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.

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).

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 Green Function Methods 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.