-N. (2009). “An analytical solution for long wave refraction over a circular hump.” J. Appl. Math. Comput., 30(1–2), 315–333.
Zhu, S.-P., Liu, H.-W., and Chen, K. (2000). “A general DRBEM model for wave refraction and diffraction.” Eng. Anal. Bound. Elem., 24(5), 377–390.
Zhu, S.-P., and Zhang, Y. L. (1996). “Scattering of long waves around a circular island mounted on a conical shoal.” Wave Motion, 23(4), 353–362.
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Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 138 • Issue 6 • November 2012
Pages: 425 - 434
Copyright
© 2012 American Society of Civil Engineers.
History
Received: Jan 4, 2011
Accepted: Mar 1, 2012
Published online: Mar 3, 2012
Published in print: Nov 1, 2012
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