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

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

Diffraction of water waves by a segmented permeable breakwater

The linearized theory of water waves is used to examine the diffraction of an incident wave by a permeable breakwater that consists of a number of distinct elements. Under the assumption that the wavelength is much greater than the thickness, each element is replaced by a thin structure and the permeability is modeled by a suitable boundary condition applied on its surface. The diffracted wave field is obtained by the solution of an integral equation and results are presented to illustrate the effects of permeability and of the characteristics of the incident wave

Water-wave propagation through an infinite array of cylindrical structures

An investigation is made into water-wave propagation through an array of vertical cylinders extending to infinity and periodic in both horizontal directions. Methods are presented for the calculation of the frequency ranges for which wave propagation without change of amplitude is possible ('passing bands'), and for which propagation without change of amplitude is not possible ('stopping bands'). Some of the techniques may be used to determine the change of wave amplitude for frequencies within the stopping bands. Approximate and numerical techniques are used to show how this infinite-array problem is related to trapped modes, Rayleigh-Bloch waves, and the problem of wave diffraction by a grating made up of a finite number of cylinder rows

Approximations to wave propagation through doubly-periodic arrays of scatterers

The propagation of waves through a doubly-periodic array of identical rigid scatterers is considered in the case that the field equation is the two-dimensional Helmholtz equation. The method of matched asymptotic expansions is used to obtain the dispersion relation corresponding to wave propagation through an array of scatterers of arbitrary shape that are each small relative to both the wave length and the array periodicity. The results obtained differ from those obtained from homogenization in that there is no requirement that the wave length be much smaller than the array periodicity, and hence it is possible to examine phenomena, such as band gaps, that are associated with the array periodicity

Second-order wave diffraction by a submerged circular-cylinder

Expressions are derived for the amplitudes of the second-harmonic waves generated when a uniform wave train is normally incident upon a two-dimensional body, submerged in water of illfinite depth. These amplitudes are given in terms of integrals over the free surface of products of first-order quantities. For a submerged, circular cylinder, it is shown analytically that there is no second-order reflected wave at any frequency. This extends the classical result that there is no reflection a t first-order for this body

Sloshing frequencies of longitudinal modes for a liquid contained in a trough

The sloshing under gravity is considered for a liquid contained in a horizontal cylinder of uniform cross-section and symmetric about a vertical plane parallel to its generators. Much of the published work on this problem has been concerned with twodimensional, transverse oscillations of the fluid. Here, attention is paid to longitudinal modes with variation of the fluid motion along the cylinder. There are two known exact solutions for all modes ; these are for cylinders whose cross-sections are either rectangular or triangular with a vertex semi-angle of in. Numerical solutions are possible for an arbitrary geometry but few calculations are reported in the open literature. In the present work, some general aspects of the solutions for arbitrary geometries are investigated including the behaviour at low and high frequency of longitudinal modes. Further, simple methods are described for obtaining upper and lower bounds to the frequencies of both the lowest symmetric and lowest antisymmetric modes. Comparisons are made with numerical calculations from a boundary element method

Motion trapping structures in the three-dimensional water-wave problem

Trapped modes in the linearized water-wave problem are free oscillations of finite energy in an unbounded fluid with a free surface. It has been known for some time that such modes are supported by certain structures when held fixed, but recently it has been demonstrated that in two dimensions trapped modes are also possible for freely-floating structures that are able to respond to the hydrodynamic forces acting upon them. For a freely-floating structure such a mode is a coupled oscillation of the fluid and the structure that, in the absence of viscosity, persists for all time. Here previous work on the two-dimensional problem is extended to give motion trapping structures in the three-dimensional water-wave problem that have a vertical axis of symmetry

Trapped modes in the water-wave problem for a freely-floating structure

Trapped modes in the linearized water-wave problem are free oscillations of an unbounded fluid with a free surface that have finite energy: it had been known for some time that such modes are supported by certain structures when held fixed. This paper investigates the problem of a freely-floating structure that is able to move in response to the hydrodynamic forces acting upon it and it is shown that trapped modes also exist in this problem. For a freely-floating structure a trapped mode is a coupled oscillation of the fluid and the structure

Green's functions for water waves in porous structures

Representations for Green's functions suitable for water-wave problems involving porous structures are obtained by integrating solutions to appropriate heat conduction problems with respect to time. By utilizing different representations for these heat equation solutions for small and large times, the changeover being determined by an arbitrary positive parameter a, a one-parameter family of formulas for the required Green's function is derived and by varying a the convergence characteristics of this new representation can be altered. Letting a --> 0 results in known eigenfunction expansions. The results of computations are presented showing the accuracy and effciency of the resulting formulas

A uniqueness criterion for linear problems of wave-body interaction

The question of uniqueness for linearized problems describing interaction of submerged bodies with an ideal unbounded fluid is far from its final resolution. In the present work a new criterion of uniqueness is suggested based on Green’s integral identity and maximum principles for elliptic differential equations. The criterion is formulated as an inequality involving integrals of the Green function over the bodies’ wetted contours. This criterion is quite general and applicable for any number of submerged bodies of fairly arbitrary shape (satisfying an exterior sphere condition) and in any dimension; it can also be generalised to more complicated elliptic problems. Very simple bounds are also derived from the criterion, which deliver uniqueness sets in the space of parameters defined by submergence of the system of bodies and the frequency of oscillation. Results of numerical investigation and comparison with known uniqueness criteria are presented