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Phase portraits of the mappings g(z) (left) and h(z) (right) by the formulae (2.19) and (2.20), respectively, where λ k = 1/2. In these figures, the black lines represent branch cuts. See Fig. 1 (right) for colour reference.
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The important open canonical problem of wave diffraction by a penetrable wedge is considered in the high-contrast limit. Mathematically, this means that the contrast parameter, the ratio of a specific material property of the host and the wedge scatterer, is assumed small. The relevant material property depends on the physical context and is differ...
Contexts in source publication
Context 1
... an example, Fig. 4 illustrates a phase portrait of (2.19) and (2.20) evaluated by MATLAB, where λ k = ...
Context 2
... cosine to manipulate the branch cuts, to g(z) and h(z) where z indicates the integer part. In the principal logarithm branch Re i ln(ie −iz ) ∈ (−π , π ]. This means our final definitions of g(z) and h(z) are, Both these definitions satisfy all of the required conditions including the requirement that they are the inverse of each other. See Fig. 4 for an example phase plot illustration of (A.9) and (A.10) where λ k = ...
Citations
... Note that (2.7) and (2.8) are only valid when λ = 1, and we refer to (Nethercote et al., 2020) for the general case. ...
We provide a description of the far-field encountered in the diffraction problem resulting from the interaction of a monochromatic plane-wave and a right-angled no-contrast penetrable wedge. To achieve this, we employ a two-complex-variable framework and use the analytical continuation formulae derived in (Kunz & Assier, QJMAM, 76(2), 2023) to recover the wave-field's geometrical optics components, as well as the cylindrical and lateral diffracted waves. We prove that the corresponding cylindrical and lateral diffraction coefficients can be expressed in terms of certain two-complex-variable spectral functions, evaluated at some given points.
... In previous articles [Nethercote et al., 2020a[Nethercote et al., , 2022a, we have extensively used COMSOL for comparison. However, there are limitations to the comparison since COMSOL is not able to find a solution with thousands Figure 5: Real part of total field for six different test cases. ...
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.
... An important parameter when studying the diffraction by a penetrable wedge is the contrast parameter λ which is defined as the ratio of either the electric permittivities ε 1,2 , magnetic permeabilities μ 1,2 , or densities ρ 1,2 corresponding to the material inside and outside the wedge, respectively, depending on the physical context, cf. section 2. The case of λ 1 (high contrast) is, for instance, studied in (6) and (7). In (6), Lyalinov adapts the Sommerfeld-Malyuzhinets technique to penetrable scatterers and concludes with a far-field approximation taking the geometrical optics components and the diffracted cylindrical waves into account whilst neglecting the lateral waves' contribution. ...
We study the problem of diffraction by a right-angled no-contrast penetrable wedge by means of a two-complex-variable Wiener–Hopf approach. Specifically, the analyticity properties of the unknown (spectral) functions of the two-complex-variable Wiener–Hopf equation are studied. We show that these spectral functions can be analytically continued onto a two-complex dimensional manifold, and unveil their singularities in C2. To do so, integral representation formulae for the spectral functions are given and thoroughly used. It is shown that the novel concept of additive crossing holds for the penetrable wedge diffraction problem, and that we can reformulate the physical diffraction problem as a functional problem using this concept.
... Beyond their importance to mathematical physics as one of the building blocks of the geometrical theory of diffraction [13], wedge diffraction problems also have applications to climate change modelling, as they are related to the scattering of light waves by atmospheric particles such as ice crystals, which is one of the big uncertainties when calculating the Earth's radiation budget (see [8,11,12] and [23]). A comprehensive literature overview dedicated to penetrable wedge diffraction problems can be found in the introductions of [15] and [19]. For a review on perfect wedge diffraction, we refer to [18]. ...
... We refer to [15] Section 2.1 for a more detailed discussion. Note that (2.8) and (2.9) are only valid when λ = 1, and we refer to [19] for the general case. Finally, we note that specifying the behaviour of the fields near the wedge's tip and at infinity is required to guarantee unique solvability of the problem described by equations (2.1)-(2.4), ...
... (A. 19) But sinceỹ 1 (resp.ỹ 2 ) is analytic onγ 2 (resp.γ 1 ), we find y 1 (τ r ) =ỹ 1 (τ l ), ∀τ ∈γ 1 , (A.20) y 2 (τ r ) =ỹ 2 (τ l ), ∀τ ∈γ 2 , (A. 21) giving f ≡ 0. ...
We study the problem of diffraction by a right-angled no-contrast penetrable wedge by means of a two-complex-variable Wiener-Hopf approach. Specifically, the analyticity properties of the unknown (spectral) functions of the two-complex-variable Wiener-Hopf equation are studied. We show that these spectral functions can be analytically continued onto a two-complex dimensional manifold, and unveil their singularities in $\mathbb{C}^2$. To do so, integral representation formulae for the spectral functions are given and thoroughly used. It is shown that the novel concept of additive crossing holds for the penetrable wedge diffraction problem and that we can reformulate the physical diffraction problem as a functional problem using this concept.
... Other innovative approaches suitable for high contrast penetrable wedge problems were given by Lyalinov [21] and, more recently, by Nethercote et al. [25]. In [21] Lyalinov uses the Sommerfeld-Malyuzhinets technique to obtain a system of two coupled Malyuzhinets equations which were solved approximately, giving the leading order far-field behaviour, whereas in [25] Nethercote et al. combine the Wiener-Hopf and Sommerfeld-Malyuzhinets method. ...
... Other innovative approaches suitable for high contrast penetrable wedge problems were given by Lyalinov [21] and, more recently, by Nethercote et al. [25]. In [21] Lyalinov uses the Sommerfeld-Malyuzhinets technique to obtain a system of two coupled Malyuzhinets equations which were solved approximately, giving the leading order far-field behaviour, whereas in [25] Nethercote et al. combine the Wiener-Hopf and Sommerfeld-Malyuzhinets method. In [25] a solution to the penetrable wedge is given as an infinite series of impenetrable wedge problems. ...
... In [21] Lyalinov uses the Sommerfeld-Malyuzhinets technique to obtain a system of two coupled Malyuzhinets equations which were solved approximately, giving the leading order far-field behaviour, whereas in [25] Nethercote et al. combine the Wiener-Hopf and Sommerfeld-Malyuzhinets method. In [25] a solution to the penetrable wedge is given as an infinite series of impenetrable wedge problems. Each of these impenetrable wedge problems is solved exactly, and the resulting infinite series for the penetrable wedge can be evaluated rapidly and efficiently using asymptotic and numerical methods. ...
In this paper, we revisit Radlow's innovative approach to diffraction by a penetra ble wedge by means of a double Wiener-Hopf technique. We provide a constructive way of obtaining his ansatz and give yet another reason for why his ansatz cannot be the true solution to the diffraction problem at hand. The two-complex-variable Wiener-Hopf equation is reduced to a system of two equations, one of which contains Radlow's ansatz plus some correction term consisting of an explicitly known integral operator applied to a yet unknown function, whereas the other equation, the compatibility equation, governs the behaviour of this unknown function.
... This means thatδ < δ = π 2θw when θ w ∈ π 2 , π and λ is sufficiently small, implying that the edge singularity is stronger than the impenetrable case. However, we believe that these last two equations are somewhat erroneous because they do not match with ours (see (1.57) from the introduction and equations (2.12) and (2.13) in Nethercote et al. (2019a) given in the next chapter). Evidence of our validity over Lyalinov can be shown by considering Numerical solutions of (4.13) are inconsistent with these values whereas numerical solutions of (1.57) are. 4 Lyalinov sought a general solution for the total fields in the form of a Sommerfeld integral (see Nethercote et al. (2019b), Section 2 for definition). ...
... Considering Figure 7 in Nethercote et al. (2019a), we shall deform the integration contour to C(z) which is defined as the path consisting of two straight lines joining the three points ζ = 0, sign(Im {z})z and i∞. This means that the formula for q (j) (z) is rewritten as, q (j) (z) = 1 2πi C(z)δ sin δ ζ h (ζ) cos δ ζ + sin δ z s (j) (θ w + h(ζ)) − s (j) (θ w − h(ζ)) −δ sin δ ζ h (ζ) cos δ ζ − sin δ z s (j) (−θ w + h(ζ)) − s (j) (−θ w − h(ζ)) dζ. ...
... In this last subsection, we discuss an alternate method to evaluate the Sommerfeld integrals. Nethercote et al. (2019a) discusses a strategy that involves fully deforming the Sommerfeld contours to the local steepest descent contours (SDCs). ...
In this thesis, various canonical problems of plane wave diffraction by infinite two-dimensional wedges are studied in both acoustic and electromagnetic physical settings. The thesis is divided into two main parts. The first, which is essentially an extensive review of extant methods, focuses on wedge diffraction with homogeneous Dirichlet or Neumann boundary conditions. It involves studying the well-known solution to Sommerfeld's half-plane diffraction problem and provides an extensive review of the literature on the perfect wedge problem, including analytic methods such as the Sommerfeld-Malyuzhinets and Wiener-Hopf techniques, as well as asymptotic techniques such as Keller's geometrical theory of diffraction. The second part of the thesis is dedicated to the problem of diffraction by a penetrable wedge. To this day, there is no clear analytical solution to this important canonical problem, but there have been numerous attempts at computational and asymptotic solutions by many reputable authors using extensions of techniques applied to perfect wedge diffraction. Because it is penetrable, the material properties between the wedge scatterer and the exterior host differ. It is hence possible to define the so-called contrast parameter as the ratio of specific material properties (depending on the physical context) between the host and the scatterer. Throughout the thesis, this parameter is considered to be small and the contrast is said to be high. This assumption allows construction of an asymptotic iterative scheme, which enables the penetrable wedge problem to be written as an infinite sequence of impenetrable wedge problems. All but the first of these impenetrable wedge problems are solved using a combination of the Sommerfeld-Malyuzhinets and Wiener-Hopf techniques. The result is a sequence of complex nested integrals which are evaluated using a subtle interplay of interpolation, asymptotic expansions and advanced complex analysis. For several test cases, including those illustrated in this thesis (that were chosen for mathematical convenience and consistency with the literature), the numerical results were in good agreement with alternative approaches.
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.
