Semi-infinite arrays of isotropic point scatterers. A unified approach

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

Scattering by a semi-infinite periodic array and the excitation of surface waves

The two-dimensional problem of acoustic scattering of an incident plane wave by a semi-infinite array of either rigid or soft circular scatterers is solved. Solutions to the corresponding infinite array problems are used, together with a novel filtering approach, to enable accurate solutions to be computed efficiently. Particular attention is focussed on the determination of the amplitude of the Rayleigh–Bloch waves that can be excited along the array. In general, the far field away from the array consists of sum of a finite number of plane waves propagating in different directions (the number depending on the observation angle) and a circular wave emanating from the edge of the array. In certain resonant cases (characterised by one of the scattered plane waves propagating parallel to the array), a different far field pattern occurs, involving contributions that are neither circular waves nor plane waves. Uniform asymptotic expansions that vary continuously across all of the shadow boundaries that exist are given for both cases

Scattering by a semi-infinite lattice and the excitation of Bloch waves

The interaction of a time-harmonic plane wave with a semi-infinite lattice of identical circular cylinders is considered. No assumptions about the radius of the cylinders, or their scattering properties, are made. Multipole expansions and Graf’s addition theorem are used to reduce the boundary value problem to an infinite linear system of equations. Applying the z transform and disregarding interaction effects due to certain strongly damped modes then leads to a matrix Wiener–Hopf equation with rational elements. This is solved by a straightforward method that does not require matrix factorisation. Implementation of the method requires that the zeros of the matrix determinant be located numerically, and once this is achieved, all far field quantities can be calculated. Numerical results that show the proportion of energy reflected back from the edge are presented for several different lattice geometries. 1

A new approximation method for scattering by long finite arrays

The scattering of water waves by a long array of evenly spaced, rigid, vertical circular cylinders is analysed under the usual assumptions of linear theory. These assumptions permit the reduction of the problem to that of solving the Helmholtz equation in two dimensions, with appropriate circular boundaries. Our primary goal is to show how solutions obtained for semi-infinite arrays can be combined to provide accurate and numerically efficient solutions to problems involving long, but finite, arrays. The particular diffraction problem considered here has been chosen both for its theoretical interest and for its applicability. The design of offshore structures supported by cylindrical columns is commonplace and understanding how the multiple interactions between the waves and the supports affect the field is clearly important. The theoretical interest comes from the fact that, for wavelengths greater than twice the geometric periodicity, the associated infinite array can support Rayleigh–Bloch surface waves that propagate along the array without attenuation. For a long finite array, we expect to see these surface waves travelling back and forth along the array and interacting with the ends. For particular sets of parameters, near-trapping has previously been observed and we provide a quantitative explanation of this phenomenon based on the excitation and reflection of surface waves by the ends of the finite array

Asymptotic approximations for Bloch waves and topological mode steering in a planar array of Neumann scatterers

We study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet-Bloch waves through an infinite array of scatterers. The approach also lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on some topical examples from topological wave physics

One-dimensional reflection by a semi-infinite periodic row of scatterers

AbstractThree methods are described in order to solve the canonical problem of the one-dimensional reflection by a semi-infinite periodic row of identical scatterers. The exact reflection coefficient R is determined. The first method is associated with shifting the domain by a single period and subsequently considering two scatterers, one being a single scatterer and the second being the entire semi-infinite array. The second method determines the reflection coefficient RN associated with a finite array of N scatterers. The limit as N→∞ is then taken. In general RN does not converge to R in this limit, although we summarize various arguments that can be made to ensure the correct limit is achieved. The third method considers direct approaches. In particular, for point masses, the governing inhomogeneous ordinary differential equation is solved using the discrete Wiener–Hopf technique