Analytical Solution for Long-Wave Reflection by a Rectangular Obstacle with Two Scour Trenches
Publication: Journal of Engineering Mechanics
Volume 137, Issue 12
Abstract
In this paper, linear long-wave reflection by a rectangular obstacle with two scour trenches of power function profile is explored. A closed-form analytical solution in terms of the first and second kinds of Bessel functions is obtained, which finds two classic analytical solutions as its special cases, i.e., the wave reflection from a rectangular obstacle and from an infinite step. The phenomenon of zero reflection coefficient for a single rectangular obstacle with the same depths in front of and behind the obstacle still remains for a rectangular obstacle with two scour trenches as long as the bathymetry is symmetrical about the obstacle. The periodicity of the reflection coefficient as a function of the relative length of the middle rectangular obstacle disappears if two scour trenches are attached to the middle rectangular obstacle. Finally, the wave reflection by a rectangular obstacle with two scour trenches generally increases when the trenches become wide and deep. The wave reflection by a degenerated single slope increases when the slope becomes deep.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The first author would like to acknowledge the support by the Innovation Project of Guangxi Graduate Education (UNSPECIFIED2010106080701M369) and the Innovation Project of Guangxi University for Nationalities (UNSPECIFIEDgxun-chx2010085). The second author is supported by the Natural Science Foundation of P.R. China (NNSFC10962001) and Guangxi Natural Science Foundation (UNSPECIFIED2010GXNSFA013115, UNSPECIFIED2011GXNSFD018006). The third author is supported by the Natural Science Foundation of P.R. China (NNSFC51061130547).
References
Chang, H.-K., and Liou, J.-C. (2007). “Long wave reflection from submerged trapezoidal breakwaters.” Ocean Eng., 34(1), 185–191.
Cho, Y.-S., and Lee, C. (2000). “Resonant reflection of waves over sinusoidally varying topographies.” J. Coastal Res., 16(3), 870–876.
Cho, Y.-S., Lee, J.-I., and Kim, Y.-T. (2004). “Experimental study of strong reflection of regular water waves over submerged breakwaters in tandem.” Ocean Eng., 31(10), 1325–1335.
Dean, R. G. (1964). “Long wave modification by linear transitions.” J. Waterway, Harb. Coastal Eng. Div., 90(1), 1–29.
Dingemans, M. W. (1997). Water wave propagation over uneven bottoms, Part 1: Linear wave propagation, World Scientific, Singapore.
Epstein, P. S. (1930). “Reflection of waves in an inhomogeneous absorbing medium.” Proc. Natl. Acad. Sci. U.S.A., 16(10), 627–637.
Fujima, K., Yuliadi, D., Goto, C., Hayashi, K., and Shigemura, T. (1995). “Characteristics of long waves trapped by conical island.” Coast. Eng. Jpn., 38(2), 111–132.
Homma, S. (1950). “On the behaviour of seismic sea waves around circular island.” Geophys. Mag., 21, 199–208.
Jeffreys, H. (1944). “Motion of waves in shallow water. Note on the offshore bar problem and reflexion from a bar.” Wave Rep. 3, Ministry of Supply, London.
Jung, T.-H., and Cho, Y.-S. (2009). “Analytical approach for long wave solution to an arbitrarily varying topography.” J. Coastal Res., 25(1), 216–223.
Jung, T. H., Lee, C., and Cho, Y. S. (2010). “Analytical solutions for long waves over a circular island.” Coastal Eng., 57(4), 440–446.
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.
Kajiura, K. (1961). “On the partial reflection of water waves passing over a bottom of variable depth.” Proc., Tsunami Committee Meeting, Vol. 24, International Union of Geodesy and Geophysics, Karlsruhe, Germany, 206–230.
Kirby, J. T., and Dalrymple, R. A. (1983). “Propagation of obliquely incident water waves over a trench.” J. Fluid Mech., 133, 47–63.
Lamb, H. (1932). Hydrodynamics, 6th Ed., Dover, New York.
Lin, P. Z., 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.
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. Z. (2005). “Discussion of ‘Wave transformation by two-dimensional bathymetric anomalies with sloped transitions’ [Coast. Eng. 50 (2003) 6184].” Coastal Eng., 52(2), 197–200.
Liu, H.-W., and Xie, J. J. (2011). “Discussion of ‘Analytic solution of long wave propagation over a submerged hump’ by Niu and Yu (2011).” Coastal Eng., 58(9), 948–952.
MacCamy, R. C., and Fuchs, R. A. (1954). “Wave forces on piles: A diffraction theory.” Technical Memorandum 69, U.S. Army Corps of Engineers, Beach Erosion Board, Washington, DC.
McIver, P. (1998). “The dispersion relation and eigenfunction expansions for water waves in a porous structure.” J. Eng. Math., 34(3), 319–334.
Mei, C. C. (1989). The applied dynamics of ocean surface waves, World Scientific, Singapore.
Michalsen, D. R., Haller, M. C., and Suh, K. D. (2008). “Wave reflection from nearshore depressions.” J. Waterway, Port, Coastal, Ocean Eng., 134(1), 1–11.
Newman, J. N. (1965). “Propagation of water waves past long dimensional obstacles.” J. Fluid Mech., 23(01), 23–29.
Niu, X. J., and Yu, X. P. (2011a). “Analytic solution of long wave propagation over a submerged hump.” Coastal Eng., 58(2), 143–150.
Niu, X. J., and Yu, X. P. (2011b). “Long wave scattering by a vertical cylinder with idealized scour pit.” J. Waterway, Port, Coastal, Ocean Eng., (Mar. 31, 2011).
Sumer, B. M., Whitehouse, R. J. S., and Tøum, A. (2001). “Scour around coastal structures: A summary of recent research.” Coastal Eng., 44(2), 153–190.
Teng, B., Cheng, L., Liu, S. X., and Li, F. J. (2001). “Modified eigenfunction expansion methods for interaction of water waves with a semi-infinite elastic plate.” Appl. Ocean Res., 23(6), 357–368.
Yoshida, K. (1948). “On the partial reflection of long waves.” Geophys. Notes Tokyo Univ. Geophys. Inst., 31.
Yu, X. P., and Zhang, B. Y. (2003). “An extended analytic 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., and Mitchell, L. (2009). “Diffraction of ocean waves around a hollow cylindrical shell structure.” Wave Motion, 46(1), 78–88.
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.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 137 • Issue 12 • December 2011
Pages: 919 - 930
Copyright
© 2011 American Society of Civil Engineers.
History
Received: Feb 16, 2011
Accepted: Jun 23, 2011
Published online: Jun 25, 2011
Published in print: Dec 1, 2011
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.