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Reflection and Transmission of an Incident Progressive Wave by Obstacles in Homogeneous Shallow Water |
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PP: 1959-1971 |
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doi:10.18576/isl/120419
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Author(s) |
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K. K. Al Arfaj,
M. A. Helal,
M. S. Abou-Dina,
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Abstract |
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| The influence of a suspended fixed obstacle on an incident progressive wave inside an ideal homogeneous shallow water is studied in two dimensions. The fluid occupies an infinite channel of a constant depth, and a fixed obstacle of a small horizontal extent is partially submerged without contact with the bottom of the channel. An asymptotic double series expansion for the solution is used. The procedure enables us to calculate analytic expressions for the local perturbations up to the second order. The results of the first-order approximation indicate that no reflections exist. The second-order approximation of the solution is found to be the superposition of a progressive wave and local perturbations. For approximations of order higher than two, a secular term which increases monotonically with time and distance appears in the expressions for the progressive wave. This unacceptable result is due to a certain aspects in the mathematical procedure used. For this reason, the procedure is modified by using a suitable transformation of variables which reduces the determination of the transmitted wave to the solution of the KdV equation. As an illustration, the special case of the incident uniform flow is considered and the stream lines of the resulting flow are drawn.
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