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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.

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A formula for the effective wavenumber in a dilute random array of identical scatterers in two dimensions is derived, based on Lax's quasi-crystalline 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.

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 d of the underlying space and the lattice dimension d Λ . Lattice sums are related to, and can be calculated from, the quasi-periodic Green's function and this object serves as the starting point of the analysis. LATTICE SUMS FOR THE HELMHOLTZ EQUATION 631to mathematically challenging problems. Attempts to evaluate the sum in (1.1) in closed form have so far been unsuccessful, though some remarkable identities have been derived in the process [23,113].The lattice sum in (1.1) arises from a sum of singular solutions of Laplace's equation. Here we will be concerned with related sums in which the underlying physics is governed by the Helmholtz equationwhich arises naturally in many contexts, particularly when investigating linear wave phenomena at a given frequency. For example, in acoustics u represents fluctuations in pressure, whereas in elasticity it would be some component of the displacement vector, and in electromagnetism a component of the electric or magnetic field. The quantity k (assumed real and positive) is the wavenumber, related to the frequency ω via some appropriate dispersion relation. In diffraction theory we are often faced with trying to solve (1.2) in a region exterior to some scatterer(s) (and maybe inside as well) subject to boundary conditions on the surface of the scatterer(s). The case where the scatterer is periodic represents an important class of such problems; examples include the study of diffraction gratings, scattering by periodic surfaces, and wave propagation through composite materials. Equation (1.2) is also equivalent to the time-independent Schrödinger equation in the presence of a constant potential, in which case k is related to the total energy of the particle under consideration, and as a result the Helmholtz equation finds application in the study of electron scattering in solids. A common simplification used is the so-called muffin-tin approximation (due to Slater [99]), in which the potential is supposed to be spherically symmetric within spheres surrounding each atom and constant in the region inbetween these spheres. This then leads naturally to the need to solve (1.2) on a periodic domain.The study of wave propagation in periodic structures has a long history and leads to a whole range of interesting mathematical problems. The classic text by Brillouin [11] arguably still serves as the best introduction to the subject, though many significant advances have been made since it was written. For example, the mathematical theory of diffraction gratings (based on functional analysis) is developed in books such as [120] and there is an ever-growing literature describing the rapidly developing field of photonic and phononic crystals; an excellent text is [39]. One of the consequences of periodicity is that a problem on...

A two-dimensional acoustical waveguide described by two infinite parallel lines a distance 2d apart has a circle of radius a < d positioned symmetrically between them. The potential satisfies the two-dimensional Helmholtz equation in the fluid region between the circle and the lines, and the normal gradient of the potential vanishes on both. For motions which are antisymmetric about the centreline of the guide there exists a cutoff frequency below which no propagation down the guide is possible. It is proved that for a circle of sufficiently small radius there exists a trapped mode, having a frequency close to the cutoff frequency, which is antisymmetric about the centreline of the guide and symmetric about a line through the centre of the circle perpendicular to the centreline. The method used is due to Ursell (1951) who established the existence of a trapped surface wave mode in the vicinity of a long totally submerged horizontal circular cylinder of small radius in deep water. Numerical computations in the present work reveal that a single trapped mode appears to exist for all values of a ≤ d and not just when the circle is small. The present method, when used to attempt to construct a solution antisymmetric about both the centreline and a line perpendicular to it through the centre of the circle does not lead to a trapped mode. The trapped modes can equally well be regarded as surface-wave modes, as in an infinitely long tank of water with a free surface, into which has been placed symmetrically, a vertical rigid circular cylinder extending throughout the depth. Numerical evidence for the existence of such trapped modes when the cylinder is of rectangular cross-section was presented in Evans & Linton (1991).

The acoustic transmission loss of a finite periodic array of long rigid cylinders, without and with porous absorbent covering, is studied both theoretically and in the laboratory. A multiple scattering model is extended to allow for the covering and its acoustical properties are described by a single parameter semi-empirical model. Data from laboratory measurements and numerical results are found to be in reasonable agreement. These data and predictions show that porous covering reduces the variation of transmission loss with frequency due to the stop/pass band structure observed with an array of rigid cylinders with similar overall radius and improves the overall attenuation in the higher frequency range. The predicted sensitivities to covering thickness and effective flow resistivity are explored. It is predicted that a random covered array also gives better attenuation than a random array of rigid cylinders with the same overall radius and volume fraction. Porous covering in arrays of cylinders2

Abstract. We solve the two-dimensional problem of acoustic scattering by a semi-infinite periodic array of identical isotropic point scatterers, i.e., objects whose size is negligible compared to the incident wavelength and which are assumed to scatter incident waves uniformly in all directions. This model is appropriate for scatterers on which Dirichlet boundary conditions are applied in the limit as the ratio of wavelength to body size tends to infinity. The problem is also relevant to the scattering of an E-polarized electromagnetic wave by an array of highly conducting wires. The actual geometry of each scatterer is characterized by a single parameter in the equations, related to the single-body scattering problem and determined from a harmonic boundary-value problem. Using a mixture of analytical and numerical techniques, we confirm that a number of phenomena reported for specific geometries are in fact present in the general case (such as the presence of shadow boundaries in the far field and the vanishing of the circular wave scattered by the end of the array in certain specific directions). We show that the semi-infinite array problem is equivalent to that of inverting an infinite Toeplitz matrix, which in turn can be formulated as a discrete Wiener-Hopf problem. Numerical results are presented which compare the amplitude of the wave diffracted by the end of the array for scatterers having different shapes.