The scattering of surface waves by a vertical plane barrier

Abstract

In this paper we use a method due to Williams (l) to discuss the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth.1. Introduction. In 1966 Williams (l) presented a new approach to the problem of the scattering of two-dimensional surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending from above the free surface to a finite depth. This problem had previously been considered by Ursell (2), with whose results Williams agreed. Subsequently Faulkner (3,4) extended Wnliams's method, in conjunction with the Wiener-Hopf technique, to discuss the scattering of three-dimensional waves. However, the second of Faulkner's papers, which considers the case when the barrier extends downwards indefinitely from a finite depth, contains an error of formulation which vitiates his results.In this paper we first extend Williams's method to the two-dimensional case where the barrier extends indefinitely downwards from a finite depth and then give a correct formulation of the problem considered by Faulkner.2. Statement of the problem. We consider the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth I. We take a non-dimensional Cartesian coordinate system with the axis of y vertically downwards, the mean free surface of the water being the plane y = 0, and such that the barrier is given by x = 0, y ^ 1, -oo < z < oo. Assuming the motion to be irrotational and simple harmonic in time, the velocity potential may be taken as the real part of x(z, y, z) e~i ai: , where (6) % is a single-valued harmonic function in the region occupied by the water satisfying the boundary conditions

Oblique diffraction of surface waves by a submerged vertical plate

“…For example, Faulkner [10,11] used the WienerHopf technique to study oblique water wave diffraction by a submerged plane vertical barrier as well as by a partially immersed vertical barrier and obtained the reflection coefficient in each case for large wave number. Jarvis and Taylor [12] pointed out an error of formulation in [ 11 ] involving the submerged barrier, which they corrected, thus obtaining asymptotically the reflection coefficient for large wave number by analysing an integral equation of the second kind with Cauchy kernel, to which the problem was reduced. The mathematical analysis in [10][11][12] appears to be rather cumbersome.…”

“…Jarvis and Taylor [12] pointed out an error of formulation in [ 11 ] involving the submerged barrier, which they corrected, thus obtaining asymptotically the reflection coefficient for large wave number by analysing an integral equation of the second kind with Cauchy kernel, to which the problem was reduced. The mathematical analysis in [10][11][12] appears to be rather cumbersome. Later, Evans and Morris [3] utilized an approximate method involving the use of a one-term Galerkin approximation to the solutions of two appropriate integral equations to obtain very accurate upper and lower bounds for the reflection and transmission coefficients for all angles of incidence and wave numbers for the problem of oblique water wave diffraction by a fixed vertical barrier partially immersed in deep water.…”

On oblique waves forcing by a porous cylindrical wall

“…The scattering of surface waves obliquely incident on partially immersed or completely submerged vertical barriers and plates in infinite fluid were investigated by Faulkner [1,2], Jarvis and Taylor [3], Evans and Morris [4], Rhodes-Robinson [5] and Mandal and Goswami [6]. Levine [7] considered the scattering of surface waves obliquely incident on a submerged circular cylinder.…”

Theoretical modelling of oscillating water column wave energy converters

“…Levine & Rodemich (1958) studied the case and showed that this problem has an exact but complicated solution, although they did not obtain the detailed solution. Jarvis (1971) solved the complementary problem of two identical semi-infinite barriers extending downwards from a point a beneath the surface, and he also obtained the curves of reflection and transmission coefficients as a function of frequency, barrier spacing, and submergence depth of barriers.…”

“…They showed how highly accurate complementary bounds could be computed with relative ease. Recently, Soumen et al (2009) reinvestigated the two-barrier problem of Jarvis (1971). The solved a system of Abel integral equations by…”