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Scattering of water waves by a submerged nearly circular cylinder

Published online by Cambridge University Press:  17 February 2009

B. N. Mandal  and
Sudeshna Banerjea
B. N. Mandal
Affiliation:
Physics and Appl. Math. Unit, Indian Statistical Institute, 203, B. T. Road, Calcutta 700 035, India.
Sudeshna Banerjea
Affiliation:
Department of Mathematics, Jadaupur University, Calcutta 700 032, India.
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Abstract

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The problem of scattering of surface water waves by a horizontal circular cylinder totally submerged in deep water is well studied in the literature within the framework of linearised theory with the remarkable conclusion that a normally incident wave train experiences no reflection. However, if the cross-section of the cylinder is not circular then it experiences reflection in general. The present paper studies the case when the cylinder is not quite circular and derives expressions for reflection and transmission coefficients correct to order ∈, where ∈ is a measure of small departure of the cylinder cross-section from circularity. A simplified perturbation analysis is employed to derive two independent boundary value problems (BVP) up to first order in ∈. The first BVP corresponds to the problem of water wave scattering by a submerged circular cylinder. The reflection coefficient up to first order and the first order correction to the transmission coefficient arise in the second BVP in a natural way and are obtained by a suitable use of Green' integral theorem without solving the second BVP. Assuming a Fourier expansion of the shape function, these are evaluated approximately. It is noticed that for some particular shapes of the cylinder, these vanish. Also, the numerical results for the transmission coefficients up to first order for a nearly circular cylinder for which the reflection coefficients up to first order vanish, are given in tabular form. It is observed that for many other smooth cylinders, the result for a circular cylinder that the reflection coefficient vanishes, is also approximately valid.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 3 , January 1995 , pp. 372 - 380
Copyright
Copyright © Australian Mathematical Society 1995

References

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