Introduction

An infinite array is a structure in which the scattering body repeats periodically to infinity in both directions. Making use of the periodicity of the structure as well as that of the incident wave, the problem can be reduced to having to solve for one body. The scattered potential of all other bodies is obtained by simple phase shift.

The infinite array is often used as an approximation for a finite array as its solution is very much simpler. Besides allowing the approximation of quantities associated with particular bodies in the array (the forces upon the body, e.g.), it also directly provides information about the far field away from the array.

The infinite-array problem is also often met in other applications, for example in acoustic of electromagnetic scattering, where it is also termed diffraction grating.

There is a vast literature on this problem dating back to early twentieth century work. Recently, a solution was suggested in Peter, Meylan, and Linton 2006 which, in particular, applies to arbitrary scatterers.

Associated with infinite arrays is the phenomenon of Rayleigh-Bloch Waves, which are waves which are bound to the infinite array and travel along it.

Literature Survey

There are two approaches to the solution of wave scattering by an infinite array, methods based on Infinite Array Green Function and methods based on Interaction Theory. Also there are two problems which may be considered. The first is to determine the scattering by an incident plate wave and the second is to determine what waves are supported by the structure in the absense of wave forcing (called in the water wave context Rayleigh-Bloch Waves).

The problem of determining the scattering of waves by periodic arrays of obstacles subject to wave forcing has received considerable research attention and spans a broad range of physical disciplines including solid-state physics, acoustics, optics, etc. In many applications, the interest centres on arrangements which are periodic in two directions (for example, the study of crystallography).

In the present context of water wave propagation and its interaction with flexible surface structures, Chou Chou98 has investigated the effect of an infinite doubly-periodic array of elastic plates on wave propagation. In the modelling of the plate equations, Chou incorporates both bending stiffness and tension effects, so that the discussion of the results not only includes the case of pure bending of elastic plates in the absence of compression forces (as considered here), but also, by setting the stiffness to zero, pure tensional effects which would describe, for example, periodic arrays of taught membranes. However, in contrast to the work described here, the doubly-periodic configuration allows significant simplification in the solution procedure by applying Floquet's theorem to reduce the problem to one on a finite domain with periodic boundary conditions. Moreover, problems involving infinite doubly-periodic structures only offer information about the possibility of wave propagation throughout the array (in the form of so-called pass-bands or stop-bands) and cannot address the diffraction of plane waves from infinity.

For arrays which are periodic in one direction only, the situation is different and diffraction grating effects occur. Thus, for an incident plane wave of a particular given wave frequency, a finite number of distinct plane waves propagating away from the array at certain discrete angles will occur. In the context of water waves and fixed periodic arrays, Twersky Twersky52 was able to solve the problem of a periodic array of vertical circular cylinders. The uniformity of the configuration in the depth coordinate implies that the resulting equations also describe two-dimensional acoustic wave scattering, in this case by circular cylinders. The problem of Twersky Twersky52, who used Schlomilch series to sum slowly-convergent series involving Hankel functions, was re-considered by Linton and Evans Linton_evans93 who used a so-called multipole method. For periodic arrays of rectangular cylinders extending uniformly through the depth, Fernyhough and Evans Fernyhough95, used domain decomposition and mode matching to derive an integral equation formulation to the problem. In order to consider more general cylinder profiles, boundary integral methods are inevitable and require the use of a periodic Green function. In its most basic form, the Green function consists of a series involving Hankel functions which is slowly convergent and unsuitable for numerical computation. Hence Linton Linton98 compared a number of different representations of the periodic Green function, designed to increase the convergence characteristics. Porter and Evans Porter_evans99 used the work of Linton Linton98 to compute so-called Rayleigh-Bloch waves (or trapped waves) along a periodic array of cylinders of arbitrary cross-section. A number of papers in recent years have concentrated on similar ideas, to those of Porter and Evans Porter_evans99, a primary motivation being the connection between large wave responses in large finite arrays of cylinders and the trapped waves in infinite periodic arrays (see Maniar and Newman Maniar_newman97).

Infinite Array Green Function methods

After Removing the Depth Dependence and the water wave problem reduces to Helmholtz's Equation. In this context a method to solve using the Infinite Array Green Function was presented by Porter and Evans. For the more complicated problem where the depth dependence cannot be removed the Infinite Array Green Function method was used by Wang, Meylan, and Porter 2006 to solve for an Infinite Array of Floating Elastic Plates. The majoy challenge is to deal with very slowly convergent series (series which are not absolutely convergent).

Interaction Theory for Infinite Arrays methods

We can use Interaction Theory to solve for and infinite array. In general, we still need to solve for the individual scatterers using the Green Function Solution Method and we also have to consider slowly convergent series. Interaction Theory for Infinite Arrays do have some advantages and probably offer a superior method to solve the problem.