M. McIver

An example of non-uniqueness in the two-dimensional linear water wave problem

An example of non-uniqueness in the two-dimensional, linear water wave problem is obtained by constructing a potential which does not radiate any waves to infinity and whose streamline pattern represents the flow around two surface-piercing bodies. The potential is constructed from two wave sources which are positioned in the free surface in such a way that the waves radiated from each source cancel at infinity. A numerical calculation of the streamline pattern indicates that there are at least two streamlines which represent surface-piercing bodies, each of which encloses a source point. A proof of the existence of these lines is then given.

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

•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,

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

It has been known for about ten years that, within the framework of the linearised water-wave problem, certain fixed structures can support fluid oscillations of finite energy known as “trapped modes”. In this work the open question of the existence of trapped modes for a freely-floating structure without moorings is considered. For the case of a structure able to move in heave, the conditions necessary for the existence of such a trapped mode are discussed.

The existence of Rayleigh{Bloch surface waves

Rayleigh-Bloch surface waves arise in many physical contexts including water waves and acoustics. They represent disturbances travelling along an infinite periodic structure. In the absence of any existence results, a number of authors have previously computed such modes for certain specific geometries. Here we prove that such waves can exist in the absence of any incident wave forcing for a wide class of structures.

Construction of trapped modes for wave guides and diffraction gratings

This paper describes various methods for the construction of trapped mode solutions within wave guides and along diffraction gratings; in the latter case the solutions are often called Rayleigh–Bloch waves. One of the main results is the explicit construction of a number of new trapped mode solutions within a wave guide that correspond to eigenvalues that are embedded in the continuous spectrum of the relevant operators. The method of construction is to take ‘trial’ solutions of the field equation that satisfy the required conditions on the guide walls and have the correct decay at large distances and then to identify lines that can be interpreted as the boundary of a structure within the wave guide. The same idea is also investigated for Rayleigh–Bloch waves within the context of diffraction gratings. Sensible choices of trial function are made by reference to solutions for a grating of circular cylinders, obtained by a multipole expansion method, and an approximate solution for more general geometries.

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.

Trapped modes for off-centre structures in guides

The existence of trapped modes near obstacles in two-dimensional waveguides is well established when the centerline of the guide is a line of symmetry for the geometry. In this paper we examine cases where no such line of symmetry exists. The boundary condition on the obstacle is of Neumann type and both Neumann and Dirichlet conditions on the guide walls are treated. A variety of techniques (variational methods, boundary integral equations, slender-body theory, modified residue calculus theory) are used to investigate trapped-mode phenomena in a number of different frequency bands.

Excitation of trapped water waves by the forced motion of structures

A numerical and analytical investigation is made into the response of a fluid when a two-dimensional structure is forced to move in a prescribed fashion. The structure is constructed in such a way that it supports a trapped mode at one particular frequency. The fluid motion is assumed to be small and the time-domain equations for linear water-wave theory are solved numerically. In addition, the asymptotic behaviour of the resulting velocity potential is determined analytically from the relationship between the time- and frequency-domain solutions. The trapping structure has two distinct surface-piercing elements and the trapped mode exhibits a vertical ‘pumping’ motion of the fluid between the elements. When the structure is forced to oscillate at the trapped-mode frequency an oscillation which grows in time but decays in space is observed. An oscillatory forcing at a frequency different from that of the trapped mode produces bounded oscillations at both the forcing and the trapped-mode frequency. A transient forcing also gives rise to a localized oscillation at the trapped-mode frequency which does not decay with time. Where possible, comparisons are made between the numerical and asymptotic solutions and good agreement is observed. The calculations described above are contrasted with the results from a similar forcing of a pair of semicircular cylinders which intersect the free surface at the same points as the trapping structure. For this second geometry no localized or unbounded oscillations are observed. The trapping structure is then given a sequence of perturbations which transform it into the two semicircular cylinders and the time-domain equations solved for a transient forcing of each structural geometry in the sequence. For small perturbations of the trapping structure, localized oscillations are produced which have a frequency close to that of the trapped mode but with amplitude that decays slowly with time. Estimates of the frequency and the rate of decay of the oscillation are made from the time-domain calculations. These values correspond to the real and imaginary parts of a pole in the complex force coefficient associated with a frequency-domain potential. An estimate of the position of this pole is obtained from calculations of the added mass and damping for the structure and shows good agreement with the time-domain results. Further time-domain calculations for a different trapping structure with more widely spaced elements show a number of interesting features. In particular, a transient forcing leads to persistent oscillations at two distinct frequencies, suggesting that there is either a second trapped mode, or a very lightly damped near-trapped mode. In addition a highly damped pumping mode is identified.

Trapping of waves by a submerged elliptical torus

An investigation is made into the trapping of surface gravity waves by totally submerged three-dimensional obstacles and strong numerical evidence of the existence of trapped modes is presented. The specific geometry considered is a submerged elliptical torus. The depth of submergence of the torus and the aspect ratio of its cross-section are held fixed and a search for a trapped mode is made in the parameter space formed by varying the radius of the torus and the frequency. A plane wave approximation to the location of the mode in this space is derived and an integral equation and a side condition for the exact trapped mode are obtained. Each of these conditions is satisfied on a different line in the plane and the point at which the lines cross corresponds to a trapped mode. Although it is not possible to locate this point exactly, because of numerical error, existence of the mode may be inferred with confidence as small changes in the numerical results do not alter the fact that the lines cross. If the torus makes small vertical oscillations, it is customary to try to express the fluid velocity as the gradient of the so-called heave potential, which is assumed to have the same time dependence as the body oscillations. A necessary condition for the existence of this potential at the trapped mode frequency is derived and numerical evidence is cited which shows that this condition is not satisfied for an elliptical torus. Calculations of the heave potential for such a torus are made over a range of frequencies, and it is shown that the force coefficients behave in a singular fashion in the vicinity of the trapped mode frequency. An analysis of the time domain problem for a torus which is forced to make small vertical oscillations at the trapped mode frequency shows that the potential contains a term which represents a growing oscillation.

Periodic structures in waveguides

We consider structures of period 2 spanning a two–dimensional waveguide of width 2N. Scattering problems, where Neumann conditions are imposed on the boundary of the structure and either Neumann or Dirichlet conditions are applied on the guide walls, are decomposed into N + 1 independent problems. The existence of at least N trapped modes is proved for the Neumann guide case, and for the Dirichlet case we prove that at least N − 1 such modes exist, this number increasing to N if a certain geometrical condition is satisfied.