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 1998 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. 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 1952 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 1952, who used Schlomilch series to sum slowly-convergent series involving Hankel functions, was re-considered by Linton and Evans Linton and Evans 1993 who used a so-called multipole method. For periodic arrays of rectangular cylinders extending uniformly through the depth, Fernyhough and Evans Fernyhough and Evans 1995, 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 Linton 1998 compared a number of different representations of the periodic Green function, designed to increase the convergence characteristics. Porter and Evans Porter and Evans 1999 used the work of Linton 1998 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 and Evans 1999, 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 and Newman 1997).

Merge the following

Introduction

The scattering of water waves by floating or submerged bodies is of wide practical importance in marine engineering. Although water waves are nonlinear, if the wave amplitude is sufficiently small, the problem can be well approximated by linear theory and linear wave theory remains the basis of most engineering design. It is also the standard model for many marine geophysical phenomena such as the wave forcing of ice floes.

The problem of the wave scattering by an infinite array of periodic and identical scatterers is a common model for wave scattering by a large but finite number of periodic scatterers, such as may be found in the construction of large off-shore structures. The periodic-array problem has been investigated by many authors. The infinite periodic-array problem in the context of water waves was considered by Spring75,Miles83,linton93,Falcao02 although the mathematical techniques for handling such arrays have a much older provenance dating back to early twentieth century work on diffraction gratings, e.g.~vonIgnatowsky14. All of the methods developed were for scattering bodies that have simple cylindrical geometry. This leads to a great simplification because the solution to the scattering problem can be found by separation of variables. If we want to consider scattering by a periodic array of scatterers of arbitrary geometry we require a modification to these scattering theories.

For a finite number of bodies of arbitrary geometry in water of finite depth, an interaction theory was developed by kagemoto86. This theory was based on Graf's addition theorem for Bessel functions which allows the incident wave on each body from the scattered wave due to all the other bodies to be expressed in the local cylindrical eigenfunction expansion. kagemoto86 did not present a method to determine the diffraction matrices for bodies of arbitrary geometry and this was given by goo90. The interaction theory was extended to infinite depth by JFM04. In this present paper we use this interaction theory to derive a solution for the problem of a periodic array of arbitrary shaped scatterers.

The use of the interaction theory of kagemoto86 for a periodic array requires us to find an efficient way to sum the slowly convergent series which arise in the formulation and to find an expression for the far field waves in terms of the amplitudes of the scattered waves from each body. The efficient computation of these kinds of slowly convergent series is due to twersky61,linton98 and the calculation of the far field is based on twersky62.

Recently, motivated by modelling of wave scattering in the marginal ice zone (MIZ), \citet*{JEM05} considered the scattering of a periodic array of elastic plates in water of infinite depth. Their method was based on an integral-equation formulation using a periodic Green's function. Beside its application to problems of finite depth, the work presented here is significantly more efficient than the method of JEM05, especially if multiple calculations are required for fixed types of bodies. Such multiple calculations are required by MIZ scattering models. Furthermore, confidence that the numerics are correct is one of the requirements for a successful wave scattering model. The results of \citeauthor{JEM05} provide a very strong numerical check for the numerics developed using the model presented here.

The MIZ is an interfacial region which forms at the boundary of the open and frozen ocean. It consists of vast fields of ice floes whose size is comparable to the dominant wavelength so that the MIZ strongly scatters the incoming waves. To understand wave propagation and scattering in the MIZ we need to understand the way in which large numbers of interacting ice floes scatter waves. One approach to this problem is to build up a model MIZ out of rows of periodic arrays of ice floes. A process of averaging over different arrangements will be required but from this, a kind of quasi two-dimensional model for wave scattering can be constructed. The accurate and efficient solution of the arbitrary periodic array scattering problem is the cornerstone of such a MIZ model. The standard model for an ice floe is a floating elastic plate of negligible draft \cite[]{Squire_Review}. A method of solving for the wave response of a single ice floe of arbitrary geometry in water of infinite depth was presented in JGR02. Furthermore, much research has been carried out on this model because of its additional application to very large floating structures such as a floating runway. Concerning this application, the current research is summarized in \citet*{kashiwagi00,watanabe_utsunomiya_wang04}.

The paper is organized as follows. We first give a precise formulation of the problem under consideration and recall the cylindrical eigenfunction expansions of the water velocity potential. Following the ideas of general interaction theories, we then derive a system of equations for the unknown coefficients of the scattered wavefield in the eigenfunction expansion. In this system, the diffraction transfer matrix as well as some slowly convergent series appear. The far field is then determined in terms of these coefficients and we explicitly show how the diffraction transfer matrices of arbitrary bodies and the slowly convergent series appearing in the system of equations can be efficiently calculated. The application of our method to the acoustic scattering by a periodic array of cylinders with arbitrary cross-section as well as the water-wave scattering by an array of fixed, rigid and flexible plates of shallow draft is discussed. Finally, we compare our results numerically to some computations from the literature and make some comparisons of arrays of fixed, rigid and flexible plates.

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 1999. 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.