Exact Solution to the Modified Mild-Slope Equation for Wave Scattering by a Cylinder with an Idealized Scour Pit
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 139, Issue 5
Abstract
In this paper, wave scattering by a vertical cylinder with a scour pit governed by the modified mild-slope equation (MMSE) is studied analytically. The scour pit around the cylinder is assumed to be axi-symmetric and idealized with its radial profile being a power function. This assumption permits transformation of the two-dimensional MMSE into an ordinary differential equation (ODE) in the radial direction through the technique of variable separation. By employing a newly derived explicit form of the resultant ODE of the MMSE in the scour pit region, an exact solution to the MMSE is constructed in terms of a Fourier-cosine series and Taylor series. To validate this new analytic solution to the MMSE, a comparison among the present solution, analytic solution to the long wave equation, and analytic solution to the Helmholtz equation is made and a good agreement is obtained. It is found that the present MMSE model is valid for a maximum bottom slope of approximately 0.927. Based on the present solution to the MMSE, the effect of dimensions of the scour pit, including both depth and width, on wave run-up around the cylinder is investigated. Finally, the influence of the wavelength of incident waves from shallow to deep water on wave run-up around the cylinder is also investigated.
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Acknowledgments
This work is supported by the Natural Science Foundation of P. R. China (10962001 and 51149007), Guangxi Natural Science Foundation (2010GXNSFA013115 and 2011GXNSFD018006), and Scientific Research Foundation of Guangxi Universities (201102ZD014). All the authors would like to gratefully acknowledge some very useful suggestions from two anonymous referees.
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Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 139 • Issue 5 • September 2013
Pages: 413 - 423
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Jul 24, 2012
Accepted: Jan 9, 2013
Published online: Jan 11, 2013
Published in print: Sep 1, 2013
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