SCATTERING OF WATER-WAVES BY A SUBMERGED NEARLY CIRCULAR-CYLINDER

The problem of scattering of surface water waves by a horizontal circu lar cylinder totally submerged in deep water is well studied in the li terature within the framework of linearised theory with the remarkable conclusion that a normally incident wave train experiences no reflect ion. However, if the cross-section of the cylinder is not circular the n it experiences reflection in general. The present paper studies the case when the cylinder is not quite circular and derives expressions f or reflection and transmission coefficients correct to order epsilon, where epsilon is a measure of small departure of the cylinder cross-se ction from circularity. A simplified perturbation analysis is employed to derive two independent boundary value problems (BVP) up to first o rder in epsilon. The first BVP corresponds to the problem of water wav e scattering by a submerged circular cylinder. The reflection coeffici ent up to first order and the first order correction to the transmissi on coefficient arise in the second BVP in a natural way and are obtain ed by a suitable use of Green's integral theorem without solving the s econd BVP. Assuming a Fourier expansion of the shape function, these a re evaluated approximately. It is noticed that for some particular sha pes of the cylinder, these vanish. Also, the numerical results for the transmission coefficients up to first order for a nearly circular cyl inder for which the reflection coefficients up to first order vanish, are given in tabular form. It is observed that for many other smooth c ylinders, the result for a circular cylinder that the reflection coeff icient vanishes, is also approximately valid.