C. M. Linton

The interaction of waves with arrays of vertical circular cylinders

The scattering of water waves by an array of N bottom-mounted vertical circular cylinders is solved exactly (under the assumption of linear water wave theory) using the method proposed by Spring & Monkmeyer in 1974. A major simplification to this theory has been found which makes the evaluation of quantities such as the forces on the cylinders much simpler. New formulae are given for the first and mean second-order forces together with one for the free-surface elevation in the vicinity of a particular cylinder. Comparisons are made between the exact results shown here and those generated using the approximate method of McIver & Evans (1984). The behaviour of the forces on the bodies in the long-wave limit is also examined for the special case of two cylinders with equal radii.

Handbook of Mathematical Techniques for Wave/Structure Interactions

INTRODUCTION The Water-Wave Problem The Linearised Equations Interaction of a Wave with a Structure Reciprocity Relations Energy of the Fluid Motion EIGENFUNCTION EXPANSIONS Introduction Construction of Vertical Eigenfunction Two-Dimensional Problems Three-Dimensional Problems Matched Eigenfunction Expansions MULTIPOLE EXPANSIONS Introduction Isolated Obstacles Multiple Bodies INTEGRAL EQUATIONS Source Distribution Greens Theorem Thin Obstacles Interior Problems Free-Surface Problems Numerical Evaluation of Greens functions Diffraction by a Gap in a Breakwater Diffraction by an Insular Breakwater Embedding Formulae Numerical Solutions THE WIENER-HOPF AND RELATED TECHNIQUES The Weiner-Hopf Technique Residue Calculus Theory ARRAYS The Wide-Spacing Approximation SMALL OBJECTS Introduction Breakwater with a Gap Vertical Cylinder Heaving Cylinder Eigenvalue Problems VARIATIONAL METHODS Scattering and Radiation Problems Eigenvalue Problems APPENDICES Bessel Functions Multipoles Principle Value and Finite Part Integrals

The Green's Function for the Two-Dimensional Helmholtz Equation in Periodic Domains

Analytical techniques are described for transforming the Greens function for the two-dimensional Helmholtz equation in periodic domains from the slowly convergent representation as a series of images into forms more suitable for computation. In particular methods derived from Kummers transformation are described, and integral representations, lattice sums and the use of Ewalds method are discussed. Greens functions suitable for problems in parallel-plate acoustic waveguides are also considered and numerical results comparing the accuracy of the various methods are presented.

Lattice Sums for the Helmholtz Equation

A survey of different representations for lattice sums for the Helmholtz equation is made. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension

The interaction of waves with horizontal cylinders in two-layer fluids

d

Multiple scattering by random configurations of circular cylinders: Second-order corrections for the effective wavenumber

of the underlying space and the lattice dimension

Trapped modes in two-dimensional waveguides

d_\Lambda

Effects of porous covering on sound attenuation by periodic arrays of cylinders

. Lattice sums are related to, and can be calculated from, the quasi-periodic Greens function and this object serves as the starting point of the analysis.

Trapped modes in open channels

•We consider two-dimensional problems based on linear water wave theory concerning the interaction of waves with horizontal cylinders in a fluid consisting of a layer of finite depth bounded above by a free surface and below by an infinite layer of fluid of greater density. For such a situation time-harmonic waves can propagate with two different wavenumbers K and k. In a single-layer fluid there are a number of reciprocity relations that exist connecting the various hydrodynamic quantities that arise. These relations are systematically extended to the two-fluid case. It is shown that for symmetric bodies the solutions to scattering problems where the incident wave has wavenumber K and those where it has wavenumber k are related so that the solution to both can be found by just solving one of them. The particular problems of wave scattering by a horizontal circular cylinder in either the upper or lower layer are then solved using multipole expansions. Linear water wave theory is a widely used technique for determining how a wave is diffracted by a fixed or floating structure. The underlying assumption of the theory is that the amplitudes of any waves or body motions are small compared to the other length scales in the problem. At first order, this means that it is only necessary to consider the diffraction of a wave of a single frequency and direction, as linear superposition yields the diffraction pattern for an irregular sea. Furthermore, the velocity potential may be split into a part which describes the scattering of waves by a fixed structure and a part which describes the radiation of waves by the body into otherwise calm water. The radiation potential may be further split into a number of sub-potentials, each of which corresponds to the body moving in a separate mode of motion. The resulting potentials may be solved for separately but they are not all independent and a series of reciprocity relations which connect various scattering and radiation quantities have been derived by many authors over the years. These relations may be obtained by applying Green’s theorem to two different potentials and a systematic derivation of all the first-order reciprocity relations is given by Newman (1976). They yield important information about the hydrodynamic loading on a body and the scattered wave field, and form a valuable check on the accuracy of any numerical wave diffraction code. The reciprocity relations which exist have been derived for single-layer fluids, both in two and three dimensions and for finite- and infinite-depth fluids. More recently, however, interest has been extended to bodies which are immersed in two-layer fluids, each fluid having a different density. One reason for this is the suggestion by Friis,

The Radiation and Scattering of Surface Waves by a Vertical Circular Cylinder in a Channel

A formula for the effective wavenumber in a dilute random array of identical scatterers in two dimensions is derived, based on Laxs quasicrystalline approximation. This formula replaces a widely-used expression due to Twersky, which is shown to be based on an inappropriate choice of pair-correlation function.