B. S. Taylor

Abstract: In this paper we use a method due to Williams(1) 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.

Abstract: Scattering of waves by vertical barriers in infinite-depth water has received much attention due to the ability to solve many of these problems exactly. However, the same problems in finite depth require the use of approximation methods. In this paper we present an accurate method of solving these problems based on a Galerkin approximation. We will show how highly accurate complementary bounds can be computed with relative ease for many scattering problems involving vertical barriers in finite depth and also for a sloshing problem involving a vertical barrier in a rectangular tank.

Abstract: The problem of oblique water wave diffraction by two equal thin, parallel, fixed vertical barriers with gaps present in uniform finite-depth water is investigated here. Three types of barrier configurations are considered. A one-term Galerkin approximation is used to evaluate upper and lower bounds for reflection and transmission coefficients for each configuration. These bounds are seen to be very close numerically for all wave numbers and as such their averages produce good numerical estimates for these coefficients. Only the bounds for the reflection coefficient are numerically computed. These are also numerically compared with the results obtained by using multiterm Galerkin approximations involving Chebyshev polynomials for a wide range of parameters. Numerical results for the reflection coefficients for the three barrier configurations are presented graphically. It is seen that total reflection occurs only for the surface-piercing barriers while total transmission occurs for all the three configurations considered here. It is also observed that the introduction of an equal second barrier to a submerged barrier increases the reflection coefficient considerably in some frequency bands and as such submerged double barrier configurations are preferable to a submerged single barrier for the purpose of reflecting more wave energy into the open sea.

Abstract: An appropriate one-term Galerkin approximation is used to evaluate very accurate upper and lower bounds for the reflection and transmission coefficients in the problems of oblique water wave diffraction by a thin vertical barrier present in water of uniform finite depth. Four different configurations of the barrier are considered. The barrier may be partially immersed, or it may be submerged from a finite depth and extending down to the seabed, or it may be in the form of a submerged plate which does not extend down to the bottom, or it may be in the form of a thin vertical wall with a submerged gap. Very accurate upper and lower bounds for the reflection and transmission coefficients for different values of the various parameters are obtained numerically. The results for the reflection coefficient are displayed in tables. Comparison with known results obtained by another method is also made.

Abstract: Uniqueness in the linearised water-wave problem is considered for a fluid layer of constant depth containing two, three or four vertical barriers. The barriers are parallel, of infinite length in a horizontal direction, and may be surface-piercing and/or bottom mounted and may have gaps. The case of oblique wave incidence is included in the theory. A solution for a particular geometry is unique if there are no trapped modes, that is no free oscillations of finite energy. Thus, uniqueness is established by showing that an appropriate homogeneous problem has only the trivial solution. Under the assumption that at least one barrier does not occupy the entire fluid depth, the following results have been proven: for any configuration of two barriers the homogeneous problem has only the trivial solution for any frequency within the continuous spectrum; for an arbitrary configuration of three barriers the homogeneous problem has only the trivial solution for certain ranges of frequency within the continuous spectrum; for three-barrier configurations symmetric about a vertical line, it is shown that there are no correspondingly symmetric trapped modes for any frequency within the continuous spectrum; for four-barrier configurations symmetric about a vertical line, the homogeneous problem has only the trivial solution for certain ranges of frequency within the continuous spectrum. The symmetric four-barrier problem is investigated numerically and strong evidence is presented for the existence of trapped modes in both finite and infinite depth. The trapped mode frequencies are found for particular geometries that are in agreement with the uniqueness results listed above.

Abstract: An analytical approach is proposed here to study scattering of deep water waves by a submerged or a surface piercing vertical porous barrier. It involves a connection between two wave potentials of which one is the solution of a boundary value problem associated with wave scattering by the porous barrier and the other is the solution of a complementary type problem where barrier and gap positions are interchanged and solid barrier takes the position of the porous barrier. The connection also involves an auxiliary or a connection wave potential. The potential for the solid barrier problem involves incident wave forcing while the auxiliary potential describes a solid barrier type problem that involves a non-physical forcing. The solution procedure of Ursell (Ursell, 1947) is chosen to solve these boundary value problems explicitly in the case of normal wave incidence as it also determines necessary exact behavior of the potential at the barrier edge. The reflection coefficients are also connected and the reflection amplitudes of the normally incident wave against the vertical porous barriers are obtained analytically. Numerical results for reflection and transmission coefficients are presented.