Diffraction of Acoustic Waves by a Wedge of Point Scatterers

Abstract

Analytical methods are fundamental in studying acoustics problems. One of the important tools is the Wiener-Hopf method, which can be used to solve many canonical problems with sharp transitions in boundary conditions on a plane/plate. However, there are some strict limitations to its use, usually the boundary conditions need to be imposed on parallel lines (after a suitable mapping). Such mappings exist for wedges with continuous boundaries, but for discrete boundaries, they have not yet been constructed. In our previous article, we have overcome this limitation and studied the diffraction of acoustic waves by a wedge consisting of point scatterers. Here, the problem is generalised to an arbitrary number of periodic semi-infinite arrays with arbitrary orientations. This is done by constructing several coupled systems of equations (one for every semi-infinite array) which are treated independently. The derived systems of equations are solved using the discrete Wiener-Hopf technique and the resulting matrix equation is inverted using elementary matrix arithmetic. Of course, numerically this matrix needs to be truncated, but we are able to do so such that thousands of scatterers on every array are included in the numerical results. Comparisons with other numerical methods are considered, and their strengths/weaknesses are highlighted.

Array scattering resonance in the context of Foldy’s approximation

“…(3.5) This was also the conclusion of [14,29]. Interestingly, if we were to evaluate A (s-inf) n using the approximate Wiener-Hopf kernel (see appendix A.3 of [17] for details), we would incorrectly obtain non-zero scattering coefficients due to the approximate kernel being finite at the branch points. Despite this, the Hankel summation given by (3.1) still numerically yields plane wave-like behaviour for the resonant scattered field (this is illustrated in figure 3a) but with an amplitude that depends on the truncation value M. We can show this if we take the Hankel summation (3.1) with the sum terms replaced by their asymptotic expansions as m → ∞,…”

“…Now we will examine the different resonant cases in the more interesting and less studied setting of a semi-infinite array. Recall the scattered field solution (figure 1b) given by (2.2) and (2.22) of [17] which was found via the discrete Wiener-Hopf technique,…”

“…In the context of Foldy's approximation, the total field due to a plane wave incident on an infinite array of small circular scatterers with homogeneous Dirichlet boundary conditions [17] (figure 1a) can be expressed as…”

“…Alternatively, one can keep simple unit cells but change the array configuration to study the effects of edges. This was the intention of our previous work [17] where we studied a wedge interface consisting of point scatterers. The notation and results of [17] will be used in this work as a starting point for infinite and semi-infinite array problems.…”

Asymptotic analysis of in-plane dynamic problems for elastic media with rigid clusters of small inclusions

“…Analytical procedures for modelling the influence of inclusions on dynamic properties of materials are also useful in designing composites for practical applications. Amenable to a range of approaches such as Floquet-Bloch theory, transfer matrix methods and hybrid approaches involving multiple scattering techniques and the Wiener-Hopf method [15], the associated models also shed light on possible designs of new devices with exotic properties including energy amplification [16], wave guiding [17][18][19], shielding [20,21], neutrality [22], localization [23] and cloaking [24,25]. Further, in this direction, at specific frequencies approximate theories such as homogenization [26,27] can be applied to capture the effective response of composites with periodic or statistically determined microstructures [28].…”