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.

Complementary approximations to wave scattering by vertical barriers

A review is presented of wave-energy devices and hydrodynamic properties of idealized equipment for extracting power from waves. The governing equations involve the fluid hydrodynamic theory applied to machines with zero forward speed which can absorb energy from the neighboring wave field. A mixture of waves of different amplitudes, periods, wavelengths, and directions with randomly distributed phases coexist at a given time a mathematical model of the sea surface assumes it to be an infinite superposition of wave trains of various amplitudes and frequencies. A theory was developed for the oscillation of two-dimensional energy-absorbing cylindrical sections which can be utilized for estimating hydrodynamic characteristics of fully three-dimensional ship hulls. Finally, three-dimensional wave-energy absorbers are represented by expressions in terms of the force amplitude, direction of motion, and the damping coefficient.

Power From Water Waves

Surface wave interactions with systems of oscillating bodies and pressure distributions

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.

Oblique Wave Diffraction by Parallel Thin Vertical Barriers with Gaps

This work is concerned with the linearized theory of water waves applied to the motion of a floating structure that restricts in some way the motion of a portion of the free surface (an example of such a structure is a floating torus). When a structure of this type is held fixed in incident monochromatic waves, or forced to move time harmonically with a prescribed velocity, the amplitude of the fluid motion will have local maxima at certain frequencies of the forcing. These resonances correspond to poles of the scattering and radiation potentials when extended to the complex frequency domain. It is shown in this work that, in general, the positions of these poles in the scattering and radiation potentials will not coincide with the positions of the poles that appear in the velocity potential for the coupled problem obtained when the structure is free to move. The poles of the potential for the coupled problem are associated with the solution for the structural velocities of the equation of motion. When physical quantities such as the amplitude of the fluid motion are examined as a function of (real) frequency, there will in general be a shift in the resonant frequencies in going from the radiation and scattering problems to the coupled problem. The magnitude of this shift depends on the geometry of the structure and how it is moored.

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Complex resonances in the water-wave problem for a floating structure

Two-dimensional waves are incident upon a pair of vertical flat plates intersecting the free surface in a fluid of infinite depth. An asymptotic theory is developed for the resulting wave reflexion and transmission, assuming that the separation between the plates is small. The fluid motion between the plates is a uniform vertical oscillation, matched to the outer wave field by a local flow at the opening beneath the plates. It is shown that the reflexion and transmission coefficients undergo rapid changes, ranging from complete reflexion to complete transmission, in the vicinity of a critical wavenumber where the fluid column between the obstacles is resonant.

J. N. Newman

Papers

Interaction of water waves with two closely spaced vertical obstacles