Acoustic scattering by a semi-infinite angular sector with impedance boundary conditions

Acoustic scattering by a semi-infinite angular sector with impedance boundary conditions Abstract In this work we study the problem of diffraction of an acoustic plane wave by a semi-infinite angular sector with impedance boundary conditions on its surface. It is studied by means of incomplete separation of variables. With the aid of Watson–Bessel integral representation the problem is reduced to a boundary value problem on the unit sphere with an operator-impedance boundary condition on a cut of the sphere. The latter problem is further studied by means of the traditional methods of extensions of sectorial sesquilinear forms. The Sommerfeld integral representation is obtained from that of Watson–Bessel with the aim to develop the far-field asymptotics. Analytic properties of the corresponding Sommerfeld transformant are also discussed. For a narrow impedance sector, an asymptotic formula for the diffraction coefficient of the spherical wave propagating from the vertex is derived. 1. Introduction 1.1. Motivation and some comments on the literature To our knowledge, until present time the problem of diffraction by an impedance sector has not been discussed in the literature. It seems that the reason is in the analytic difficulties arising in the study of the problem in spite of the fact that the impedance boundary conditions are more realistic in practice in comparison with those ideal ones. We modify and adapt the approach recently developed by Lyalinov (2013), for the case of the sector with Dirichlet boundary conditions, in order to describe the far-field asymptotics in the problem of diffraction by a plane angular sector with impedance boundary conditions on its surface. Consider the unit sphere with the centre at the sector’s vertex then the sector and the sphere are intersected across a segment (‘cut’) AB of a big circle, Fig. 1. We assume that the angular measure 2a of the corresponding arc satisfies the restrictions 0 < 2a < π. One of the most interesting cases is the quarter-plane corresponding to 2a = π/2. Fig. 1. View largeDownload slide Diffraction by an impedance sector. Fig. 1. View largeDownload slide Diffraction by an impedance sector. In many aspects our approach has common features with those used for the problems of diffraction by cones with ideal (see e.g. Bowman et al., 1987; Felsen & Marcuvitz, 1973; Borovikov, 1966; Jones, 1964; Jones, 1997; Cheeger & Taylor, 1982; Smyshlyaev, 1991; Smyshlyaev, 1990; Babich et al., 2000; Bonner et al., 2005) or impedance boundary conditions. Diffraction by an impedance cone is studied in the papers of Bernard (1997), Bernard & Lyalinov (2001), and Bernard et al. (2008). Some additional results and references can be found in Lyalinov et al. (2010) and Lyalinov & Zhu (2012). As regards papers on diffraction by a sector with the ideal boundary conditions reader might be forwarded to discussion of the literature in our work (Lyalinov, 2013, see also Kraus & Levine, 1961; Hansen, 1991; Meister & Speck, 1988; Radlow, 1961; Albani, 2007; Valyaev & Shanin, 2012; Abawi et al., 1997; Assier & Peake, 2012a; Assier & Peake, 2012b; Budaev & Bogy, 2005). It is worth noting that the far-field asymptotics is of basic importance in numerous applications of such canonical problem like diffraction by an impedance sector. However, in order to develop the corresponding asymptotic expressions it is necessary to obtain the corresponding analytic tools, integral representations, to justify them and give a constructive way to obtain a solution and then to develop efficient numerical procedures. In this work we pay much attention to such tools implying that a detailed study of the far-field asymptotics will be discussed in a forthcoming publication. Nevertheless, we demonstrate the efficiency of the developed formulae by deducing a practically useful expression for the diffraction coefficient of the spherical wave from the vertex of a narrow impedance sector. In the modern interpretation a diffraction problem might be considered satisfactorily solved provided the following conditions are fulfilled. First, it is demonstrated that it has a unique solution and all necessary representations (integral or others) for the solution are established. Second, the far-field asymptotics, including those in the transition zones, are derived from such representations and the expressions for the diffraction coefficients are obtained. Finally, the numerical elaboration of the solution is given and the numerical results for the far field are represented. From this point of view, for instance, the problem of diffraction by a right-circular cone with ideal boundary conditions can be considered solved. As regard the problem of diffraction by an impedance sector, the study represented in this work basically deals with the first step of the solution in the above-mentioned sense. On the other hand, a formula for the diffraction coefficient in the so-called oasis (defined below) is derived and its asymptotic expression for a narrow impedance sector is given. 1.2. Description of the approach In the following sections we formulate the problem of diffraction by an impedance sector, Fig. 1. The wave field satisfies the Helmholtz equation, the impedance boundary conditions, Meixner’s conditions at the edges and at the vertex. Provided the scattered wave satisfies the radiation condition at infinity, the classical solution of the problem is unique. However, in the case of incidence of a plane wave the scattered field does not satisfy the radiation condition and we then discuss asymptotic behaviour of the wave field at infinity. We introduce a set of characteristic domains of the unit sphere centred at the vertex of the sector. Each point of such a domain on the sphere is attributed to some direction of observation. In different characteristic domains the far-field asymptotics consists of different wave field components. In order to separate the radial variable we make use of the Watson–Bessel integral representation for the solution and formulate a problem for the unknown ‘spectral’ function on the unit sphere with the cut $$\overline {AB}$$. An elliptic (Helmholtz type) equation for the spectral function depends on the variable of separation ν, whereas the boundary condition on the sides of the cut $$\overline {AB}$$ is nonlocal with respect to ν and has a form of the mixed boundary condition with an operator-impedance in it. We then carefully study such a problem attributing an m-sectorial operator to it. This operator is introduced by means of the corresponding sesquilinear form admitting a closed extension.1 Then the meromorphic continuation for spectral function w.r.t. ν is discussed. The latter exploits the corresponding Green’s theorem on the unit sphere with the cut and leads to an integral equation of the second kind for the spectral function on the cut. The Sommerfeld integral representation for the wave field is derived from that of Watson–Bessel and the properties of the Sommerfeld transformant (an analytic function in the integrand) are discussed. In particular, domain of regularity of the transformant is described, which enables one to point out domains on the complex plane, where singularities of the Sommerfeld transformant are located. The singularities of the integrand give rise to different far-field components provided the Sommerfeld integral is evaluated by means of the saddle point technique. Indeed, some of these singularities may be captured in the process of deformation of the Sommerfeld double loop contour into the steepest descent paths (Lyalinov, 2013). The contribution of the saddle points is responsible for the spherical wave from the vertex. Remark that, provided the observation point is in the space region, where only the spherical wave from the vertex is observed in the scattered far field (in the so-called ‘oasis’), these singularities are not intersected. For a narrow sector (2a ≪ 1) an approximate asymptotic expression for the diffraction coefficient of the spherical wave from the vertex in the oasis is deduced by means of the above-mentioned results. 2. Formulation of the problem Let us use the spherical coordinates (r, ϑ, φ) attributed to the Cartesian ones by the correlations   \begin{align*} X_{1}=r\cos\varphi\sin\vartheta,\,\, X_{2}=r\sin\varphi\sin\vartheta,\,\, X_{3}=r\cos\vartheta. \end{align*} We consider a plane wave2 which is incident from the direction specified by ω0 = (ϑ0, φ0) (Fig. 1)   \begin{align} U_{i}(r,\vartheta,\varphi)=\exp\left\{-\mathrm{i}kr\cos\theta_{i}(\omega,\omega_{0})\right\}, \end{align} (2.1) where ω = (ϑ, φ) corresponds to the direction of observation and   \begin{align*} \cos\theta_{i}(\omega,\omega_{0})=\cos\vartheta\cos\vartheta_{0}+ \sin\vartheta\sin\vartheta_{0} \cos[\varphi-\varphi_{0}],\end{align*} θi(ω, ω0) coincides with the geodesic distance between two points ω and ω0 denoted also θ(ω, ω0), θ(ω, ω0) = θi(ω, ω0). The wave field U(r, ϑ, φ) + Ui(r, ϑ, φ) is the sum of the scattered and incident fields, U fulfils the Helmholtz equation   \begin{align} (\triangle +k^{2})U(r,\vartheta,\varphi)=0, \end{align} (2.2) k > 0 is the wave number. Let $$\varSigma =S^{2}\setminus \overline {AB}$$ be the exterior of the cut on S2 and σ = ∂Σ, σ = S ∩ S2 its boundary, σ = σ+ ∪ σ− and σ± are two sides of the cut AB. The impedance boundary condition   \begin{align} \left.\frac{1}{r}\frac{\partial (U_{i}+U)}{\partial\mathscr{N}_{\pm}}\right|_{S_{\pm}}\,- \, \mathrm{i}k\eta_{\pm}\,(U_{i}+U)|_{S_{\pm}}=0, \end{align} (2.3) is satisfied on the sector S, S± are two faces of the sector $$S=\{(r,\omega ): \, r\geqslant 0, \omega \in \overline {AB}\}$$ corresponding to σ± on S2. The vectors $$\mathscr {N}_{\pm }$$ are in the tangent plane to S2 at the points of σ±, are orthogonal to σ± and point out to the ‘exterior’ of Σ. The surface impedances η± = ε± + iχ± do not depend on k, ε± > 0, which means absorption of the wave energy on the sector’s faces.3 The case of reactive faces (ε± = 0) can be considered as a limiting one. The Meixner’s edge conditions are assumed near the edges ∂Si, i = 1, 2 (and outside some close vicinity of the vertex)   \begin{align} U\sim C_{1,2}^{0}(z)+ C_{1,2}(z,\phi)\rho^{1/2}+\dots,\,\,\rho\to 0 \end{align} (2.4) uniformly bounded with respect to z, ϕ, assuming that ρ, ϕ, z are natural local cylindrical coordinates attributed to the edges ∂Si, i = 1, 2, where the index i is omitted for the coordinates. We connect notations A and B with the edges ∂S1 and ∂S2 correspondingly. The conditions at the vertex of the sector take the form   \begin{align} |U|\,\leqslant\, C \,r^{-1/2+\epsilon},\,\,|\nabla U|\,\leqslant \,C\, r^{-3/2+\epsilon},\,\,r\to 0 \end{align} (2.5) which are valid uniformly with respect to the angular variables for some positive ɛ, see Section 5.7 in Van Bladel (1991). We are looking for a classical solution of the problem, i.e. $$U\in C^{2}_{\text {loc}}(R^{3}\setminus \overline {S})$$ and such that $$\frac {1}{r}\frac {\partial U}{\mathscr {N}_{\pm }}$$ and U exist and are continuous on S±. Now we turn to the behaviour of the solution at infinity. 2.1. Uniqueness Let us now assume that the scattered field satisfies the radiation condition (which is not the case for the incidence of a plane wave)   \begin{align} \int_{S_{R}\setminus S}\,|\partial_{r} U-\mathrm{i}k U|^{2}\,\mathrm{d}s\,\to\,0\,, \,\,\, \,R\,\to\, \infty\,,\,\,\varepsilon_{\pm}>0\,, \end{align} (2.6) where $$S_{R}=\{(r,\omega ):\,r=R\,,\,\, \omega \in S^{2}\setminus \overline {\sigma }\}$$, ω = (ϑ, φ). Theorem 2.1 The classical solution of the homogeneous problem (2.2)--(2.6) (i.e. with Ui = 0) is trivial, U ≡ 0. Consider a ball Bδ of small radius δ centred at the vertex O with the boundary Sδ, ∂Bδ := Sδ, Fig. 2. Also we introduce semi-infinite cylinders $$C^{i}_{\delta }$$ of the radius δ having the axis ∂Si, i = 1, 2. We denote dδ the set $$B_{\delta }\cup C^{1}_{\delta }\cup C^{2}_{\delta }$$ with the external boundary ∂dδ, see Fig 2. The domain dδ is a δ −neighbourhood of the edges ∂S1 ∪ ∂S2. Let BR be a ball of the large radius R, ∂BR := SR, then Dδ, R = BR ∖ (S ∪ dδ) is a part of this ball being exterior to S and to dδ. Remark that the boundary of the domain Dδ, R is ∂Dδ, R consisting of ∂dδ, R := ∂dδ ∩ BR, $$S^{\pm }_{\delta ,R}:=B_{R}\cap ( S_{\pm }\setminus d_{\delta })$$ and Sδ, R := SR ∖ (dδ ∪ S). Fig. 2. View largeDownload slide Domains in the proof of uniqueness. Fig. 2. View largeDownload slide Domains in the proof of uniqueness. We apply the Green’s identity   \begin{align*} \int_{D_{\delta,R}}\triangle U\,\overline{U}\mathrm{d}x \,=\,\int_{\partial D_{\delta,R}} \partial_{n} U\,\overline{U}\,\mathrm{d}s \,-\,\int_{D_{\delta,R}}\,| \nabla U |^{2}\,\mathrm{d}x \end{align*} to U and $$\overline {U}$$ in Dδ, R, exploit the Helmholtz equation, boundary conditions and take the imaginary part. Simple calculations lead to   \begin{align*} k\sum_{\pm}\,\Re{(\eta_{\pm})}\,\int_{S^{\pm}_{\delta,R}}\,| U |^{2}\,\mathrm{d}s\, +\,\Im\left( \int_{\partial d_{\delta,R}} \partial_{n} U\,\overline{U}\,\mathrm{d}s\right)\,+\, \,\Im\left( \int_{ S_{\delta,R}} \partial_{r} U\,\overline{U}\,\mathrm{d}s\right)\,\,=\,0, \end{align*} where ∂n is the differentiation along the normal directed outward w.r.t. Dδ, R. Let δ → 0, we apply the Meixner’s conditions at the edges and vertex. The integral over ∂dδ, R vanishes. We then verify the chain of inequalities   \begin{align}\displaystyle 0\,&\leqslant\,k\sum_{\pm}\,\varepsilon_{\pm}\,\int_{S^{\pm}_{0,R}}\,| U |^{2}\,\mathrm{d}s\,=\,-\, \Im\left( \int_{ S_{R}\setminus S} \partial_{r} U\,\overline{U}\,\mathrm{d}s\right)\,\,\\ &=\, \,-\,\Im\left( \int_{ S_{R}\setminus S}\left(\partial_{r} U\,-\,\mathrm{i}k U\right)\overline{U}\,\mathrm{d}s\right)\,-\, k\int_{ S_{R}\setminus S} |U|^{2}\,\mathrm{d}s\,\nonumber\\&\leqslant\, \left( \int_{ S_{R}\setminus S}\left|\partial_{r} U\,-\,\mathrm{i}k U\right|{}^{2}\,\mathrm{d}s\right)^{1/2}\,\left( \int_{ S_{R}\setminus S} |U|^{2}\,\mathrm{d}s\right)^{1/2}-\, k\int_{ S_{R}\setminus S} |U|^{2}\,\mathrm{d}s\nonumber. \end{align} (2.7) The latter expression is non-negative and from the radiation condition we have   \begin{align*} \displaystyle k\left(\int_{ S_{R}\setminus S} |U|^{2}\,\mathrm{d}s\right)^{1/2}\,\leqslant\,\left( \int_{ S_{R}\setminus S}\left|\partial_{r} U\,-\,\mathrm{i}k U\right|{}^{2}\,\mathrm{d}s\right)^{1/2}\,\,\rightarrow\,\,0 \end{align*} as $$R\to \infty $$. We find that   \begin{align*} k\lim_{R\to\infty}\int_{ S_{R}\setminus S} |U|^{2}\,\mathrm{d}s\,=\,0\, \end{align*} and from (2.7)   \begin{equation*} \,\int_{S_{\pm}}\,| U |^{2}\,\mathrm{d}s\,=0, \end{equation*}  \begin{equation*} {\hskip28pt}U|_{S_{\pm}}\,=\,0. \end{equation*} Making use of the boundary conditions, we find   \begin{align*} \left.\frac{1}{r}\frac{\partial U}{\partial\mathscr{N}_{\pm}}\right|_{S_{\pm}}\,=0. \end{align*} As a result, we deal with the homogeneous Cauchy problem for an elliptic (Helmholtz) equation satisfied by U with the trivial initial conditions on the surface S±. Such a solution is known to be trivial, U ≡ 0. As we have already mentioned the radiation condition (2.6) is not valid for the scattered field excited by a plane incident wave. It must also be modified appropriately as ε+ = 0 or ε− = 0. In such cases surface waves can be excited and propagate along the sector’s surface at infinity without attenuation. 2.2. The far-field asymptotics In order to describe the asymptotic behaviour of the scattered field as $$r\to \infty $$ it is useful to define some sub-domains on the unit sphere S2 with the cut $$\sigma = \overline {AB}$$. Let $$\varOmega _{\delta _{0}}$$ be a close vicinity of the cut on the unit sphere   \begin{align*} \displaystyle \varOmega_{\delta_{0}}=\,\left\{\omega\in S^{2}:\,\,\,\text{dist}(\omega,\sigma)<\delta_{0}\right\}\!, \end{align*} where δ0 is some small positive constant. In what follows we assume that $$\omega _{0}\in S^{2}\setminus \varOmega _{\delta _{0}}$$, i.e. the incident plane wave arrives from the directions being not very close to $$\overline {AB}$$. Consider the geodesic distance θ(ω, ω0) between two points ω and ω0 on the sphere S2 (see, Lyalinov, 2013, Section 2 for details). In the same manner, introduce the ‘broken’ geodesic θr(ω, ω0). This geodesic has simple geometrical meaning: this is a broken geodesic of the minimal length which originates at the source ω0 then reflects on the boundary $$\sigma =\overline {AB}$$ in accordance with geometrical optics laws and arrives at the point ω. The ‘incident’ parts of such broken geodesics fill in the spherical triangle ω0AB, whereas the ‘reflected’ parts fill in the spherical triangle FrBA which is further denoted Ωr =: Ωr(ω0), (see Figs 3–5). Fig. 3. View largeDownload slide The triangular domain Ωr with the vertexes ABFr on S2. Fig. 3. View largeDownload slide The triangular domain Ωr with the vertexes ABFr on S2. Fig. 4. View largeDownload slide The triangular domain $$\varOmega _{r}^{*}$$ with the vertexes ABFi on S2. Fig. 4. View largeDownload slide The triangular domain $$\varOmega _{r}^{*}$$ with the vertexes ABFi on S2. Fig. 5. View largeDownload slide The domain ΩA on S2. Fig. 5. View largeDownload slide The domain ΩA on S2. Specify the spherical triangular domain $$\varOmega _{r}^{*}$$ coinciding with the triangle ABFi in Fig. 3. The domain $$\varOmega _{r}^{*}$$ is the mirror image of Ωr with respect to the boundary σ for the same fixed position of ω0. We also make use of two other domains ΩA(ω0) (see Fig. 5) and ΩB(ω0). Consider the ray (geodesic) ω0A which arrives at the edge point A and produces a set of ‘diffracted’ rays (geodesics) outgoing from A in all direction. For each point ω there exists such a diffracted ray with length ψA(ω), (ψA(ω) < π) that arrives at this point. We define ΩA(ω0) as a domain on the sphere such that $$\varOmega _{A}(\omega _{0})=\{\omega \in S^{2}:\,\,0\leqslant \theta _{A}(\omega ,\omega _{0}):=\theta _{i}(A,\omega _{0})+\psi _{A}(\omega )<\pi \}$$, see also Section 2 in Lyalinov (2013). The domains ΩA and ΩB on S2 intersect with Ωr and $$\varOmega _{r}^{*}$$ and with each other. The domain $$\varOmega _{r}^{*}$$ corresponds to directions in which geometrical shadow of the incident wave is observed. Ωr forms a set of directions of propagation of the space rays reflected from the sector S. The domain ΩA (or ΩB) corresponds to the directions for which the diffracted cylindrical wave from the edge A (or from B) is present in the far-field asymptotics. These simple facts follow from the analysis also presented in Lyalinov (2013). Remark It is worth noting that we can also introduce domain ΩAB (and analogously ΩBA), see Lyalinov (2013) as well as the domains $$\varOmega _{ABA},\varOmega _{BAB},\varOmega _{ABAB}\dots , $$ etc. which are defined and interpreted analogously. They correspond to multiple diffractions. We introduce the domain Ω0 = S2 ∖ (ΩA ∪ ΩB ∪ Ωr) on the sphere S2, which is called ‘oasis’. The scattered far-field U (total minus incident) in this domain of directions consists of the spherical wave propagating from the vertex of the sector   \begin{align} U(r,\vartheta,\varphi)=D(\omega,\omega_{0})\frac{\exp(\mathrm{i}kr)}{-\mathrm{i}kr}\left(1+ O\left(\frac{1}{kr}\right)\right),\,\,\, kr\to \infty. \end{align} (2.8) In the asymptotics (2.8), which is non-uniform with respect to $$\omega \in \overline {\varOmega }_{0}$$, the diffraction coefficient D(ω, ω0) is one of the most important characteristics of the scattered field.4 In the exterior of the oasis the structure of asymptotics is more complex and contains also other wave components in the far field. Consider the directions from $$(S^{2}\setminus (\varOmega _{r}^{*}\cup \varOmega _{r}\cup \varOmega _{B}))\cap \varOmega _{A}$$ in which the spherical wave, the cylindrical wave from the edge A as well as surface waves (possibly also other multiply diffracted) are observed in the far field   \begin{align} U(r,\vartheta,\varphi)&=D(\omega,\omega_{0})\frac{\exp(\mathrm{i}kr)}{-\mathrm{i}kr}\left(1+ O\left(\frac{1}{kr}\right)\right)\nonumber \\[10pt] &\quad+ d_{A}(\omega,\omega_{0})\,\frac{\exp\left(-\mathrm{i}kr\cos\theta_{A}(\omega,\omega_{0})\right)} {\sqrt{-\mathrm{i}kr\sin\psi_{A}}}\left(1+ O\left(\frac{1}{kr\sin\psi_{A}}\right)\right)+V_{s}(r,\vartheta,\varphi) +\dots, \end{align} (2.9) where dots denote the waves multiply diffracted from the edges, provided the corresponding directions belong also to $$\varOmega _{BA}, \varOmega _{ABA},\dots $$ .5 The yet unknown function dA(ω, ω0) in (2.9) is connected with the diffraction coefficient of the cylindrical wave from the edge A. The summand Vs(r, ϑ, φ) is the sum of the surface waves generated by the interaction of the incident waves with the vertex or edges. It is only essential provided the observation point is in the close vicinity of the sector, $$\omega \in \varOmega _{\delta _{0}}$$ and as ε± = 0, χ± < 0,   \begin{align}{}\displaystyle V_{s}(r,\vartheta,\varphi)&= \nonumber D_{s}(\omega,\omega_{0})\,\frac{\exp(-\mathrm{i}kr\cos\theta_{s}(\omega,\omega_{0}))}{\sqrt{-\mathrm{i}kr}} \left(1+ O\left(\frac{1}{kr}\right)\right)\\ &\quad+ {D_{s}^{A}}(\omega,\omega_{0})\,\exp\left(-\mathrm{i}kr{\cos\theta^{A}_{s}}(\omega,\omega_{0})\right) + {D_{s}^{B}}(\omega,\omega_{0})\,\exp\left(-\mathrm{i}kr{\cos\theta^{B}_{s}}(\omega,\omega_{0})\right) +\dots\, \end{align} (2.10) with complex valued eikonals $$\theta _{s}(\omega ,\omega _{0}),\theta _{s}^{A,B}(\omega ,\omega _{0})$$. These eikonals solve the equations   \begin{align} (\nabla_{\omega} \theta_{s})^{2}\,=\,1,\,\,\,\,\,\left(\nabla_{\omega} \theta_{s}^{A,B}\right)^{2}\,=\,1, \end{align} (2.11) satisfying the conditions   \begin{align*} \Im(\theta_{s})|_{\sigma}&=0,\,\,\,\Im(\theta_{s})>0,\,\,\,0\leqslant\Re(\theta_{s}) <\pi\!,\,\,\\ \Im(\theta_{s}^{A,B})|_{\sigma}&=0,\,\,\,\Im(\theta_{s}^{A,B})>0\,\,\,0\leqslant\Re(\theta_{s}^{A,B}) <\pi\,\, \end{align*} as ε± = 0, χ± < 0. In the domain $$(S^{2}\setminus (\varOmega _{r}^{*}\cup \varOmega _{r}\cup \varOmega _{A}))\cap \varOmega _{B}$$ the asymptotics has the same form as in (2.9) with the change of the subscript A on to B in the second summand which describes the cylindrical wave from the edge B. In the directions ω from Ωr ∩ (ΩB ∪ΩA) the leading terms consist of the reflected, spherical and diffracted (from the edges A and B) waves (as well as surfaces ones)   \begin{align} U(r,\vartheta,\varphi)=&\ R(\omega,\omega_{0}) \exp\left(-\mathrm{i}kr\cos\theta_{r}(\omega,\omega_{0})\right)+ D(\omega,\omega_{0})\frac{\exp(\mathrm{i}kr)}{-\mathrm{i}kr}\left(1+ O\left(\frac{1}{kr}\right)\right)\nonumber\\&+\ d_{A}(\omega,\omega_{0})\,\frac{\exp\left(-\mathrm{i}kr\cos\theta_{A}(\omega,\omega_{0})\right)}{\sqrt{-\mathrm{i}kr\sin\psi_{A}}} \left(1+ O\left(\frac{1}{kr\sin\psi_{A}}\right)\right)\\\nonumber&+\ d_{B}(\omega,\omega_{0})\,\frac{\exp\left(-\mathrm{i}kr\cos\theta_{B}(\omega,\omega_{0})\right)}{\sqrt{-\mathrm{i}kr\sin\psi_{B}}} \left(1+ O\left(\frac{1}{kr\sin\psi_{B}}\right)\right)+V_{s}(r,\vartheta,\varphi) \,+ \dots, \end{align} (2.12) where R(ω, ω0) is the reflection coefficient in the first summand $$U_{r}(\omega ,\omega _{0})=R \exp (-\mathrm {i}kr\cos \theta _{r}(\omega ,\omega _{0}))$$ of (2.12) which is the reflected wave. The summand Vs(r, ϑ, φ) has the same meaning as in the formula (2.9). The wave field U + Ui in the shadow of the incident wave, i.e. as $$\omega \in \varOmega _{r}^{*}\cap (\varOmega _{B}\cup \varOmega _{A})$$, reads   \begin{align} U(r,\vartheta,\varphi)+U_{i}(r,\vartheta,\varphi)=&\ D(\omega,\omega_{0})\frac{\exp(\mathrm{i}kr)}{-\mathrm{i}kr}\left(1+ O\left(\frac{1}{kr}\right)\right)\nonumber\\&+\ d_{A}(\omega,\omega_{0})\,\frac{\exp\left(-\mathrm{i}kr\cos\theta_{A}(\omega,\omega_{0})\right)} {\sqrt{-\mathrm{i}kr\sin\psi_{A}}}\left(1+ O\left(\frac{1}{kr\sin\psi_{A}}\right)\right)\nonumber\\ &+\ d_{B}(\omega,\omega_{0})\,\frac{\exp\left(-\mathrm{i}kr\cos\theta_{B}(\omega,\omega_{0})\right)} {\sqrt{-\mathrm{i}kr\sin\psi_{B}}}\left(1+ O\left(\frac{1}{kr\sin\psi_{B}}\right)\right)\nonumber\\ &+\ V_{s}(r,\vartheta,\varphi) + \dots, \end{align} (2.13) $$kr\to \infty $$. We do not describe the asymptotics of the far field in the transition domains, where the expressions depend on special transition functions, see e.g. Lyalinov (2013), Sections 6 and 7. It is worth mentioning the work of Shanin (2011), where similar asymptotics for diffraction by an ideal circular cone were established. 3. Watson–Bessel integral representation of the solution and reduction to the problem for the spectral function We turn to the separation of the radial variable of the solution which exploits the Watson–Bessel integral representation. We begin with that for the incident wave. 3.1. Watson–Bessel integral representation for the incident wave We make use of the known Watson–Bessel integral representation for the incident wave (2.1)   \begin{align} U_{i}(r,\vartheta,\varphi)= 4\mathrm{i}\sqrt{\frac{\pi}{2}}\int\limits_{C_{\phi}}\nu \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}u_{\nu}^{i}(\omega,\omega_{0}) \frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\,d \nu, \end{align} (3.1) where $$u_{\nu }^{i}(\omega ,\omega _{0})=-\frac {P_{\nu -1/2}(-\cos \theta _{i}(\omega ,\omega _{0}))}{4\cos \pi \nu },$$  \begin{align} \left(\triangle_{\omega}+(\nu^{2}-1/4)\right)u_{\nu}^{i}(\omega,\omega_{0})=\delta(\omega-\omega_{0}), \end{align} (3.2) Pν−1/2(x) is the Legendre function and Cϕ with ϕ ∈ [0, π/2) is shown in Fig. 6. The contour Cϕ is traditionally taken for ϕ = 0; however, for some reductions it is possible also to use Cϕ with ϕ ∈ (0, π/2). The ‘spectral’ function $$u_{\nu }^{i}(\omega ,\omega _{0})$$ corresponding to the incident plane wave admits the estimate  \begin{align}\displaystyle \left|u_{\nu}^{i}(\omega,\omega_{0})\right|\,<\,C\,\frac{1}{\sqrt{|\nu|}}\exp\left\{-|\nu|\,|\sin\phi|\,\theta(\omega,\omega_{0})\right\}\!, \end{align} (3.3) Fig. 6. View largeDownload slide The contours C0 and Cϕ. Fig. 6. View largeDownload slide The contours C0 and Cϕ. as $$|\nu |\to \infty $$ and ν ∈ Cϕ. Remark that $$J_{\nu }(kr)\sim \frac {[kr/2]^{\nu }}{\varGamma (\nu +1)},\,\,\,\Re \,\nu \to +\infty $$, $$|\arg \nu |\,<\,\pi /2$$. The estimate of the integrand in (3.1) on Cϕ as $$|\nu |\to \infty $$, ν = |ν|eiϕ, $$0\leqslant \phi < \pi /2$$ is given by6  \begin{align} \left|\nu\,\mathrm{e}^{-\mathrm{i}\nu\pi/2}u_{\nu}^{i}(\omega,\omega_{0}) {J_{\nu}(kr)}\right|\,<\,& C\, \exp\left\{-|\nu|(\log|\nu|-1)\cos\phi- |\nu|(\sin\phi[{\arg(kr/2)-\pi/2-\phi}]\right. \nonumber\\[-5pt] &\qquad\quad\left.+\,|\sin\phi|\theta(\omega,\omega_{0})-\cos\phi\log|kr/2|)\right\}. \end{align} (3.4) It is assumed in (3.4) that k may be complex with |argk| < π/2 although we consider argk = 0 in this work. Actually the constants in the estimates (3.4) and (3.3) are different; however, for convenience we take the maximal one and denote it by C. It is also useful to have these estimates not only for real k > 0 but for $$|\arg k|<\pi /2$$ as well. The integral in (3.1) then rapidly converges. It is worth noting that the function $$u_{\nu }^{i}(\omega ,\omega _{0})$$ is even with respect to ν and is holomorphic in the strip Π1/2,   \begin{align*} \varPi_{\delta}=\left\{\nu\in\mathrm{C}:\,\,\,|\Im(\nu)|<\delta\right\}\!.\end{align*} Its simple poles are located at zeros of $$\cos \pi \nu $$. 3.2. Watson–Bessel integral representation for the scattered field and separation of the radial variable In order to separate the radial variable for the problem at hand we look for the solution in the integral form   \begin{align} U(r,\vartheta,\varphi)= 4\mathrm{i}\sqrt{\frac{\pi}{2}}\int\limits_{C_{0}}\nu \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}\,u_{\nu}(\omega,\omega_{0}) \frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\,d \nu \end{align} (3.5) with unknown ‘spectral’ function uν(ω, ω0). We begin with some formal reductions enabling us to motivate our approach and to make reasonable assumptions about the basic properties of the spectral function to be constructed. Let uν(ω, ω0) satisfy the equation   \begin{align} \left(\triangle_{\omega}+(\nu^{2}-1/4)\right)u_{\nu}(\omega,\omega_{0})=\,0\, \end{align} (3.6) in the classical sense, i.e. uν(⋅, ω0) ∈ C2(Σ), Σ = S2 ∖ σ then, as is well known (see e.g. Smyshlyaev, 1991 and others), the scattered field U(r, ϑ, φ) in (3.5) solves the Helmholtz equation (2.2). Now we turn to the boundary condition (2.3)   \begin{align*} &\left.\frac{1}{r}\frac{\partial (U_{i}+U)}{\partial\mathscr{N}_{\pm}}\right|_{\omega\in\sigma_{\pm}}\,- \, \mathrm{i}k\eta_{\pm}\,(U_{i}+U)|_{\omega\in\sigma_{\pm}}\\ &\quad= 4\mathrm{i}\sqrt{\frac{\pi}{2}}\int\limits_{{C_{0}^{b}}}\nu \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}\left(\left. \frac{\partial \widehat{u}_{\nu}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\, \frac{(-\mathrm{i})J_{\nu}(kr)}{(-\mathrm{i}kr)^{3/2}}\,-\,\mathrm{i}\eta_{\pm}\, \widehat{u}_{\nu}(\omega,\omega_{0})|_{\sigma_{\pm}}\,\frac{J_{\nu}(kr)}{(-\mathrm{i}kr)^{1/2}}\right)\,d \nu\,=\,0, \end{align*} where $${C_{0}^{b}}$$ is shown in Fig. 7, $$\widehat {u}_{\nu }(\omega ,\omega _{0})={u}_{\nu }(\omega ,\omega _{0})+{u}_{\nu }^{i}(\omega ,\omega _{0}) $$. We assumed that uν is holomorphic7 and even w.r.t. ν in some vicinity of the imaginary axis, meromorphic in C and some of its possible sigularities are located inside the contour $${C_{0}^{b}}$$. Fig. 7. View largeDownload slide Integration contour $${C_{0}^{b}}$$ and its deformation. Fig. 7. View largeDownload slide Integration contour $${C_{0}^{b}}$$ and its deformation. We take into account the identity zJν−1(z) + zJν+1(z) = 2νJν(z). After some reductions one has   \begin{align*} 4\sqrt{\frac{\pi}{2}}\int\limits_{{C_{0}^{b}}}\,d \nu\, \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}\left(\left. \frac{\partial \widehat{u}_{\nu}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\, \frac{\mathrm{i}(J_{\nu-1}(kr)+J_{\nu+1}(kr))}{2\sqrt{-\mathrm{i}kr}}\,+\,\eta_{\pm}\,\nu\, \widehat{u}_{\nu}(\omega,\omega_{0})\Bigg|_{\sigma_{\pm}}\,\frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\right)\,=\,0, \end{align*} then appropriately change the variables of integration so that   \begin{align*} &2\mathrm{i}\sqrt{\frac{\pi}{2}}\left(\int\limits_{{C_{0}^{b}}-1}\!d \nu\, \,\mathrm{e}^{-\mathrm{i}(\nu+1)\pi/2}\left. \frac{\partial \widehat{u}_{\nu+1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\, \frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\!+\! \int\limits_{{C_{0}^{b}}+1}\!d \nu\, \,\mathrm{e}^{-\mathrm{i}(\nu-1)\pi/2}\left. \frac{\partial \widehat{u}_{\nu-1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\, \frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\,\right.\\ &\qquad\qquad\left. -\, 2\mathrm{i}\eta_{\pm}\,\int\limits_{{C_{0}^{b}}}\,d \nu\,\nu\,\mathrm{e}^{-\mathrm{i}\nu\pi/2} \widehat{u}_{\nu}(\omega,\omega_{0})\bigg|_{\sigma_{\pm}}\,\frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\,\right)\,=\,0, \end{align*} where the contours of integration $${C_{0}^{b}}+1$$ and $${C_{0}^{b}}-1$$ are those in Fig. 7 obtained by shifting $${C_{0}^{b}}$$ along the real axis to the right- or left-hand side accordingly and $${C_{0}^{b}}$$ consists of the parts $$(\infty -\mathrm {i}b,-\mathrm {i}b],$$ [−ib, ib] and $$[\mathrm {i}b, \infty +\mathrm {i}b)$$. The contours $${C_{0}^{b}}+1$$ and $${C_{0}^{b}}-1$$ are explicitly described by $$\{{C_{0}^{b}}+1\}=(\infty -\mathrm {i}b+1,-\mathrm {i}b+1]\cup [-\mathrm {i}b+1,\mathrm {i}b+1]\cup [\mathrm {i}b+1, \infty +\mathrm {i}b+1)$$ and $$\{{C_{0}^{b}}-1\}=(\infty -\mathrm {i}b-1,-\mathrm {i}b-1]\cup [-\mathrm {i}b-1,\mathrm {i}b-1]\cup [\mathrm {i}b-1, \infty +\mathrm {i}b-1)$$. Let us now assume that uν(ω, ω0) is taken meromorphic w.r.t. ν ∈ C with the poles inside some strip |I(ν)| < b, holomorphic in Πδ for some small δ > 0 and such that $$\left . \frac {\partial \widehat {u}_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}$$ is regular (holomorphic) in the strip Πδ+1. We are now able to deform the contours $${C_{0}^{b}}+1$$ and $${C_{0}^{b}}-1$$ into that $${C_{0}^{b}}$$ in Fig. 7 thus having   \begin{align*} 2\sqrt{\frac{\pi}{2}}\,\int\limits_{{C_{0}^{b}}}\,d \nu\,\mathrm{e}^{-\mathrm{i}\nu\pi/2}\left( \left.\frac{\partial \widehat{u}_{\nu+1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\,-\, \left.\frac{\partial \widehat{u}_{\nu-1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}} +2\eta_{\pm}\nu\, \widehat{u}_{\nu}(\omega,\omega_{0})|_{\sigma_{\pm}}\,\right)\frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\,\,=\,0. \end{align*} We conclude that, provided $$\widehat {u}_{\nu } (\omega ,\omega _{0})=u_{\nu }(\omega ,\omega _{0})+u_{\nu }^{i}(\omega ,\omega _{0})$$ satisfies the boundary condition on σ = σ+ ∪ σ−  \begin{align} \left.\frac{\partial \widehat{u}_{\nu+1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\,-\, \left.\frac{\partial \widehat{u}_{\nu-1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}} +\,2\nu\,\eta_{\pm}\, \widehat{u}_{\nu}(\omega,\omega_{0})|_{\sigma_{\pm}}\,=\,0, \end{align} (3.7) the wave field U + Ui fulfils the boundary condition (2.3). Remark The condition (3.7) is nonlocal with respect to the spectral variable ν, which means that, contrary to the Helmholtz equation, the radial variable in the mixed boundary condition (2.3) is not separable in a traditional meaning. In order to transform the boundary condition to an alternative form we take into account a simple Lemma which actually follows from the known technique developed for a class of functional equations, Babic et al. (2008), Chapter 7. Lemma 3.1 Let H(ν) be holomorphic as ν ∈ Πδ and $$|H(\nu )|\leqslant C\mathrm {e}^{-\kappa |\nu |}\,,\,|\nu |\to \infty \,\,, \kappa>0$$ in this strip, H(ν) = −H(−ν). Then an even solution s(ν) of the equation   \begin{align*} s(\nu+1)-s(\nu-1)= -2\mathrm{i}H(\nu), \end{align*} which is regular (holomorphic) in the strip ν ∈Π1+δ and exponentially vanishes as $$|\nu |\to \infty $$ there, is given by   \begin{align*} s(\nu)&=\frac{1}{4}\int\limits_{{-\textrm i}\infty}^{\mathrm{i}\infty}\, d\tau\, H(\tau)\,\left( \frac{\sin(\pi\tau/2)}{\cos(\pi\tau/2)-\sin(\pi\nu/2)}\,+\, \frac{\sin(\pi\tau/2)}{\cos(\pi\tau/2)+\sin(\pi\nu/2)}\, \right)\\&= \frac{1}{4}\int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\, d\tau\, H(\tau)\, \frac{\sin\pi\tau}{\cos\pi\tau+\cos\pi\nu}\,,\,\,\,\,\,\,\nu\in \varPi_{1+\delta}. \end{align*} Assuming exponential decreasing of $$\widehat {u}_{\nu }$$ as $$|\nu |\to \infty ,$$ν ∈ Πδ and making use of this Lemma, we arrive at a new form of the boundary condition (3.7)   \begin{align} \left.\frac{\partial \widehat{u}_{\nu}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\,= \frac{\eta_{\pm}\,}{2\mathrm{i}} \int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\, d\tau\, \frac{\tau\,\sin\pi\tau\,\widehat{u}_{\tau}(\omega,\omega_{0})|_{\sigma_{\pm}}}{\cos\pi\tau+\cos\pi\nu}\, \,,\,\,\,\,\nu\in \varPi_{\delta}. \end{align} (3.8) Remark Provided $$\widehat {u}_{\nu }(\omega ,\omega _{0})|_{\sigma _{\pm }}$$ is holomorphic as ν ∈ Πδ, then from the Lemma and condition (3.8) $$\left .\frac {\partial \widehat {u}_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$ is holomorphic as ν ∈ Π1+δ. In order to formulate a basic statement of this Section we first postulate a set of conditions specifying a desired class for solution of the problem (3.6), (3.8). 3.3. Conditions specifying a class of solutions for spectral function uν(ω, ω0) We make use of some known definitions (Dieudonne, 1960; Chapter 7) dealing with a meromorphic in the complex plane C (or holomorphic in a domain D ⊂ C) mapping fν from C (or from D) into a Banach space B, i.e. with such meromorphic (or holomorphic) function of ν that fν : ν↦fν(⋅), where fν(⋅) ∈ B. Let the spectral function uν(ω, ω0) satisfy the conditions The spectral function is such a mapping that uν : ν↦uν(⋅, ω0) , ($$u_{\nu }(\cdot , \omega _{0})\in \,C^{2}_{loc}(\varSigma ) \,$$) is meromorphic in the complex plane C for all $$\omega _{0}\notin \varOmega _{\delta _{0}}$$, where ω0 is a parameter. For all regular ν it admits the estimate (Meixner’s condition on the unit sphere) $$u_{\nu }(\omega ,\omega _{0})= c_{0}\,+\,c_{1}(\chi _{A})\,\psi _{A}^{1/2}+\dots $$, ψA → 0, where ψA is the geodesic distance from A to ω = (ψA, χA). A similar condition is valid for the point B. The trace of the spectral function on σ is holomorphic in Πδ, i.e. the mapping uν|σ : ν↦uν(⋅|σ, ω0) is holomorphic in ν ∈ Πδuν(⋅|σ, ω0) ∈ C(σ) for all $$\omega _{0}\notin \varOmega _{\delta _{0}}$$. The set Q of poles of the meromorphic function uν is contained in the strip |I(ν)| < b for some b > 0. The spectral function is even with respect to ν, uν(ω, ω0) = u−ν(ω, ω0). There exist the traces $$ \left .\frac {\partial \widehat {u }_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$ on σ which are holomorphic functions of ν ∈ Π1+δ with the values in C(σ), $$\omega _{0}\notin \varOmega _{\delta _{0}}$$. Notice that, in view of the boundary conditions (3.7), (3.8), these traces admit meromorphic continuation on the complex plane. The spectral function satisfies the estimate ($$|\nu |\to \infty $$, ν = |ν|eiϕ, − π/2 < ϕ < π/2)   \begin{align} |u_{\nu}(\omega,\omega_{0})|\,\leqslant\,C\, \frac{1}{\sqrt{|\nu|}}\exp\left\{ -|\nu|\,\left(|\sin\phi|\,\tau_{0}(\omega,\omega_{0})-\cos\phi \,|\tau_{1}(\omega,\omega_{0})|\right.\right\}\, \end{align} (3.9) for some τ0, τ1, where τ0 > 0. This estimate is valid on the contour Cϕ in Fig. 6. Remark that analogous properties may be verified for $$u_{\nu }^{i}(\omega ,\omega _{0})$$. It is worth commenting on the origin of the estimate (3.9). The equation (3.6) has high-frequency (or quasi-classical) structure as $$\nu \to \infty $$. The solution of the equation in this case may be determined as a sum of the ‘ray’ expansions   \begin{align*} u_{\nu}(\omega)\asymp&\, \sum_{j} U^{\,j}(\nu,\omega)\,,\\ U^{\,j}(\nu,\omega)=&\,\mathrm{e}^{\mathrm{i}\nu\tau_{j}(\omega)}\sum_{m=1}^{\infty} v_{m}(\omega)/\nu^{1/2+m}. \end{align*} The boundary condition for uν(ω) also depends on the large parameter and is not self-adjoint because, in particular, it depends on complex η±. As a result, amongst solutions τj(ω) of the eikonal equations (∇ωτj(ω))2 = 1 there are those complex valued having the structure $$\tau _{j}(\omega )={\tau ^{j}_{0}}(\omega )+\mathrm {i}{\tau ^{j}_{1}}(\omega )$$ with the real valued $${\tau ^{j}_{0}}(\omega )>0\,$$ and $$\,{\tau ^{j}_{1}}(\omega )$$. Assuming that the ray solutions admit continuation to ν = |ν|eiϕ, $$|\nu |\to \infty $$, 0 < ϕ < π/2, we may expect   \begin{align*} |u_{\nu}(\omega)|\leqslant\text{Const}\,|\nu|^{-1/2} \left|\mathrm{e}^{\mathrm{i}\nu(\tau_{0}(\omega)+\mathrm{i}\tau_{1}(\omega))}\right|, \end{align*} where $$\tau _{0}(\omega )=\min _{j}({\tau _{0}^{j}}(\omega ))$$ and $$|\tau _{1}(\omega )|=\max _{j}(|{\tau _{1}^{j}}(\omega )|)$$. This accounts for the condition 6. It is worth noting that rapid convergence of the Watson–Bessel integral representation (3.5) follows from the estimate (see also (3.4)) given by   \begin{align} \left|\nu\,\mathrm{e}^{-\mathrm{i}\nu\pi/2}u_{\nu}(\omega,\omega_{0}) {J_{\nu}(kr)}\right|<&\, \,C\, \exp\left\{-|\nu|(\log|\nu|-1)\cos\phi--|\nu|(\sin\phi[{\arg(kr/2)-\pi/2-\phi}]\right.\nonumber\\ &\left.\quad\qquad+\ |\sin\phi|\tau_{0}(\omega,\omega_{0}))+|\nu|\cos\phi[ |\tau_{1}(\omega,\omega_{0})|+\log|kr/2|]\right\} \end{align} (3.10) on Cϕ as $$|\nu |\to \infty $$, ν = |ν|eiϕ, − π/2 < ϕ < π/2. Taking into account the discussion in this section, we arrive at Theorem 3.1 Let uν(ω, ω0) be a solution of the problem (3.6), (3.8) from the class of functions described by the conditions 1–6. Then Watson–Bessel integral representation (3.5) for U(r, θ, φ) is the desired classical solution of the problem. We mentioned that the radiation condition (2.6) is not valid for the solution in the case of the plane wave incidence. On the other hand, we should emphasize that description of the far-field asymptotics for the diffraction problem at hand will not be exhaustively considered in this work, however, we shall discuss the far-field expression in the oasis Ω0. Nevertheless, we shall develop an efficient formalism based on the Sommerfeld integral representation which is well adapted for such a description (Lyalinov, 2013). In the following section we shall consider an approach that enables one to show a way of construction of the spectral function solving the problem (3.6), (3.8). 4. Study of the problem for the spectral function In this section we consider the problem (3.6), (3.8) and restrict it on $$\nu \in [0,\mathrm {i}\infty )$$ exploiting also that uν(ω, ω0) is even. Then it is useful to study the inhomogeneous equation instead of (3.6) and, vice versa, homogeneous boundary condition so that   \begin{align} \left(\triangle_{\omega}+(\nu^{2}-1/4)\right)w_{\nu}(\omega)=&\,F(\nu,\omega)\,,\,\,\,\,\omega\in \varSigma\nonumber\\ \left.\frac{\partial w_{\nu}(\omega)}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}=&\, \frac{\eta_{\pm}\,}{\mathrm{i}} \int\limits_{0}^{\mathrm{i}\infty}\, d\tau\, \frac{\tau\,\sin\pi\tau\,w_{\tau}(\omega)|_{\sigma_{\pm}}}{\cos\pi\tau+\cos\pi\nu}\,. \end{align} (4.1) This problem is not traditional because of the non-local dependence on the parameter of separation ν. It should be noticed that the problems (3.6), (3.8) and (4.1) are actually connected by a simple change of the unknown function uν(ω, ω0) = wν(ω, ω0) + v(ν, ω, ω0), where v is uniquely defined. We shall write wν(ω) instead of wν(ω, ω0) omitting dependence on the parameter ω0. We are looking for solution wν(ω) of (4.1) which is from $$C([0,\mathrm {i}\infty ),C^{2}(\varSigma ))$$, having also continuous value of $$\frac {\partial w_{\nu }(\omega )}{\partial \mathscr {N}_{\pm }}$$ on σ such that the boundary condition in (4.1) is satisfied. Having such a solution, we shall continue it appropriately onto the whole complex plane w.r.t ν and also ensure the conditions 1–6. for the spectral function uν(ω, ω0). It is convenient to make use of the variable x instead of ν and the new unknown function $${\mathscr {U}}(x,\omega )$$ defined by the expressions   \begin{align*} \,x=\,1/\cos\pi\nu\,,\,\,\, \nu=\,\mathrm{i} d(x),\,\,\,d(x):=&\,\frac{1}{\pi}\,\text{arccosh}(1/x)= \frac{1}{\pi}\, \log\left(\frac{1}{x} +\sqrt{\frac{1}{x^{2}}-1}\right)\,,\,\,\,\,x\in [0,1]\,,\\ {\mathscr{U}}(x,\omega)\,=&\,\sqrt{d(x)}\,\cos\left(\mathrm{i}\pi d(x)\right)\,w_{\mathrm{i}d(x)}(\omega). \end{align*} Remark that   \begin{align*} 0\leqslant\,\frac{1}{\pi}\,\log(1/x)\leqslant d(x)\,\leqslant\,d_{0}(x)\end{align*} with $$d_{0}(x)=\frac {1}{\pi }\,\log (2/x)$$. As a result, we write the problem (4.1) in the form   \begin{align} \left(-\triangle_{\omega}+d^{2}(x)+1/4\right){\mathscr{U}}(x,\omega)\,=&\,f(x,\omega)\,,\,\,\,\,\omega\in \varSigma\nonumber\\[15pt] \left.\frac{\partial{\mathscr{U}}(x,\omega)\,}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\,+ \eta_{\pm}\,\mathscr {A} \mathscr {U}(x,\omega|_{\sigma_{\pm}})\,=&\,0, \end{align} (4.2) where $$f(x,\omega )\,=\, -\sqrt {d(x)}\cosh (\pi d(x) F(\mathrm {i}d(x),\omega )$$ and the operator-impedance $${\mathscr {A}}$$ is given by   \begin{align}\displaystyle \mathscr {A} \mathscr {U}(x,\omega|_{\sigma_{\pm}})\,=\,\frac{1}{\pi}\,{\int\limits_{0}^{1}}\,\frac{\sqrt{d(x)d(y)}}{x+y}\, {\mathscr{U}}(y,\omega|_{\sigma_{\pm}})\,\mathrm{d}y. \end{align} (4.3) The operator $${\mathscr {A}}$$ in (4.3) is formally symmetric. It is worth commenting on the reduction of the problem (4.1) to that (4.2). We are looking for the solution of the problem (4.1) which meromorphically depends on the variable ν. However, the study of the meromorphic operator-function   \begin{align*} {\mathscr{M}}_{\nu} \,=\,\left\{\triangle_{\omega}+(\nu^{2}-1/4)\,,\,\,\, \left.\left(\frac{\partial\,\,\,\,}{\partial\,\mathscr{N}_{\pm}}\,-\, \eta_{\pm}\,{\mathscr{A}}_{0}\,\right)\right|_{\sigma_{\pm}}\,\right\},\end{align*} and of the equation   \begin{align*} {\mathscr{M}}_{\nu}\, w_{\nu}(\omega) \,= \,{\mathscr{F}}_{\nu} (\omega) \end{align*} is not a simple task. In the latter problem the operator $${\mathscr {A}}_{0}$$ is defined by the right-hand side of the boundary condition in (4.1) and $${\mathscr {F}}_{\nu } (\omega )=\{F_{\nu }(\omega ),0\}$$. Instead, we consider the reduced problem (4.2). The solution of the latter problem $${\mathscr {U}}(x,\omega )$$ is defined on the segment x ∈ (0, 1) and is then analytically continued as a meromorphic function wν(ω) onto the complex plane ν ∈ C taking into account the change of the variable x↦ν and of $${\mathscr {U}}(x,\omega )\mapsto w_{\nu }(\omega )$$. Our further goal is to study unique solvability of the problem (4.2) in an appropriate functional space. To this end, we introduce a Hilbert space and attribute an operator, acting in this space, with the problem (4.2). It is performed by use of the traditional technique based on a sesquilinear form connected with the problem (4.2). The corresponding sectorial form is taken densely defined and proved to be closable, see Kato (1972), Chapter 6. The latter circumstance enables one to define an m-sectorial operator A attributed to (4.2), see also Assier et al. (2016). It turns out that this operator is boundedly invertible, A−1 = (A−Λ)−1|Λ=0 is correctly defined because Λ = 0 belongs to the resolvent set of the operator A, where Λ is the spectral parameter. 4.1. Definition of the sectorial form attributed to the problem (4.2) Consider the Hilbert space H = L2((0, 1);L2(Σ)) of such functions u of x ∈ (0, 1) with the values in L2(Σ) denoting them u = u(x, ω). The Hilbert norm $$\|v\|_{H}=\sqrt {<v,v>}$$ is specified by the scalar product   \begin{align*} \displaystyle <u,v>\,=\,{\int\limits_{0}^{1}}\,d\,x\,\left(\int\limits_{\varSigma}\, u(x,\omega)\overline{v(x,\omega)} \,\mathrm{d}\,\omega\right)\,. \end{align*} In order to give a motivated expression for the sesquilinear form tA attributed to the problem (4.2) we apply the Green’s identity to the differential operator and make use of the boundary condition for u  \begin{align*}<(-\triangle_{\omega}+d^{2}+1/4) u,v>\,=&\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma}\,(-\triangle_{\omega}+d^{2}(x)+1/4) u(x,\omega)\overline{v(x,\omega)}\mathrm{d}\,\omega\right)\\[6pt] =&\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\!\left(\int\limits_{\varSigma}\,(\nabla_{\omega} u(x,\omega)\cdot \nabla_{\omega} \overline{v(x,\omega)}+(d^{2}(x)+1/4) u(x,\omega)\overline{v(x,\omega)})\mathrm{d}\,\omega \right)\!-\,\\[6pt] &-{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{+}}\mathrm{d}\sigma\, \frac{\partial u(x,\sigma)\,}{\partial\mathscr{N}_{+}}\overline{v(x,\sigma)} \,-\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{-}}\mathrm{d}\sigma\, \frac{\partial u(x,\sigma)\,}{\partial\mathscr{N}_{-}}\overline{v(x,\sigma)}\\[6pt] =&\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma}\,\mathrm{d}\omega( \nabla_{\omega} u(x,\omega)\cdot \nabla_{\omega} \overline{v(x,\omega)}+ (d^{2}(x)+1/4) u(x,\omega)\overline{v(x,\omega)}) \right)\\[6pt] &+\, \frac{\eta_{+}}{\pi}{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{+}}\mathrm{d}\sigma\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{\sqrt{d(x)d(y)}}{x+y}\, u(y,\sigma) \overline{v(x,\sigma)}\\[6pt] &+\, \frac{\eta_{-}}{\pi}{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{-}}\mathrm{d}\sigma\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{\sqrt{d(x)d(y)}}{x+y}\, u(y,\sigma) \overline{v(x,\sigma)}\,, \end{align*} where u and v are taken such that the reductions above are justified. Let us now define a sesquilinear form   \begin{align} t_{A}[u,v]\,=\,{t^{1}_{A}}[u,v]+{t_{A}^{2}}[u,v]+{t_{A}^{3}}[u,v]\,, \end{align} (4.4) where   \begin{align*} {t^{1}_{A}}[u,v]\,&=\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int_{\varSigma}\, \left(\nabla_{\omega} u(x,\omega)\cdot \nabla_{\omega} \overline{v(x,\omega)} \,+\, \frac{1}{4}u(x,\omega)\overline{v(x,\omega)}\right)\mathrm{d}\omega \right)\!,\\ {t_{A}^{2}}[u,v]&=\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma} d^{2}(x) u(x,\omega)\overline{v(x,\omega)}\mathrm{d}\,\omega \right)\!,\\ {t_{A}^{3}}[u,v]\,&=\,\mathop{\sum}_{\pm}\,\frac{\eta_{\pm}}{\pi}{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{\sqrt{d(x)d(y)}}{x+y}\, u(y,\sigma) \overline{v(x,\sigma)} , \end{align*} with the domains   \begin{align*} D\left({t_{A}^{1}}\right)&=\left\{v:\,\,v\in L_{2}\left((0,1);H^{1}(\varSigma)\right)\subset H\right\}\!,\\ D\left({t_{A}^{2}}\right)&=\big\{v:\,\,v\in H=L_{2}((0,1);L_{2}(\varSigma))\,\,\text{such that}\,\,\,d\,v\in L_{2}((0,1);L_{2}(\varSigma))\, \big\}\!,\\ D\left({t_{A}^{3}}\right)&=\left\{w:\,w\in H,\,\,\,w|_{\sigma}\in L_{2}((0,1);L_{2}(\sigma))\,\, \text{such that}\,\,\,\sqrt{d}\,w|_{\sigma}\in L_{2}((0,1);L_{2}(\sigma))\, \right\}\, \end{align*} correspondingly. Then the domain of tA is   \begin{align}\displaystyle D(t_{A})\,=\, \bigcap_{i=1}^{3} D\left({t_{A}^{i}}\right) \end{align} (4.5) and tA is densely defined. It is worth mentioning that H1(Σ) is the usual Sobolev space and is boundedly embedded into L2(σ). The norm in L2((0, 1);H1(Σ)) is given by   \begin{align*} \|v\|^{2}_{1}\,=\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma}\,(|\nabla_{\omega} u(x,\omega)|^{2}+ |u(x,\omega)|^{2}\mathrm{d}\,\omega\right)\, \end{align*} and this space is boundedly embedded into L2((0, 1);L2(σ)). The form $${t_{A}^{3}}$$ may be also written as   \begin{align*} {t_{A}^{3}}[u,v]\,=\,\mathop{\sum}_ {\pm}\,\eta_{\pm}{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, \, ({\mathscr{A}} u)(x,\sigma) \overline{v(x,\sigma)} \,=\,\mathop{\sum}_{\pm}\,\eta_{\pm}({\mathscr{A}} u,v)_{\pm}, \end{align*} where the operator-impedance $${\mathscr {A}}\geqslant 0$$ is an integral operator, has the kernel $$\frac {\sqrt {d(x)d(y)}}{x+y}\,$$ and is formally symmetric. We aim to show that the form tA is sectorial (Kato, 1972, Chapter 6) with the vertex Γ which is a real number and with the half-angle Θ. We verify that the range R(tA) of the form, which is the set of values of the quadratic form tA[u, u] provided u ∈ D(tA) and ∥u∥H = 1, is located in the sector $$|\text {arg}(\varLambda -\varGamma )|\leqslant \,\Theta ,$$$$0\leqslant \Theta <\pi /2$$ on the complex plane of the variable Λ. To that end, we represent the form as   \begin{align*} t_{A}\,=\,\Re(t_{A})\,+\,\mathrm{i}\Im(t_{A}), \end{align*} where (see Kato, 1972, Chapter 6) real part $$\Re (t_{A})$$ of the form is defined by   \begin{align*} \Re(t_{A})[u,v]=\frac{t_{A}[u,v]+ \overline{t_{A}[v,u]}}{2}\end{align*} and similarly the imaginary part   \begin{align*} \Im(t_{A})[u,v]=\frac{t_{A}[u,v]- \overline{t_{A}[v,u]}}{2\mathrm{i}}\end{align*} for any u, v ∈ D(tA). Simple calculations lead to the expressions   \begin{align*} \Re(t_{A})[u,v]&= {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma}\, \nabla_{\omega} u(x,\omega)\cdot \nabla_{\omega} \overline{v(x,\omega)} \,+\, \left(d^{2}(x)+\frac{1}{4}\right)\,u(x,\omega)\overline{v(x,\omega)} \mathrm{d}\omega \right)\,\\&\quad+ \,\mathop{\sum}_{\pm}\,\varepsilon_{\pm}({\mathscr{A}} u,v)_{\pm}, \Im(t_{A})[u,v]=\,\mathop{\sum}_{\pm}\,\chi_{\pm}({\mathscr{A}} u,v)_{\pm}\, \end{align*} recalling that η± = ε± + iχ±,   \begin{align*} ({\mathscr{A}} u,v)_{\pm}\,=\,\frac{1}{\pi}\,{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{\sqrt{d(x)d(y)}}{x+y}\, u(y,\sigma) \overline{v(x,\sigma)} \,. \end{align*} It is obvious that   \begin{align*} \Re(t_{A})[u,u]\,\geqslant\,\frac{1}{4}\|u\|_{H}^{2} \end{align*} so that the vertex $$\varGamma \geqslant \frac {1}{4}$$ and the form $$\Re (t_{A})$$ is positive definite. For the imaginary part we find   \begin{align*} |\Im(t_{A})[u,u]|\,&\leqslant\,\mathop{\sum}_{\pm}\,|\chi_{\pm}|({\mathscr{A}} u,u)_{\pm}\,\leqslant\, \mathop{\sum}_{\pm}\,\frac{|\chi_{\pm}|}{\varepsilon_{\pm}}\, \left(\varepsilon_{\pm}({\mathscr{A}} u,u)_{\pm}\,+\,\|\nabla_{\omega} u\|_{H}^{2}+\,\|d\,u\|_{H}^{2}\right)\,\\&\leqslant\, \left(\frac{|\chi_{+}|}{\varepsilon_{+}}+\frac{|\chi_{-}|}{\varepsilon_{-}}\right)\, \left(\Re(t_{A})[u,u]\,-\,\frac{1}{4}\|u\|_{H}^{2}\right) \end{align*} for any u ∈ D(tA). As a result, we have the estimate for the half-angle Θ of the sectorial form tA  \begin{align*} 0\,\leqslant\,\tan\Theta\,\leqslant\, \left(\frac{|\chi_{+}|}{\varepsilon_{+}}+\frac{|\chi_{-}|}{\varepsilon_{-}}\right)\!. \end{align*} It is crucial to prove that the sesquilinear sectorial form tA is closable, i.e. admits a closed extension, which is equivalent to that for $$\Re (t_{A})$$. 4.2. The form tA is closable In order to prove that tA admits a closure it is sufficient to show that its summands $${t_{A}^{i}},\,\,i=1,2,3$$ are closable (i.e. admit closed extensions, see Theorem 1.31 in Chapter 6 of Kato, 1972). To do this we shall use the following simple statement (see, Birman & Solomyak, 1987, Section 10.1(4)) Lemma 4.1 Let h be a densely defined sesquilinear form and G be an h −dense in D(h) set. Also let the conditions h[un − um, un − um] → 0 and ∥un∥H → 0 (as $$n,m\to \infty $$) be followed by   \begin{align*} h[u_{n},g]\,\to\, 0\,,\,\,\,n\to\infty \end{align*} for any g ∈ G then the form h is closable. Introduce the set G by   \begin{align*} G\,=\,\left\{\,g\in D(t_{A})\,:\, \,\,\,\,{d_{0}^{2}} g\in H=L_{2}((0,1);L_{2}(\varSigma)),\,\,\, {d_{0}^{2}} g\big|_{\sigma}\in L_{2}((0,1);L_{2}(\sigma))\,\right\}\!, \end{align*} $$d_{0}(x)=\frac {1}{\pi }\log (2/x)$$. Remark that the form $${t_{A}^{1}}$$ is closable because it is directly connected with the form   \begin{align*} \tau^{1}[u,v]\,=\, \,\int\limits_{\varSigma}\, \left(\nabla_{\omega} u(\omega)\cdot \nabla_{\omega} \overline{v(\omega)} \,+\, \frac{1}{4}u(\omega)\overline{v(\omega)}\right)\mathrm{d}\omega \, \end{align*} which is known to be closable in H1(Σ), see e.g. Assier et al. (2016). That the form $${t^{1}_{A}}$$ is closable implies the following: from ∥un∥→ 0 and $${t_{A}^{1}}[u_{n}-u_{m},u_{n}-u_{m}]\to 0$$ ($$n,m\to \infty $$) it follows that (see Theorem 1.17 in Chapter 6, Kato, 1972)   \begin{align} {t_{A}^{1}}[u_{n},u_{n}]\to 0\,,\,\,\,\text{as}\,\,\, n\to \infty. \end{align} (4.6) The form $${t_{A}^{2}}$$ is closable because   \begin{align*} \left|{t_{A}^{2}}[u_{n},g]\right|\,&\leqslant\,{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma} \big|d^{2}(x) u(x,\omega)\big|\,|g(x,\omega)|\mathrm{d}\,\omega \right)\,\\[15pt] &\leqslant\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma} |u(x,\omega)|\,\left|{d_{0}^{2}}(x) g(x,\omega)\right|\mathrm{d}\,\omega \right)\, \leqslant\,\|u_{n}\|_{H}\,\|{d_{0}^{2}} g\|_{H}\,\to\,0\,, \end{align*} as ∥un∥H , → 0 , $$n\to \infty $$ for any g ∈ G. The corresponding estimate for $${t_{A}^{3}}$$ requires a bit more work   \begin{align*} \left|{t_{A}^{3}}[u_{n},g]\right|\,&\leqslant\, \,\mathop{\sum}_{\pm}\,\frac{|\eta_{\pm}|}{\pi}{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{\sqrt{d(x)d(y)}}{{d_{0}^{2}}(x)(x+y)}\, |u_{n}(y,\sigma)|\, \left|{d_{0}^{2}}(x)g(x,\sigma)\right| \,\\&\leqslant\, \,\mathop{\sum}_{\pm}\,\frac{|\eta_{\pm}|}{\pi}\left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, \left[\int\limits_{0}^{1}\,\mathrm{d}y\,\frac{\sqrt{d(y)}}{\sqrt{d_{0}(x)}^{3}(x+y)}\, |u_{n}(y,\sigma)|\right]^{2}\,\right\}^{1/2}\\ &\quad\ \ \left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, |{d_{0}^{2}}(x)g(x,\sigma)|^{2} \right\}^{1/2}\,\\&\leqslant\, \,\mathop{\sum}_{\pm}\,C\,\frac{|\eta_{\pm}|}{\pi}\, \left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{d(y)}{d_{0}(x)^{3}(x+y)^{2}}\, \right\}^{1/2}\,\\&\quad\times \left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, | u_{n}(x,\sigma)|^{2} \right\}^{1/2}\, \,\left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, \left|{d_{0}^{2}}(x)g(x,\sigma)\right|^{2} \right\}^{1/2}\,, \end{align*} where we have multiply used the Cauchy–Schwarz inequality. From (4.6) and that L2((0, 1), H1(Σ)) is boundedly embedded into L2((0, 1), L2(σ)) (un ∈ L2((0, 1), H1(Σ)) we have   \begin{align*} \left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, | u_{n}(x,\sigma)|^{2} \right\}^{1/2}\,\leqslant\,C\|u_{n}\|_{1}\,\,\leqslant\,C_{1}\, {t_{A}^{1}}[u_{n},u_{n}]\to 0\,,\,\,\,\text{as}\,\,\, n\to \infty. \end{align*} It remains to verify that   \begin{align*} {\int\limits_{0}^{1}}\,\mathrm{d}\,x\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{d(y)}{d_{0}(x)^{3}(x+y)^{2}}\,\leqslant \,\text{Const}, \, \end{align*} i.e. it is bounded. The latter double integral over the square S = (0, 1) × (0, 1) is represented as a sum of two integrals, one is over $$S_{1}=\{(x,y):\,\,\sqrt {x^{2}+y^{2}}<1\}\cap S $$ and that over S ∖ S1. The integral over $$\overline {S\setminus S_{1}}$$ is obviously bounded because the integrand is continuous in this domain. The integral over the sector S1 is estimated by making use of the polar coordinates ρ, ϕ in its evaluation and is estimated, which is a simple exercise, by a constant C1,   \begin{align*} \int\limits_{0}^{\pi/2}\,\mathrm{d}\phi\, {\int\limits_{0}^{1}}\,\frac{\mathrm{d}\rho}{\rho}\,\frac{\left|\log\rho+\log(\sin\phi)\right|}{ \left|\log(2/\rho)-\log(\cos\phi)\right|{}^{3}(\cos\phi+\sin\phi)^{2}}\,\leqslant \,\text{C}_{1}. \, \end{align*} Taking into account the estimates above we find that $$|{t_{A}^{3}}[u_{n},g]|\,\to \,0$$ as ∥un∥H → 0 , $$n\to \infty $$ for any g ∈ G. As a result, we can assert that the form tA is closable. 4.3. An m-sectorial operator A corresponding to the problem (4.2). Solvability of the problem Let us denote TA the closure of tA with D(tA) ⊂ D(TA).8 In accordance with the theorem on representation for the closed sectorial forms (see Kato, 1972, Section 2, Chapter 6, Theorem 2.1) we obtain Theorem 4.1 There exists a unique m-sectorial operator A such that Dom(A) ⊂ D(TA) and   \begin{align*} <Au,v>= T_{A}[u,v] \end{align*} with u ∈ Dom(A), v ∈ D(TA). It is important to notice that this operator is the desired operator attributed to the problem (4.2). Let u ∈ D(TA) and for any v ∈ D(TA)   \begin{align*} T_{A}[u,v]=<f,v>\,,\,\,\,\,f\in H, \end{align*} then u is called weak solution of the equation   \begin{align} Au=f. \end{align} (4.7) This solution is also a weak solution of the problem (4.2) by definition. From the theorem on representation it also follows that, provided u ∈ D(TA), f ∈ H and the equality   \begin{align*} \displaystyle T_{A}[u,v]=<f,v>\end{align*} is valid for any v from the core of the form TA (i.e. from Dom(A)) then u ∈ Dom(A) and u is a solution of the equation (4.7). The range Θ(A) of the m-sectorial operator A is a dense subset of the range Θ(TA), which implies that Λ = 0 belongs to the resolvent set of the operator A. The inverse operator A−1 = (A−Λ)−1|Λ=0 is bounded and the problem (4.2) has a weak solution $${\mathscr {U}}$$. Recall that   \begin{align*} F(\nu,\omega)|_{x=1/\cos\pi\nu}\,=\,-\,\frac{f(x,\omega)}{\sqrt{d(x)}\cosh\left(\pi d(x)\right)} \end{align*} in (4.1) and, provided f is from C([0, 1];C(σ)) ⊂ H the function F(ν, ω) is continuous with respect to ν and is from C(σ) w.r.t. ω, exponentially vanishing as $$\nu \to \mathrm {i}\infty $$. Similar simple arguments enable us to conclude that solution of the problem (3.6), (3.8) exists and is unique, uν(ω, ω0) is from $$C([0,\mathrm {i}\infty );C^{2}(\varSigma ))$$ which admits an estimate   \begin{align*} \displaystyle |u_{\nu}(\omega,\omega_{0})|\,\leqslant\, C\,\left|\frac{\text{e}^{\mathrm{i}\nu\tau_{*}}}{ \sqrt\nu}\right|\,,\,\,\,\,\nu\to\mathrm{i}\infty, \end{align*} $$0<\tau _{*}\leqslant \pi /2$$ as dist(ω0, σ) > π/2. This estimate follows from the fact that $${\mathscr {U}}\in L_{2}((0.1);H^{1}(\varSigma ))$$. Remark that $${\mathscr {U}}_{i}(x,\omega )\,=\,\sqrt {d(x)}\,\cos (\mathrm {i}\pi d(x))\,u_{\mathrm {i}d(x)}^{i}(\omega ,\omega _{0})\,$$ is from H = L2((0, 1);L2(Σ)) provided dist(ω0, σ) > π/2, which is implied. 5. Meromorphic continuation of uν(ω, ω0) In order to continue uν(ω, ω0) specified on $$\nu \in [0,\mathrm {i}\infty ) $$, as demonstrated in the previous section, onto the complex plane we need to develop some technical tools. First, we continue uν(ω, ω0) from $$[0,\mathrm {i}\infty )$$ onto iR making use of evenness w.r.t. ν. Then, consider gν(ω, ω′) the Green’s function of the operator △ω + (ν2 − 1/4) on the unit sphere   \begin{align*} g_{\nu}(\omega,\omega^{\prime})=-\frac{P_{\nu-1/2}\left(-\cos\theta(\omega,\omega^{\prime})\right)}{4\cos\pi\nu}, \end{align*} satisfying   \begin{align*} \left(\triangle_{\omega}+(\nu^{2}-1/4)\right)g_{\nu}(\omega,\omega^{\prime})=\delta(\omega-\omega^{\prime}). \end{align*} We apply Green’s identity to uν(ω, ω0) and gν(ω, ω′) and, omitting some technical details, obtain   \begin{align*} &\int\limits_{\varSigma}\,\mathrm{d}\omega^{\prime} \,\left( \triangle_{\omega^{\prime}}g_{\nu}(\omega,\omega^{\prime})\,u_{\nu}(\omega^{\prime},\omega_{0})\right)\,-\, g_{\nu}(\omega,\omega^{\prime})\,\triangle_{\omega^{\prime}}u_{\nu}(\omega^{\prime},\omega_{0}))\\&\quad=\, \mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\left(\frac{\partial g_{\nu}(\omega,\sigma)}{\partial{\mathscr{N}}_{\pm}} u_{\nu}(\sigma,\omega_{0})\,-\, \frac{\partial u_{\nu}(\sigma,\omega_{0})}{\partial{\mathscr{N}}_{\pm}} g_{\nu}(\omega,\sigma)\right)\,. \end{align*} Making use of the equations for gν(ω, ω′) and uν(ω, ω′) and the boundary condition (see (3.8)), we arrive at the representation   \begin{align} u_{\nu}(\omega,\omega_{0})\,=\, \mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\left(\frac{\partial g_{\nu}(\omega,\sigma)}{\partial{\mathscr{N}}_{\pm}} u_{\nu}(\sigma,\omega_{0})\,-\,\eta_{\pm} \,{\mathscr{A}}_{0}u_{\nu}(\sigma,\omega_{0})\,g_{\nu}(\omega,\sigma)\right)\,+\, \varPsi_{\nu}(\omega,\omega_{0}), \end{align} (5.1) where   \begin{align*} {\mathscr{A}}_{0}u_{\nu}(\sigma,\omega_{0})\,:&=\,\frac{ 1\,}{2\mathrm{i}} \int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\, d\tau\, \frac{\tau\,\sin\pi\tau\,u_{\tau}(\sigma,\omega_{0})}{\cos\pi\tau+\cos\pi\nu}\,,\\ \varPsi_{\nu}(\omega,\omega_{0})&=\,\mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\,\mathrm{d}\sigma \left(\frac{\partial u^{i}_{\nu}(\omega,\sigma)}{\partial{\mathscr{N}}_{\pm}} g_{\nu}(\sigma,\omega_{0})\,-\,\eta_{\pm } \,{\mathscr{A}}_{0}u_{\nu}^{i}(\sigma,\omega_{0})\,g_{\nu}(\omega,\sigma)\right)\!. \end{align*} The integral equation for uν(s, ω0) as s belongs to the boundary σ+ ∪ σ− is then derived and takes the form   \begin{align} \alpha_{s} u_{\nu}(s,\omega_{0})\,=\, \mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\left(\frac{\partial g_{\nu}(s,\sigma)}{\partial{\mathscr{N}}_{\pm}} u_{\nu}(\sigma,\omega_{0})\,-\,\eta_{\pm} \,{\mathscr{A}}_{0}u_{\nu}(\sigma,\omega_{0})\,g_{\nu}(s,\sigma)\right)\,+\, \varPsi_{\nu}(s,\omega_{0}), \end{align} (5.2) where $$\alpha _{s} =\frac {1}{2}$$ as s ∈ (σ+ ∪ σ−), s ∉ {A, B} and αs = 1 as s ∈ {A, B}. The properties of the single and double layer potentials have been used in (5.2). It is not difficult to show that Ψν(ω, ω0) in (5.2), (5.1) admits meromorphic continuation as ν ∈ C such that it is holomorphic as ν ∈ Πδ for some δ > 0. Remark that the integral equation really has solution, because it is derived from the uniquely solvable problem (3.6), (3.8). It is easily verified that Ψν is a holomorphic mapping ν ↦ L2(σ) as ν ∈ Πδ, Ψν(⋅, ω0) ∈ L2(σ). Now we explain that the solution uν(s, ω0) of the equation (5.2) obeys the same property. To that end, we write the equation (5.2) in an equivalent form   \begin{align} (I-{\mathscr{K}}_{0})u_{\nu} = 2\chi_{\nu,0}, \end{align} (5.3) where   \begin{align*} {\mathscr{K}}_{0} u_{\nu}(s,\omega_{0})\,&=\, \mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\frac{\partial g_{\nu}(s,\sigma)}{\partial{\mathscr{N}}_{\pm}}\, u_{\nu}(\sigma,\omega_{0})\,,\,\,s\in \sigma\setminus\{A,B\} \\2\chi_{\nu,0}(s,\omega_{0})&=\, 2\mathop{\sum}_{\pm} \eta_{\pm}\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, \,g_{\nu}(s,\sigma) {\mathscr{A}}_{0}u_{\nu}(\sigma,\omega_{0})\,+\,2\varPsi_{\nu}(s,\omega_{0}), \end{align*} so that the operator $$(I-{\mathscr {K}}_{0})$$ is boundedly invertible for any ν ∈ Πδ and $$(I-{\mathscr {K}}_{0})^{-1}$$ is holomorphic with respect to ν ∈ Πδ. Because the right-hand side χν, 0 in (5.3) is a holomorphic mapping ν ↦ L2(σ) as ν ∈ Πδ , χν, 0(⋅, ω0) ∈ L2(σ), we conclude that the solution uν of the equation (5.2) is a holomorphic mapping ν ↦ L2(σ) as ν ∈ Πδ , uν(⋅, ω0)|σ ∈ L2(σ) (see condition 2 in Section 3.3). The representation (5.1) enables one to assert that uν is also a holomorphic mapping ν ↦ L2(Σ) as ν ∈ Πδ , uν(⋅, ω0) ∈ L2(Σ). Indeed, the integrand in the right-hand side of (5.1) is holomorphic as ν ∈ Πδ for all ω ∈ Σ because gν(ω, ⋅)|σ obeys this property as well as uν(⋅, ω0)|σ in the integrand, gν(ω, ⋅) is continuous w.r.t. ω ∈ Σ. uν(s, ω0)|σ is continuous w.r.t. s assuming that $$\omega _{0}\notin \varOmega _{\delta _{0}}$$. The same regularity is valid for $${\mathscr {A}}_{0}u_{\nu }(\sigma ,\omega _{0})$$, which follows from the explicit formula for $${\mathscr {A}}_{0}u_{\nu }(\sigma ,\omega _{0})$$. After integration w.r.t. σ we obtain the desired, where we also make use of the regularity of Ψν(ω, ω0) declared above. Making use of the regularity of the mapping $$(u_{\nu }+u_{\nu }^{i})|_{\sigma }$$ as ν ∈ Πδ, from the boundary condition (3.8) we have that $$ \left .\frac {\partial \widehat {u}_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$ specifies a holomorphic mapping as ν ∈ Π1+δ. The latter directly follows from the fact that the denominator $$\cos \pi \tau +\cos \pi \nu \ne 0 $$ as τ ∈iR and ν ∈ Π1 in the integrand of (3.8), whereas $$(u_{\nu }+u_{\nu }^{i})|_{\sigma }$$ is holomorphic as ν ∈ Πδ. Further use of the integral equation (5.2) enables one to continue analytically uν onto the strip ν ∈ Π1+δ as a meromorphic mapping with the poles located in the strip |Iν| < b, b > 0. Indeed, we make use of the same argumentation, exploiting the equation written in the form (5.3) and the fact that $$(I-{\mathscr {K}}_{0})$$ is boundedly invertible, in order to verify that uν(s, ω0)|σ has only polar singularities as ν ∈ Π1+δ because $${\mathscr {A}}_{0}u_{\nu }(\sigma ,\omega _{0})$$ is regular and gν(s, σ), Ψν(s, ω0) are meromorphic in this strip with the poles located on the real axis. The condition (3.7) shows that having the meromorphic functions in the strip Π1+δ one can continue $$ \left .\frac {\partial \widehat {u}_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$ from this strip onto a neighbouring strip on the left- or right-hand sides. Because the shift of the argument ν is performed along the real axis, the corresponding poles appear only in some strip |Iν| < b, b > 0. On this way we obtain that $$ \left .\frac {\partial \widehat {u}_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$ and uν(s, ω0)|σ are meromorphic in the strip Π2+δ. The corresponding procedure can be iterated and one can prove meromorphic continuation of the mapping uν (as well as of $$ \left .\frac {\partial \widehat {u}_{\nu }}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$) from the strip Π2+δ onto the complex plane with the singularities located in some strip |Iν| < b, b > 0. The representation (5.1) enables one to assert that uν(s, ω0)|σ is a meromorphic mapping ν ↦ L2(Σ) as ν ∈ C , with the values uν(⋅, ω0) ∈ L2(Σ). 6. Sommerfeld integral representation for the wave field In order to study the far-field asymptotics of the scattered wave field U(r, φ) it is profitable to reduce the Watson-Bessel representation to the Sommerfeld integral because, contrary to that of the Watson-Bessel, it is well adapted to this goal. 6.1. Sommerfeld representation for the incident wave We transform the Watson–Bessel representation (3.1) for the incident plane wave making use of the Sommerfeld formula for the Bessel function   \begin{align*} J_{\nu}(kr)=\frac{1}{2\pi}\int\limits_{\gamma_{-}}\mathrm{e}^{-\mathrm{i}kr\cos\alpha}\, \mathrm{e}^{\mathrm{i}\nu\pi/2-\mathrm{i}\nu\alpha}\,d \alpha, \end{align*} where γ− is the lower part of the double-loop Sommerfeld contour (Fig. 8). Fig. 8. View largeDownload slide Sommerfeld double-loop contour, γ = γ+ ∪ γ−. Fig. 8. View largeDownload slide Sommerfeld double-loop contour, γ = γ+ ∪ γ−. Substitute the latter into (3.1) and change the orders of integration, which is justified,   \begin{align} U_{i}(r,\omega,\omega_{0})&= 4\mathrm{i}\sqrt{\frac{\pi}{2}}\int\limits_{C_{0}}\, d \nu\,u_{\nu}^{i}(\omega,\omega_{0}) \frac{\nu \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}}{\sqrt{-\mathrm{i}kr}}\left( \frac{1}{2\pi}\int\limits_{\gamma_{-}}\mathrm{e}^{-\mathrm{i}kr\cos\alpha}\,\mathrm{e}^{\mathrm{i}\nu\pi/2-\mathrm{i}\nu\alpha}\,d \alpha\, \right)\,\nonumber\\ &=\frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma_{-}} d \alpha\,\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}} \left( -2\sqrt{2\pi}\int\limits_{C_{0}} \,\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha} u_{\nu}^{i}(\omega,\omega_{0})\,d \nu\, \right)\, \nonumber\\ &=\frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma_{-}}\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}} \varPsi_{i}(\alpha,\omega,\omega_{0})\, d \alpha, \end{align} (6.1) where we introduced notation   \begin{align*} \varPsi_{i}(\alpha,\omega,\omega_{0})\,=\,-2\sqrt{2\pi}\int\limits_{C_{0}} \,\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha} u_{\nu}^{i}(\omega,\omega_{0})\,d \nu, \end{align*} Ψi(α, ω, ω0) is regular in D− = {α ∈ C : ℑα < 0} because $$u_{\nu }^{i}(\omega ,\omega _{0})$$ is bounded on C0 for any ω ∈ S2. In view of the estimate (3.3) on the contour Cϕ one can verify that Ψi(⋅, ω, ω0) is regular in the domain   \begin{align*} D_{\theta}^{\phi} =\,\left\{\alpha\in C:\,\,\sin\phi\,\Re\alpha+\cos\phi\,\Im\alpha < \theta|\sin\phi|\right\}\cap \{\alpha\in C:\,\,-\sin\phi\,\Re\alpha+\cos\phi\,\Im\alpha < \theta|\sin\phi|\} \end{align*} because   \begin{align*} \left|\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha}u_{\nu}^{i}(\omega,\omega_{0}) \right|\,< \,C\,|\nu|^{1/2} \exp\big\{|\nu|\big(|\sin\phi|\,\Re(\alpha)+\cos\phi\,\Im(\alpha)-\theta|\sin\phi|\big)\big\}, \end{align*} θ = θ(ω, ω0). It is useful to notice that the regularity domain varies from the halfplane $$D_{\theta }^{0}$$ to the strip $$D_{\theta }^{\pi /2}$$ (see also Fig. 7 in Lyalinov, 2013). Consider Ψi(⋅, ω, ω0) in $$D_{\theta }^{\pi /2}$$ and introduce a regular function Φi(⋅, ω, ω0) by the equality   \begin{align*} \varPhi_{i}(\alpha,\omega,\omega_{0})\,:=\,\frac{1}{2} \varPsi_{i}(\alpha,\omega,\omega_{0})\,= \mathrm{i}\sqrt{2\pi}\int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty} \,\nu\, \sin(\nu\alpha) u_{\nu}^{i}(\omega,\omega_{0})\,d \nu, \end{align*} which is obviously odd in α, Φi(α, ω, ω0) = −Φi(−α, ω, ω0). We easily verify that Φi(⋅, ω, ω0) is analytically continued as a holomorphic function into C∖l±, $$l_{\pm }\!=\!\{\alpha \in C:\,\pm \Re (\alpha )\geqslant \theta (\omega ,\omega _{0}),\,\Im (\alpha )=0\}$$, i.e. it is regular in the complex plane with the cuts along the lines l±. Remark that Φi(α, ω, ω0) can be computed in a closed form Lyalinov (2013)  \begin{align*} \varPhi_{i}(\alpha,\omega,\omega_{0})=\frac{\partial \widetilde\varPhi_{i}(\alpha,\omega,\omega_{0})}{\partial \alpha} \end{align*} with   \begin{align*} \widetilde\varPhi_{i}(\alpha,\omega,\omega_{0})=-\frac{\sqrt{\pi}}{2}\,\big[\cos\alpha-\cos\theta(\omega,\omega_{0})\big]^{-1/2}, \end{align*} the branch is fixed by the condition $$\sqrt {\cos \alpha -\cos \theta (\omega ,\omega _{0})}>0$$ as − θ < α < θ. This expression is obtained by means of the formula (see Gradstein & Ryzhik, 1980, 7.216)   \begin{align*} \frac{1}{\sqrt{2}}\,[\cos\alpha-\cos\theta]^{-1/2}\,=\, \int\limits_{0}^{\infty}\mathrm{d}\tau \frac{P_{\mathrm{i}\tau-1/2}\big(\cos[\pi-\theta]\big)}{\cos\mathrm{i}\pi\tau} \,\cos\mathrm{i}\alpha\tau\,,\,\,\,\,\,\,(\mathrm{i}\alpha>0)\,. \end{align*} The Sommerfeld representation for the incident wave takes the form   \begin{align} U_{i}(r,\vartheta,\varphi)= \frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma} \,\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}}\, \varPhi_{i}(\alpha,\omega,\omega_{0}) \,d \alpha, \, \end{align} (6.2) and also   \begin{align} U_{i}(r,\vartheta,\varphi)= \frac{{\sqrt{-\mathrm{i}kr}}}{2\pi\mathrm{i}} \int\limits_{\gamma} \,{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}\, \sin\alpha\, \widetilde\varPhi_{i}(\alpha,\omega,\omega_{0}) \,d \alpha, \, \end{align} (6.3) where γ = γ+ ∪ γ− is shown in Fig. 8. 6.2. Sommerfeld representation for the scattered field The derivations given in the previous section motivate appearance of the analogous representations for the scattered field. We have   \begin{align} U(r,\vartheta,\varphi)&= 4\mathrm{i}\sqrt{\frac{\pi}{2}}\int\limits_{C_{0}}\, d \nu\,u_{\nu}(\omega,\omega_{0}) \frac{\nu \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}}{\sqrt{-\mathrm{i}kr}}\left( \frac{1}{2\pi}\int\limits_{\gamma_{-}}\mathrm{e}^{-\mathrm{i}kr\cos\alpha}\,\mathrm{e}^{\mathrm{i}\nu\pi/2-\mathrm{i}\nu\alpha}\,d \alpha\, \right)\,\nonumber\\ &=\frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma_{-}} d \alpha\,\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}} \left( -2\sqrt{2\pi}\int\limits_{C_{0}} \,\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha} u_{\nu}(\omega,\omega_{0})\,d \nu\, \right)\, \nonumber\\&=\frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma_{-}}\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}} \varPsi(\alpha,\omega,\omega_{0})\, d \alpha, \end{align} (6.4) where we introduced   \begin{align*} \varPsi(\alpha,\omega,\omega_{0})\,=\,-2\sqrt{2\pi}\int\limits_{C_{0}} \,\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha} u_{\nu}(\omega,\omega_{0})\,d \nu, \end{align*} Ψ(⋅, ω, ω0) is regular in $$D_{\tau _{1}}=\{\alpha \in C:\, \,\Im ( \alpha )< -|\tau _{1}(\omega ,\omega _{0})|\}$$ because uν(ω, ω0) satisfies the estimate (3.9) on C0 (see Fig. 7) for any ω ∈ S2. We take into account the estimate   \begin{align*} \left|\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha}u_{\nu}(\omega,\omega_{0}) \right|\,< \,C\,|\nu|^{1/2} \exp&\left\{|\nu|\left(|\sin\phi|\Re(\alpha)+\cos\phi(\Im\alpha)\right.\right.\\&\quad\left.-\ \tau_{0}(\omega,\omega_{0})|\sin\phi|\,+ \,\mathrm{sign}(\phi)\,\cos\phi|\tau_{1}(\omega,\omega_{0})|\right\}\!, \end{align*} and conclude that the regularity domain for Ψ(⋅, ω, ω0) is   \begin{align*} D_{\tau_{0}\tau_{1}}^{\phi} =\,&\left\{\alpha\in C:\,\,\sin\phi\Re(\alpha)+\cos\phi\Im(\alpha) < \tau_{0}|\sin\phi|-\cos\phi\,\tau_{1}\right\}\cap\\ &\left\{\alpha\in C:\,\,-\sin\phi\Re(\alpha)+\cos\phi\Im(\alpha) < \tau_{0}|\sin\phi|+\cos\phi\,\tau_{1}\right\}\!, \end{align*} which is verified from   \begin{align*} \varPsi(\alpha,\omega,\omega_{0})\,=\,-2\sqrt{2\pi}\int\limits_{C_{\phi}} \,\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha} u_{\nu}(\omega,\omega_{0})\,d \nu, \end{align*} where ϕ ∈ (0, π/2). Remark Provided the contour Cϕ varies from C0 into Cπ/2 the domain of regularity $$D_{\tau _{0}\tau _{1}}^{\phi }$$ of Ψ(⋅, ω, ω0) deforms from the halfplane ℑ(α) < −|τ1(ω, ω0)| onto the strip |I(α)| < τ0(ω, ω0). We consider Ψ(⋅, ω, ω0) in the strip $$D_{\tau _{0}\tau _{1}}^{\pi /2}$$ and introduce there a regular function   \begin{align*} \varPhi(\alpha,\omega,\omega_{0})\,:=\,\frac{1}{2} \varPsi(\alpha,\omega,\omega_{0})\,= \mathrm{i}\sqrt{2\pi}\int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty} \,\nu\, \sin(\nu\alpha) u_{\nu}(\omega,\omega_{0})\,d \nu, \end{align*} which is odd in α, Φ(α, ω, ω0) = −Φ(−α, ω, ω0). Simple analysis enables us to conclude that Φ(⋅, ω, ω0) is continued as a regular function into the domain   \begin{align*} D\,=\,D_{\tau_{0}\tau_{1}}^{\pi/2}\cup D_{\tau_{0}\tau_{1}}^{0}\cup \left(D_{\tau_{0}\tau_{1}}^{\pi/2}\right)^{*}\!, \end{align*} where $$\left (D_{\tau _{0}\tau _{1}}^{\pi /2}\right )^{*}$$ is symmetric to $$D_{\tau _{0}\tau _{1}}^{\pi /2}$$ with respect to the origin, (Fig. 9). In other words, we can assert that all singularities of Φ(⋅, ω, ω0) are located in C ∖ D. For the diffraction by a sector with Dirichlet boundary conditions such singularities are symmetrically distributed on the real axis Lyalinov, 2013. For the sector with the impedance boundary conditions there are some additional complex singularities which are responsible for the surface waves propagating along the surface of the sector. Fig. 9. View largeDownload slide Domain of regularity D, singularities are in C ∖ D. Fig. 9. View largeDownload slide Domain of regularity D, singularities are in C ∖ D. The Sommerfeld representations for the scattered field take the form   \begin{align} U(r,\vartheta,\varphi)= \frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma} \,\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}}\, \varPhi(\alpha,\omega,\omega_{0}) \,d \alpha, \, \end{align} (6.5) or   \begin{align} U(r,\vartheta,\varphi)= \frac{{\sqrt{-\mathrm{i}kr}}}{2\pi\mathrm{i}}1 \int\limits_{\gamma} \,{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}\, \sin\alpha\, \widetilde\varPhi(\alpha,\omega,\omega_{0}) \,d \alpha\, \, \end{align} (6.6) with   \begin{align*} \displaystyle \varPhi(\alpha,\omega,\omega_{0})=\frac{\partial \widetilde\varPhi(\alpha,\omega,\omega_{0})}{\partial \alpha}\,. \end{align*} As we remarked the singularities of $$\varPhi (\alpha ,\omega ,\omega _{0}), \widetilde \varPhi (\alpha ,\omega ,\omega _{0})$$ play a crucial role in studying the far-field asymptotics by use of the Sommerfeld integral representations. In order to describe the singularities of the Sommerfeld transformants $$\varPhi (\alpha ,\omega ,\omega _{0}), \widetilde \varPhi (\alpha ,\omega ,\omega _{0})$$ we turn to formulation of the problems for them, although the singularities themselves will be studied elsewhere. 6.3. Problems for the Sommerfeld transformants $$\varPhi (\alpha ,\omega ,\omega _{0}),\widetilde {\varPhi }(\alpha ,\omega ,\omega _{0})$$ It is obvious that the Fourier transform of (3.6) with integration along the imaginary axis, enables us to write down the equation for the Sommerfeld transformant (Lyalinov, 2013),   \begin{align} \left(\triangle_{\omega}-\partial^{2}_{\alpha}-1/4\right)\varPhi(\alpha,\omega,\omega_{0})=0 \end{align} (6.7) as ω ∈ S2 ∖ σ. Let us turn to the boundary conditions   \begin{align} \left. r^{-1}\frac{\partial\left(U + U^{\mathrm{i}}\right)}{\partial{\mathscr{N}}_{\pm}} \right|_{\sigma_{\pm}} &=\frac{1}{\sqrt{-\mathrm{i} kr}}\frac{1}{2\mathrm{i}\pi}\int\limits_{\gamma}\mathrm{e}^{-\mathrm{i} kr\cos\alpha}(-\mathrm{i} k\sin\alpha) \left.\frac{\partial}{\partial{\mathscr{N}}_{\pm}}\left(\widetilde{\varPhi}+\widetilde{\varPhi}_{i}\right)(\alpha,\omega,\omega_{0})\right|_{\sigma_{\pm}} \mathrm{d} \alpha \nonumber\\ &=\,\left.\mathrm{i} k \eta_{\pm} \left(U + U^{\mathrm{i}}\right)\right|_{S}\,=\,\,\frac{\mathrm{i} k \eta_{\pm}}{\sqrt{-\mathrm{i} kr}} \frac{1}{2\mathrm{i}\pi}\int\limits_{\gamma}\mathrm{e}^{-\mathrm{i} kr\cos\alpha} \left.\frac{\partial}{\partial \alpha}\left(\widetilde{\varPhi}+\widetilde{\varPhi}_{ i}\right)(\alpha,\omega,\omega_{0})\right|_{\sigma_{\pm}}\mathrm{d} \alpha, \end{align} (6.8) then, exploiting the Malyuzhinets theorem (see Lyalinov & Zhu, 2012, Chapter 1), we obtain   \begin{align*} \sin\alpha \left.\frac{\partial \left(\widetilde{\varPhi}+\widetilde{\varPhi}_{\mathrm{i}}\right)}{\partial{\mathscr{N}}_{\pm}}(\alpha,\omega,\omega_{0})\right|_{\sigma_{\pm}} = \left.-\eta_{\pm}\frac{\partial \left(\widetilde{\varPhi}+\widetilde{\varPhi}_{ i}\right)} {\partial \alpha}(\alpha,\omega,\omega_{0})\right|_{\sigma_{\pm}}\!, \end{align*} which is followed by   \begin{align} \left.\frac{\partial (\varPhi+\varPhi_{ i})}{\partial{\mathscr{N}}_{\pm}}(\alpha,\omega,\omega_{0})\right|_{\sigma_{\pm}} = \left.-\eta_{\pm}\frac{\partial}{\partial \alpha}\frac{(\varPhi+\varPhi_{ i})(\alpha,\omega,\omega_{0})}{\sin\alpha}\right|_{\sigma_{\pm}}\!, \end{align} (6.9) where   \begin{align*} \varPhi_{ i}(\alpha,\omega,\omega_{0})=\frac{\partial} {\partial \alpha}\widetilde{\varPhi}_{ i}(\alpha,\omega,\omega_{0}). \end{align*} The equation (6.7) and the condition (6.9) are proved to be valid in the domain D by means of the analytic continuation. For real α from this strip the equation (6.7) is of hyperbolic type. In order to have a correct problem for this equation on S2 ∖ σ with the boundary condition (6.9) as α > 0 we should add initial conditions at α = 0. For any α ∈ [0, τ0) we have   \begin{align} \varPhi(\alpha,\omega,\omega_{0})+\varPhi_{i}(\alpha,\omega,\omega_{0})= \,\mathrm{i}\sqrt{2\pi}\int\limits_{\mathrm{i} R} \,\nu\, \sin({\nu\alpha}) u_{\nu}(\omega,\omega_{0})\,d \nu\, \,-\frac{\sqrt{\pi}}{4} \frac{\sin\alpha}{\left(\cos\alpha-\cos\theta(\omega,\omega_{0})\right)^{3/2}} \end{align} (6.10) which is a regular function. Therefore, assuming that uν(ω, ω0) is known in the integrand, we arrive at the initial conditions   \begin{align} \varPhi(0,\omega,\omega_{0})= 0,\,\,\,\, \left.\frac{\partial \varPhi(\alpha,\omega,\omega_{0})}{\partial\alpha}\right|_{\alpha=0}= \mathrm{i}\sqrt{2\pi}\int\limits_{\mathrm{i} R} \,\nu^{2}\, u_{\nu}(\omega,\omega_{0})\,d \nu.\, \end{align} (6.11) The Cauchy boundary value problem (6.7), (6.9), (6.11) can be used in order to determine singularities of Φ(α, ω, ω0) as α ∈ C ∖ D. However, instead of the conditions (6.11) one can exploit the equality (6.10) for any point α ∈ [0, τ0) together with (6.7), (6.9). The analogous problem is valid for $$\widetilde \varPhi (\alpha ,\omega ,\omega _{0})$$  \begin{align} \left(\triangle_{\omega}-\partial^{2}_{\alpha}-1/4\right)\widetilde\varPhi(\alpha,\omega,\omega_{0})=0 \end{align} (6.12) as ω ∈ S2 ∖ σ and   \begin{align} \sin\alpha \left.\frac{\partial \left(\widetilde{\varPhi}+\widetilde{\varPhi}_{\mathrm{i}}\right)} {\partial N_{\pm}}\right|_{\sigma_{\pm}} = \left.-\eta_{\pm}\frac{\partial \left(\widetilde{\varPhi}+\widetilde{\varPhi}_{ i}\right)}{\partial \alpha}\right|_{\sigma_{\pm}}\!, \end{align} (6.13) and   \begin{align}\displaystyle \widetilde\varPhi(0,\omega,\omega_{0})= -\mathrm{i}\sqrt{2\pi}\int\limits_{\mathrm{i} R} \, u_{\nu}(\omega,\omega_{0})\,d \nu,\,\,\,\, \left.\frac{\partial \widetilde\varPhi(\alpha,\omega,\omega_{0})}{\partial\alpha}\right|_{\alpha=0}= 0 .\, \end{align} (6.14) The discussion above enables us to prove Theorem 6.1 There exists an analytic function Φ(α, ω, ω0) which is holomorphic in the domain D. For real α ∈ [0, τ0) this function solves the problem (6.7), (6.9), (6.11). For the strip $$|\Re (\alpha )|<\tau _{0}(\omega ,\omega _{0})$$ it is specified by the equality (see (6.10))   \begin{align*} \varPhi(\alpha,\omega,\omega_{0})\,= \,\mathrm{i}\sqrt{2\pi}\int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty} \,\nu\, \sin({\nu\alpha}) u_{\nu}(\omega,\omega_{0})\, \mathrm{d} \nu. \end{align*} A similar statement is valid for $$\widetilde \varPhi (\alpha ,\omega ,\omega _{0})$$. 7. Comments on the derivations of the far-field asymptotics. Diffraction coefficient for narrow sector in ‘oasis’ In this section we give some comments dealing with the derivations of the far-field asymptotics described in Section 2.2 although the detailed exposition of the results will appear in a future work. Fig. 10. View largeDownload slide Deformation of the Sommerfeld contour into the steepest descent paths γ+ ∪ γ−: (left) singularities are not captured, ω ∈ Ω0; (right) singularities are captured ω ∈ S2 ∖ Ω0. Fig. 10. View largeDownload slide Deformation of the Sommerfeld contour into the steepest descent paths γ+ ∪ γ−: (left) singularities are not captured, ω ∈ Ω0; (right) singularities are captured ω ∈ S2 ∖ Ω0. Solution Φ(α, ω, ω0) of the problems for the hyperbolic equation (6.7) as α ∈ [0, τ0) is unique. This solution is represented by the Fourier type integral (see, Theorem 6.1) as $$|\Re (\alpha )|<\tau _{0}(\omega ,\omega _{0})$$. However, one needs to have analytic continuation of the Sommerfeld transformant into C ∖ D, in particular, into vicinities of the singularities9 located in C ∖ D. As we mentioned, these singularities are of crucial importance for studying the far-field asymptotics (Lyalinov, 2013). Indeed, in order to evaluate the Sommerfeld integral representation (6.5) asymptotically as $$kr \to \infty $$ we ought to deform the Sommerfeld double-loop contour γ into the steepest descent paths γ±π (Fig. 10). In the process of such deformation some singularities of the transformant can be captured. Contributions of these singularities (Fig. 10, right) give rise to the corresponding components of the far field such as reflected wave, edge waves (see Lyalinov, 2013 for the Dirichlet boundary conditions) as well as the surface waves which may be excited near the impedance surface of the sector (2.9), (2.10). These surface waves are governed by complex singularities which do not exist in the case of the ideal boundary conditions. It is worth mentioning that the singularities migrate (Fig. 10) when the observation point varies its position. The saddle point ±π are responsible for the spherical wave from the vertex of the sector. The diffraction coefficient of this wave is easily calculated by use of the direct application of the saddle point technique provided the observation point (r, ω) is located in the oasis, ω ∈ Ω0. In this case the singularities are not captured (Fig. 10, left) and the asymptotics of the scattered far-field is given by (see also (2.8))  \begin{align*} \displaystyle U(r,\vartheta,\varphi)=D(\omega,\omega_{0})\frac{\exp(\mathrm{i}kr)}{-\mathrm{i}kr}\left(1+ O\left(\frac{1}{kr}\right)\right),\,\,\, kr\to \infty. \end{align*} with   \begin{align} D(\omega,\omega_{0})\,=\,-\sqrt{\frac{2}{\pi}}\,\varPhi(\pi,\omega,\omega_{0})\,=\, \,\frac{2}{\mathrm{i}}\int\limits_{\mathrm{i} R} \,\nu\, \sin(\pi\nu) u_{\nu}(\omega,\omega_{0})\, \mathrm{d} \nu\, \end{align} (7.1) or10  \begin{align*} D(\omega,\omega_{0})\,= 2\int\limits_{\mathrm{i} R} \,\nu\, \exp(-\mathrm{i}\pi\nu) u_{\nu}(\omega,\omega_{0})\, \mathrm{d} \nu, \end{align*} where the integrals converge exponentially as ω ∈ Ω0. It is obvious that the numerical determination of the spectral function uν(ω, ω0) is necessary for the calculation of the diffraction coefficient. In order to demonstrate also some practical issues from the results developed above we consider derivation of the closed form of the asymptotic expression for the diffraction coefficient in the case of a narrow impedance sector, β = 2a ≪ 1. In this case a closed asymptotic formula for the spectral function uν(ω, ω0) can be given and the integral for the diffraction coefficient (7.1) can be computed explicitly leading to a simple expression in the high order approximation as β ≪ 1. 7.1. Expression for D(ω, ω0) as β = 2a ≪ 1 There are, at least, two ways to determine such an expression. The first one is in asymptotic solution of the boundary value problem (3.6), (3.8) for the spectral function uν(ω, ω0). It is based on matching local asymptotic expansions (see e.g. Babich, 1997; Lyalinov & Zhu, 2012, pp. 122–124) for the problem on the unit sphere S2 with the narrow cut σ = σ+ ∪ σ−, mes(σ) = 2mes(σ±) = 2β ≪ 1. In this work we exploit an alternative approach, based on approximate solution of the integral equation (5.2) for uν(σ, ω0) as mes(σ±) = β ≪ 1, Bernard & Lyalinov (2001). It leads to the same desired result and also makes use of the integral representation (5.1) of the solution uν(ω, ω0), ω ∈ Ω0. Simple analysis of the integral equation (5.2) enables us to assert that $$ u_{\nu }(\omega |_{\sigma _{\pm }},\omega _{0})$$ is of O(β), therefore, taking into account that mes(σ+) = β and mes(σ−) = β, from the integral representation (5.1) we find   \begin{align} u_{\nu}(\omega,\omega_{0})&=\, \, \varPsi_{\nu}(\omega,\omega_{0})(1+O(\beta\log\beta)\nonumber\\&=\, \,- \,\frac{ 1\,}{2\mathrm{i}}\mathop{\sum}_{\pm} \eta_{\pm} \int\limits_{\sigma_{\pm}}\!\mathrm{d}\sigma\, g_{\nu}(\omega,\sigma)\!\left(\ \int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\,\mathrm{d}\tau\, \frac{\tau\,\sin\pi\tau\,u_{\tau}^{i}(\sigma,\omega_{0})}{\cos\pi\tau+\cos\pi\nu}\,\right) \left(1+O(\beta\log\beta)\right)\!,\,\,\,\,\,\omega\in\varOmega_{0}, \end{align} (7.2) where we took into account the estimate   \begin{align*} \mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\,\mathrm{d}\sigma\,g_{\nu}(\omega,\sigma) \,\left.\frac{\partial u_{\nu}^{i}}{ \partial{\mathscr{N}}_{\pm}}\right|_{\sigma}\, =\,O(\beta^{2}\log\beta). \end{align*} Substituting the integral expression from (7.2) into the formula for the diffraction coefficient (7.1) in oasis, in the leading approximation we arrive at   \begin{align*} D(\omega,\omega_{0})\,&=\mathop{\sum}_{\pm} \frac{\eta_{\pm}}{16} \int\limits_{\sigma_{\pm}}\,\mathrm{d}\sigma\,\\&\quad \left(\ \int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\,\mathrm{d} \tau\, \int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\,\mathrm{d} \nu\, \frac{\tau\,\tan\pi\tau\,\,\nu\,\tan\pi\nu\,}{\cos\pi\tau+\cos\pi\nu}\, P_{\nu-1/2}(\cos\theta(\omega,\sigma))P_{\tau-1/2}(-\cos\theta(\sigma,\omega_{0})) \right) \end{align*} The iterated integral in the brackets was computed in the work (Bernard & Lyalinov (1999) so that one has   \begin{align*} D(\omega,\omega_{0})\,= \,-\, \mathop{\sum}_{\pm} \frac{\eta_{\pm}}{4\pi} \int\limits_{\sigma_{\pm}}\, \frac{\mathrm{d}\sigma\,} {\left[\cos\theta(\omega,\sigma)+\cos\theta\left.(\sigma,\omega_{0}\right)\right]^{2}}\left(1+\,O(\beta \log\beta)\right)\, \end{align*} as θ(ω, σ) + θ(σ, ω0) > π. Integration in the latter formula is conducted along asymptotically small arcs σ± then asymptotically equivalent version of this formula takes the form   \begin{align*} D(\omega,\omega_{0})\,= \,-\, \frac{\eta_{+}+\eta_{-}}{2\pi} \frac{2a\,} {\left[\cos\theta(\omega,M)+\cos\theta\left.(M,\omega_{0}\right)\right]^{2}}\,\left(1+\,O(\beta \log\beta)\right)\!, \end{align*} θ(ω, M) + θ(M, ω0) > π and ω ∈ Ω0, M is the middle point of the arc AB. The advantage of the latter formula is that it is quite elementary and can be easily used in the engineering applications.11 It is instructive to compare this result with that obtained by Babich (1997), where he made use of matching asymptotic series in the case of Dirichlet boundary condition on the surface of a cone,   \begin{align*} D_{d}(\omega,\omega_{0})\,= \, -4\pi\, \int\limits_{-\infty}^{\infty}\,\mathrm{d} \tau\,\,\tau\,\mathrm{e}^{\pi\tau} \frac{g_{\tau} (\omega,M)\,g_{\tau}(M,\omega_{0})}{W_{\beta}-\Re{\psi(\mathrm{i}\tau-1/2)-{\mathscr{C}}- \log 2}}\, \left(1+\,O(\beta \log\beta)\right), \end{align*} θ(ω, M) + θ(M, ω0) > π, where $$W_{\beta }= - \log (a/2)=\mathrm {O}(\log \beta )$$, Wβ is the Wiener capacity, an integral characteristic of the segment of the length 2a, $${\mathscr {C}}$$ is Euler constant, $$g_{\tau } (\omega ,M)= -\frac {P_{\mathrm {i}\tau -1/2}(-\cos \theta (\omega ,M))}{4 \cosh \pi \tau }$$. As mentioned in Babich (1997) the latter expression for the diffraction coefficient Dd(ω, ω0) integral may be simplified by formally neglecting $$-\Re \psi (\mathrm {i}\tau -1/2)-{\mathscr {C}}- \log 2$$ in comparison with Wβ then computed, which leads to   \begin{align*} D_{d}(\omega,\omega_{0})\,= \,\,-\, \frac{1}{2\log\frac{a}{2}} \frac{1} {\left[\cos\theta(\omega,M)+\left.\cos\theta(M,\omega_{0}\right))\right]}\,\left(1+\,O(1/ \log\beta)\right)\!, \end{align*} as θ(ω, M) + θ(M, ω0) > π. It is obvious that the diffraction coefficient Dd(ω, ω0) for the Dirichlet case is of $$O(1/\log (a))$$ in the leading approximation, whereas D(ω, ω0) = O(a) as a → 0 for a narrow sector with the impedance boundary conditions. 8. Conclusion In this work we developed a motivated procedure to study the problem of diffraction by a semi-infinite sector with impedance boundary conditions. Such kind of the boundary conditions thwarts complete separation of variables. The separation of the radial variable in the boundary conditions leads to a condition which is non-local with respect to the parameter of separation. Nevertheless, after separation of the radial variable the problem on the unit sphere with the non-local condition on the cut AB admits an efficient study. To that end, the traditional theory of extension of sectorial sesquilinear forms has been exploited. The Watson–Bessel integral representation for the solution is not efficient for the derivation of the far-field asymptotics that is why reduction to the Sommerfeld integral representations was used. Analytic properties of the Sommerfeld transformants were considered. In particular, domains, where singularities of the transformants are localized, were indicated. Although a complete study of the singularities and of their contributions to the far-field asymptotics is postponed to a future work, we obtained a practically useful simple formula for the diffraction coefficient of the spherical wave from the vertex in ‘oasis’ in the case of scattering by a narrow impedance sector. One of the goals for the further studies is to develop a reliable numerical procedure in order to compute the spectral function uν(ω, ω0). Such a development could be based on our study of the problem for the spectral function given in this work. The situation with the impedance boundary conditions is more complex than that for the ideal ones. In particular, there is no reason to assert that the spectrum of the operator A from the Section 4.3 is discrete. Contrary to the case of the Laplace–Beltrami operator on the unit sphere with usual mixed (Robin) boundary conditions with complex impedance parameter (see Assier et al., 2016, Theorem 4.10) on the cut the argumentation based on the compact resolvent does not work in our spectral problem. However, there are some hopes that the study of the spectral properties of the non-local operator $${\mathscr {A}}$$ in the boundary condition of the problem (4.2) may be exploited in order to create an efficient numerical procedure to compute the spectral function uν(ω, ω0). This might be used in order to reduce the spectral problem with non-local operator in the boundary condition to that with simpler conditions like those complex Robin in Assier et al. (2016). It is also important to give a complete description of the far-field asymptotics. Some natural extensions of the ideas proposed in Lyalinov (2013) might be useful in this case. Acknowledgements The author is grateful to Dr Ning Yan Zhu (Stuttgart University) for his important comments on the subject of the work. It is also a pleasure to thank Prof. V.P. Smyshlyaev (University College of London) for his hospitality and fruitful discussions on diffraction by sectors during the visit of the author to UCL. The latter visit has been supported by the travel grant N 11.42.678.2017 of Saint Petersburg State University (SPbGU) for scientific collaboration. Funding The work was supported in part by the Russian Foundation of Basic Research, RFBR 17-01-00668a. Footnotes 1 It is worth remarking that in our study the use of closable sectorial forms and m-sectorial operators looks natural for the problem on the unit sphere with the cut. It has been partly inspired by a very useful work by Assier et al. (2016), where similar but simpler spectral problems for the Laplace–Beltrami operator with ideal boundary conditions on the cut have been considered. 2 The harmonic time-dependence $$\mathrm {e}^{-\mathrm {i}\widehat \omega t}$$ is assumed and suppressed throughout the paper. 3 The sign of χ± is not fixed; however, in the case χ± < 0, ε± = 0 surface waves are excited on the impedance surfaces of the sector and propagate to infinity without attenuation. 4  The asymptotics (2.8) fails provided the observation point approaches the boundary of Ω0, near the boundaries of the domains $$\varOmega _{i},\,\, i=0,r,A,B,\dots $$ some special transition functions apply to match the local asymptotics. 5 Usually these waves are neglected in comparison with the first two terms because they have higher order with respect to (kr)−1. 6 The Stirling asymptotics for the complex argument of the gamma-function is also exploited. 7 More accurate formulations will be given in the following section. 8 It is possible to describe the domain of TA more efficiently. 9 These singularities are of the branch point type, see also Lyalinov (2013). 10 For the first time this kind of formulae for the ideal circular cones has been obtained by Smyshlyaev, see e.g. Smyshlyaev (1990). 11 It is worth remarking that the identical result can be obtained by use of the matching the local asymptotic series, which is shown in Lyalinov & Zhu (2012), Chapter 5, for a convex impedance cone. References Albani, M. ( 2007) On Radlow’s quarter-plane diffraction solution. Radio Science , 42, RS6S11. Abawi, A. T., Dashen, R. F. & Levine, H. ( 1997) The eigenvalues of the Laplacian on a sphere with boundary conditions specified on a segment of agreatcircle. J. Math. Phys.  38, 1623-- 1649. Google Scholar CrossRef Search ADS   Assier, R. C. & Peake, N. ( 2012a) On the diffraction of acoustic waves by a quarter-plane, Wave Motion , 49, 64-- 82. Google Scholar CrossRef Search ADS   Assier, R. C. & Peake, N. ( 2012b) Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane, IMA Journal of Applied Mathematics , 77, 605-- 625. (doi:hxs042[IMAMAT-2011–150]/imamat/hxh000) Assier, R. C., Poon, C. & Peake, N. ( 2016) Spectral study of the Laplace-Beltrami operator arising in the problem of acoustic wave scattering by a quarter-plane, Q. Jl. Mech. Appl. Math. , 69, 281-- 317. 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M. L. ( 2014) Advanced Theory of Diffraction by a Semi-infinite Impedance Cone , Oxford: Alpha Science Ser. Wave Phenom Alpha Science. Bernard, J. -M. L. & Lyalinov, M. A. ( 1999) The leading asymptotic term for the scattering diagram in the problem of diffraction by a narrow circular impedance cone. J. Phys. A: Math. Gen.  32, L43-- L48. Google Scholar CrossRef Search ADS   Bernard, J. -M. L. & Lyalinov, M. A. ( 2001) Diffraction of acoustic waves by an impedance cone of an arbitrary cross-section. Wave Motion , 33 155-- 181. (Erratum : p.177 replace $$O(1/\cos (\pi (\nu -b)))$$ by $$O(\nu ^d\sin (\pi \nu )/\cos (\pi (\nu -b)))$$). Google Scholar CrossRef Search ADS   Bernard, J. -M. L., Lyalinov, M. A. & Zhu, N. Y. ( 2008) Analytical-numerical calculation of diffraction coefficients for a circular impedance cone. IEEE Trans. on Antennas and Propagation , 56, 1616-- 1622. Google Scholar CrossRef Search ADS   Birman, M. S., Solomyak, M. Z. 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( 1960) Foundations of Modern Analysis . New York and London: Academic Press. Felsen, L. B. & Marcuvitz, N. ( 1973) Radiation and Scattering of Waves . Englewood Cliffs, NJ: Prentice Hall. Gradstein, I. S. & Ryzhik, I. M. ( 1980) Tables of Integrals, Series and Products , 4th edn. Orlando: Academic Press. Hansen, T. B. ( 1991) Corner diffraction coefficients for the quarter-plane. IEEE Trans. on Antennas and Propagation , 39, 976-- 984. Google Scholar CrossRef Search ADS   Jones, D. S. ( 1964) The Theory of Electromagnetism . London: Pergamon Press. Jones, D. S. ( 1997) Scattering by a cone. Q. Jl. Mech. appl. Math. , 50, 499-- 523. Google Scholar CrossRef Search ADS   Kato, T. ( 1972) Perturbation Theory for Linear Operators . Moscow: Springer, M.  740. Kraus, L. & Levine, L. M. ( 1961) Diffraction by an elliptic cone. Commun. Pure Appl. Math. , 14, 49-- 68. Google Scholar CrossRef Search ADS   Lyalinov, M. A., Zhu N. Y. & Smyshlyaev, V. P. ( 2010) Scattering of a plane electromagnetic wave by a hollow circular cone with thin semi-transparent walls. IMA Journ. Appl. Math. , 75, 676-- 719. Google Scholar CrossRef Search ADS   Lyalinov, M. A. & Zhu, N. Y. ( 2012) Scattering of Waves by Wedges and Cones with Impedance Boundary Conditions. Mario Boella Series on Electromagnetism in Information & Communication . Edison, NJ: SciTech-IET. Lyalinov, M. A. ( 2013) Scattering of acoustic waves by a sector. Wave Motion , 50, 739-- 762. Google Scholar CrossRef Search ADS   Meister, E. & Speck, F. -O. ( 1988) A contribution to the quarter-plane problem in diffraction theory, J. Math. Analysis and Appl. , 130, 223-- 236. Google Scholar CrossRef Search ADS   Radlow, J. ( 1961) Diffraction by a quarter-plane, Arch. Ration. Mech. Anal. , 8, 139-- 158. Google Scholar CrossRef Search ADS   Satterwhite, R. ( 1974) Diffraction by a quater-plane, exact solution and some numerical results. IEEE Trans. on Antennas and Propagation , AP22, 500-- 503. Shanin, A. V. ( 2011) Asymptotics of waves diffracted by a cone and diffraction series on a sphere. Zapiski Nach. Sem. POMI RAN  393, 234-- 258. (In Russian, English translation Shanin, A. V. (2012) Journal of Mathematical Sciences , 185, 644-- 657.) Smyshlyaev, V. P. ( 1990) Diffraction by conical surfaces at high frequencies. Wave Motion  12, 329-- 339. Google Scholar CrossRef Search ADS   Smyshlyaev, V. P. ( 1991) High frequency asymptotics of the wave field diffracted by a plane angular sector. Izvestia Vys. Uch. Zaved. (Radiofizika),  34, 10-- 12, 1170-- 1179. Valyaev, V. & Shanin, A. ( 2012) Embedding formulae for Laplace–Beltrami problems on the sphere with a cut. Wave Motion,  49, 83-- 92. Google Scholar CrossRef Search ADS   Van Bladel, J. ( 1991) Singular Electromagnetic Fields and Sources . Oxford: Wiley. © The Author(s) 2018. 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Acoustic scattering by a semi-infinite angular sector with impedance boundary conditions

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Abstract

Abstract In this work we study the problem of diffraction of an acoustic plane wave by a semi-infinite angular sector with impedance boundary conditions on its surface. It is studied by means of incomplete separation of variables. With the aid of Watson–Bessel integral representation the problem is reduced to a boundary value problem on the unit sphere with an operator-impedance boundary condition on a cut of the sphere. The latter problem is further studied by means of the traditional methods of extensions of sectorial sesquilinear forms. The Sommerfeld integral representation is obtained from that of Watson–Bessel with the aim to develop the far-field asymptotics. Analytic properties of the corresponding Sommerfeld transformant are also discussed. For a narrow impedance sector, an asymptotic formula for the diffraction coefficient of the spherical wave propagating from the vertex is derived. 1. Introduction 1.1. Motivation and some comments on the literature To our knowledge, until present time the problem of diffraction by an impedance sector has not been discussed in the literature. It seems that the reason is in the analytic difficulties arising in the study of the problem in spite of the fact that the impedance boundary conditions are more realistic in practice in comparison with those ideal ones. We modify and adapt the approach recently developed by Lyalinov (2013), for the case of the sector with Dirichlet boundary conditions, in order to describe the far-field asymptotics in the problem of diffraction by a plane angular sector with impedance boundary conditions on its surface. Consider the unit sphere with the centre at the sector’s vertex then the sector and the sphere are intersected across a segment (‘cut’) AB of a big circle, Fig. 1. We assume that the angular measure 2a of the corresponding arc satisfies the restrictions 0 < 2a < π. One of the most interesting cases is the quarter-plane corresponding to 2a = π/2. Fig. 1. View largeDownload slide Diffraction by an impedance sector. Fig. 1. View largeDownload slide Diffraction by an impedance sector. In many aspects our approach has common features with those used for the problems of diffraction by cones with ideal (see e.g. Bowman et al., 1987; Felsen & Marcuvitz, 1973; Borovikov, 1966; Jones, 1964; Jones, 1997; Cheeger & Taylor, 1982; Smyshlyaev, 1991; Smyshlyaev, 1990; Babich et al., 2000; Bonner et al., 2005) or impedance boundary conditions. Diffraction by an impedance cone is studied in the papers of Bernard (1997), Bernard & Lyalinov (2001), and Bernard et al. (2008). Some additional results and references can be found in Lyalinov et al. (2010) and Lyalinov & Zhu (2012). As regards papers on diffraction by a sector with the ideal boundary conditions reader might be forwarded to discussion of the literature in our work (Lyalinov, 2013, see also Kraus & Levine, 1961; Hansen, 1991; Meister & Speck, 1988; Radlow, 1961; Albani, 2007; Valyaev & Shanin, 2012; Abawi et al., 1997; Assier & Peake, 2012a; Assier & Peake, 2012b; Budaev & Bogy, 2005). It is worth noting that the far-field asymptotics is of basic importance in numerous applications of such canonical problem like diffraction by an impedance sector. However, in order to develop the corresponding asymptotic expressions it is necessary to obtain the corresponding analytic tools, integral representations, to justify them and give a constructive way to obtain a solution and then to develop efficient numerical procedures. In this work we pay much attention to such tools implying that a detailed study of the far-field asymptotics will be discussed in a forthcoming publication. Nevertheless, we demonstrate the efficiency of the developed formulae by deducing a practically useful expression for the diffraction coefficient of the spherical wave from the vertex of a narrow impedance sector. In the modern interpretation a diffraction problem might be considered satisfactorily solved provided the following conditions are fulfilled. First, it is demonstrated that it has a unique solution and all necessary representations (integral or others) for the solution are established. Second, the far-field asymptotics, including those in the transition zones, are derived from such representations and the expressions for the diffraction coefficients are obtained. Finally, the numerical elaboration of the solution is given and the numerical results for the far field are represented. From this point of view, for instance, the problem of diffraction by a right-circular cone with ideal boundary conditions can be considered solved. As regard the problem of diffraction by an impedance sector, the study represented in this work basically deals with the first step of the solution in the above-mentioned sense. On the other hand, a formula for the diffraction coefficient in the so-called oasis (defined below) is derived and its asymptotic expression for a narrow impedance sector is given. 1.2. Description of the approach In the following sections we formulate the problem of diffraction by an impedance sector, Fig. 1. The wave field satisfies the Helmholtz equation, the impedance boundary conditions, Meixner’s conditions at the edges and at the vertex. Provided the scattered wave satisfies the radiation condition at infinity, the classical solution of the problem is unique. However, in the case of incidence of a plane wave the scattered field does not satisfy the radiation condition and we then discuss asymptotic behaviour of the wave field at infinity. We introduce a set of characteristic domains of the unit sphere centred at the vertex of the sector. Each point of such a domain on the sphere is attributed to some direction of observation. In different characteristic domains the far-field asymptotics consists of different wave field components. In order to separate the radial variable we make use of the Watson–Bessel integral representation for the solution and formulate a problem for the unknown ‘spectral’ function on the unit sphere with the cut $$\overline {AB}$$. An elliptic (Helmholtz type) equation for the spectral function depends on the variable of separation ν, whereas the boundary condition on the sides of the cut $$\overline {AB}$$ is nonlocal with respect to ν and has a form of the mixed boundary condition with an operator-impedance in it. We then carefully study such a problem attributing an m-sectorial operator to it. This operator is introduced by means of the corresponding sesquilinear form admitting a closed extension.1 Then the meromorphic continuation for spectral function w.r.t. ν is discussed. The latter exploits the corresponding Green’s theorem on the unit sphere with the cut and leads to an integral equation of the second kind for the spectral function on the cut. The Sommerfeld integral representation for the wave field is derived from that of Watson–Bessel and the properties of the Sommerfeld transformant (an analytic function in the integrand) are discussed. In particular, domain of regularity of the transformant is described, which enables one to point out domains on the complex plane, where singularities of the Sommerfeld transformant are located. The singularities of the integrand give rise to different far-field components provided the Sommerfeld integral is evaluated by means of the saddle point technique. Indeed, some of these singularities may be captured in the process of deformation of the Sommerfeld double loop contour into the steepest descent paths (Lyalinov, 2013). The contribution of the saddle points is responsible for the spherical wave from the vertex. Remark that, provided the observation point is in the space region, where only the spherical wave from the vertex is observed in the scattered far field (in the so-called ‘oasis’), these singularities are not intersected. For a narrow sector (2a ≪ 1) an approximate asymptotic expression for the diffraction coefficient of the spherical wave from the vertex in the oasis is deduced by means of the above-mentioned results. 2. Formulation of the problem Let us use the spherical coordinates (r, ϑ, φ) attributed to the Cartesian ones by the correlations   \begin{align*} X_{1}=r\cos\varphi\sin\vartheta,\,\, X_{2}=r\sin\varphi\sin\vartheta,\,\, X_{3}=r\cos\vartheta. \end{align*} We consider a plane wave2 which is incident from the direction specified by ω0 = (ϑ0, φ0) (Fig. 1)   \begin{align} U_{i}(r,\vartheta,\varphi)=\exp\left\{-\mathrm{i}kr\cos\theta_{i}(\omega,\omega_{0})\right\}, \end{align} (2.1) where ω = (ϑ, φ) corresponds to the direction of observation and   \begin{align*} \cos\theta_{i}(\omega,\omega_{0})=\cos\vartheta\cos\vartheta_{0}+ \sin\vartheta\sin\vartheta_{0} \cos[\varphi-\varphi_{0}],\end{align*} θi(ω, ω0) coincides with the geodesic distance between two points ω and ω0 denoted also θ(ω, ω0), θ(ω, ω0) = θi(ω, ω0). The wave field U(r, ϑ, φ) + Ui(r, ϑ, φ) is the sum of the scattered and incident fields, U fulfils the Helmholtz equation   \begin{align} (\triangle +k^{2})U(r,\vartheta,\varphi)=0, \end{align} (2.2) k > 0 is the wave number. Let $$\varSigma =S^{2}\setminus \overline {AB}$$ be the exterior of the cut on S2 and σ = ∂Σ, σ = S ∩ S2 its boundary, σ = σ+ ∪ σ− and σ± are two sides of the cut AB. The impedance boundary condition   \begin{align} \left.\frac{1}{r}\frac{\partial (U_{i}+U)}{\partial\mathscr{N}_{\pm}}\right|_{S_{\pm}}\,- \, \mathrm{i}k\eta_{\pm}\,(U_{i}+U)|_{S_{\pm}}=0, \end{align} (2.3) is satisfied on the sector S, S± are two faces of the sector $$S=\{(r,\omega ): \, r\geqslant 0, \omega \in \overline {AB}\}$$ corresponding to σ± on S2. The vectors $$\mathscr {N}_{\pm }$$ are in the tangent plane to S2 at the points of σ±, are orthogonal to σ± and point out to the ‘exterior’ of Σ. The surface impedances η± = ε± + iχ± do not depend on k, ε± > 0, which means absorption of the wave energy on the sector’s faces.3 The case of reactive faces (ε± = 0) can be considered as a limiting one. The Meixner’s edge conditions are assumed near the edges ∂Si, i = 1, 2 (and outside some close vicinity of the vertex)   \begin{align} U\sim C_{1,2}^{0}(z)+ C_{1,2}(z,\phi)\rho^{1/2}+\dots,\,\,\rho\to 0 \end{align} (2.4) uniformly bounded with respect to z, ϕ, assuming that ρ, ϕ, z are natural local cylindrical coordinates attributed to the edges ∂Si, i = 1, 2, where the index i is omitted for the coordinates. We connect notations A and B with the edges ∂S1 and ∂S2 correspondingly. The conditions at the vertex of the sector take the form   \begin{align} |U|\,\leqslant\, C \,r^{-1/2+\epsilon},\,\,|\nabla U|\,\leqslant \,C\, r^{-3/2+\epsilon},\,\,r\to 0 \end{align} (2.5) which are valid uniformly with respect to the angular variables for some positive ɛ, see Section 5.7 in Van Bladel (1991). We are looking for a classical solution of the problem, i.e. $$U\in C^{2}_{\text {loc}}(R^{3}\setminus \overline {S})$$ and such that $$\frac {1}{r}\frac {\partial U}{\mathscr {N}_{\pm }}$$ and U exist and are continuous on S±. Now we turn to the behaviour of the solution at infinity. 2.1. Uniqueness Let us now assume that the scattered field satisfies the radiation condition (which is not the case for the incidence of a plane wave)   \begin{align} \int_{S_{R}\setminus S}\,|\partial_{r} U-\mathrm{i}k U|^{2}\,\mathrm{d}s\,\to\,0\,, \,\,\, \,R\,\to\, \infty\,,\,\,\varepsilon_{\pm}>0\,, \end{align} (2.6) where $$S_{R}=\{(r,\omega ):\,r=R\,,\,\, \omega \in S^{2}\setminus \overline {\sigma }\}$$, ω = (ϑ, φ). Theorem 2.1 The classical solution of the homogeneous problem (2.2)--(2.6) (i.e. with Ui = 0) is trivial, U ≡ 0. Consider a ball Bδ of small radius δ centred at the vertex O with the boundary Sδ, ∂Bδ := Sδ, Fig. 2. Also we introduce semi-infinite cylinders $$C^{i}_{\delta }$$ of the radius δ having the axis ∂Si, i = 1, 2. We denote dδ the set $$B_{\delta }\cup C^{1}_{\delta }\cup C^{2}_{\delta }$$ with the external boundary ∂dδ, see Fig 2. The domain dδ is a δ −neighbourhood of the edges ∂S1 ∪ ∂S2. Let BR be a ball of the large radius R, ∂BR := SR, then Dδ, R = BR ∖ (S ∪ dδ) is a part of this ball being exterior to S and to dδ. Remark that the boundary of the domain Dδ, R is ∂Dδ, R consisting of ∂dδ, R := ∂dδ ∩ BR, $$S^{\pm }_{\delta ,R}:=B_{R}\cap ( S_{\pm }\setminus d_{\delta })$$ and Sδ, R := SR ∖ (dδ ∪ S). Fig. 2. View largeDownload slide Domains in the proof of uniqueness. Fig. 2. View largeDownload slide Domains in the proof of uniqueness. We apply the Green’s identity   \begin{align*} \int_{D_{\delta,R}}\triangle U\,\overline{U}\mathrm{d}x \,=\,\int_{\partial D_{\delta,R}} \partial_{n} U\,\overline{U}\,\mathrm{d}s \,-\,\int_{D_{\delta,R}}\,| \nabla U |^{2}\,\mathrm{d}x \end{align*} to U and $$\overline {U}$$ in Dδ, R, exploit the Helmholtz equation, boundary conditions and take the imaginary part. Simple calculations lead to   \begin{align*} k\sum_{\pm}\,\Re{(\eta_{\pm})}\,\int_{S^{\pm}_{\delta,R}}\,| U |^{2}\,\mathrm{d}s\, +\,\Im\left( \int_{\partial d_{\delta,R}} \partial_{n} U\,\overline{U}\,\mathrm{d}s\right)\,+\, \,\Im\left( \int_{ S_{\delta,R}} \partial_{r} U\,\overline{U}\,\mathrm{d}s\right)\,\,=\,0, \end{align*} where ∂n is the differentiation along the normal directed outward w.r.t. Dδ, R. Let δ → 0, we apply the Meixner’s conditions at the edges and vertex. The integral over ∂dδ, R vanishes. We then verify the chain of inequalities   \begin{align}\displaystyle 0\,&\leqslant\,k\sum_{\pm}\,\varepsilon_{\pm}\,\int_{S^{\pm}_{0,R}}\,| U |^{2}\,\mathrm{d}s\,=\,-\, \Im\left( \int_{ S_{R}\setminus S} \partial_{r} U\,\overline{U}\,\mathrm{d}s\right)\,\,\\ &=\, \,-\,\Im\left( \int_{ S_{R}\setminus S}\left(\partial_{r} U\,-\,\mathrm{i}k U\right)\overline{U}\,\mathrm{d}s\right)\,-\, k\int_{ S_{R}\setminus S} |U|^{2}\,\mathrm{d}s\,\nonumber\\&\leqslant\, \left( \int_{ S_{R}\setminus S}\left|\partial_{r} U\,-\,\mathrm{i}k U\right|{}^{2}\,\mathrm{d}s\right)^{1/2}\,\left( \int_{ S_{R}\setminus S} |U|^{2}\,\mathrm{d}s\right)^{1/2}-\, k\int_{ S_{R}\setminus S} |U|^{2}\,\mathrm{d}s\nonumber. \end{align} (2.7) The latter expression is non-negative and from the radiation condition we have   \begin{align*} \displaystyle k\left(\int_{ S_{R}\setminus S} |U|^{2}\,\mathrm{d}s\right)^{1/2}\,\leqslant\,\left( \int_{ S_{R}\setminus S}\left|\partial_{r} U\,-\,\mathrm{i}k U\right|{}^{2}\,\mathrm{d}s\right)^{1/2}\,\,\rightarrow\,\,0 \end{align*} as $$R\to \infty $$. We find that   \begin{align*} k\lim_{R\to\infty}\int_{ S_{R}\setminus S} |U|^{2}\,\mathrm{d}s\,=\,0\, \end{align*} and from (2.7)   \begin{equation*} \,\int_{S_{\pm}}\,| U |^{2}\,\mathrm{d}s\,=0, \end{equation*}  \begin{equation*} {\hskip28pt}U|_{S_{\pm}}\,=\,0. \end{equation*} Making use of the boundary conditions, we find   \begin{align*} \left.\frac{1}{r}\frac{\partial U}{\partial\mathscr{N}_{\pm}}\right|_{S_{\pm}}\,=0. \end{align*} As a result, we deal with the homogeneous Cauchy problem for an elliptic (Helmholtz) equation satisfied by U with the trivial initial conditions on the surface S±. Such a solution is known to be trivial, U ≡ 0. As we have already mentioned the radiation condition (2.6) is not valid for the scattered field excited by a plane incident wave. It must also be modified appropriately as ε+ = 0 or ε− = 0. In such cases surface waves can be excited and propagate along the sector’s surface at infinity without attenuation. 2.2. The far-field asymptotics In order to describe the asymptotic behaviour of the scattered field as $$r\to \infty $$ it is useful to define some sub-domains on the unit sphere S2 with the cut $$\sigma = \overline {AB}$$. Let $$\varOmega _{\delta _{0}}$$ be a close vicinity of the cut on the unit sphere   \begin{align*} \displaystyle \varOmega_{\delta_{0}}=\,\left\{\omega\in S^{2}:\,\,\,\text{dist}(\omega,\sigma)<\delta_{0}\right\}\!, \end{align*} where δ0 is some small positive constant. In what follows we assume that $$\omega _{0}\in S^{2}\setminus \varOmega _{\delta _{0}}$$, i.e. the incident plane wave arrives from the directions being not very close to $$\overline {AB}$$. Consider the geodesic distance θ(ω, ω0) between two points ω and ω0 on the sphere S2 (see, Lyalinov, 2013, Section 2 for details). In the same manner, introduce the ‘broken’ geodesic θr(ω, ω0). This geodesic has simple geometrical meaning: this is a broken geodesic of the minimal length which originates at the source ω0 then reflects on the boundary $$\sigma =\overline {AB}$$ in accordance with geometrical optics laws and arrives at the point ω. The ‘incident’ parts of such broken geodesics fill in the spherical triangle ω0AB, whereas the ‘reflected’ parts fill in the spherical triangle FrBA which is further denoted Ωr =: Ωr(ω0), (see Figs 3–5). Fig. 3. View largeDownload slide The triangular domain Ωr with the vertexes ABFr on S2. Fig. 3. View largeDownload slide The triangular domain Ωr with the vertexes ABFr on S2. Fig. 4. View largeDownload slide The triangular domain $$\varOmega _{r}^{*}$$ with the vertexes ABFi on S2. Fig. 4. View largeDownload slide The triangular domain $$\varOmega _{r}^{*}$$ with the vertexes ABFi on S2. Fig. 5. View largeDownload slide The domain ΩA on S2. Fig. 5. View largeDownload slide The domain ΩA on S2. Specify the spherical triangular domain $$\varOmega _{r}^{*}$$ coinciding with the triangle ABFi in Fig. 3. The domain $$\varOmega _{r}^{*}$$ is the mirror image of Ωr with respect to the boundary σ for the same fixed position of ω0. We also make use of two other domains ΩA(ω0) (see Fig. 5) and ΩB(ω0). Consider the ray (geodesic) ω0A which arrives at the edge point A and produces a set of ‘diffracted’ rays (geodesics) outgoing from A in all direction. For each point ω there exists such a diffracted ray with length ψA(ω), (ψA(ω) < π) that arrives at this point. We define ΩA(ω0) as a domain on the sphere such that $$\varOmega _{A}(\omega _{0})=\{\omega \in S^{2}:\,\,0\leqslant \theta _{A}(\omega ,\omega _{0}):=\theta _{i}(A,\omega _{0})+\psi _{A}(\omega )<\pi \}$$, see also Section 2 in Lyalinov (2013). The domains ΩA and ΩB on S2 intersect with Ωr and $$\varOmega _{r}^{*}$$ and with each other. The domain $$\varOmega _{r}^{*}$$ corresponds to directions in which geometrical shadow of the incident wave is observed. Ωr forms a set of directions of propagation of the space rays reflected from the sector S. The domain ΩA (or ΩB) corresponds to the directions for which the diffracted cylindrical wave from the edge A (or from B) is present in the far-field asymptotics. These simple facts follow from the analysis also presented in Lyalinov (2013). Remark It is worth noting that we can also introduce domain ΩAB (and analogously ΩBA), see Lyalinov (2013) as well as the domains $$\varOmega _{ABA},\varOmega _{BAB},\varOmega _{ABAB}\dots , $$ etc. which are defined and interpreted analogously. They correspond to multiple diffractions. We introduce the domain Ω0 = S2 ∖ (ΩA ∪ ΩB ∪ Ωr) on the sphere S2, which is called ‘oasis’. The scattered far-field U (total minus incident) in this domain of directions consists of the spherical wave propagating from the vertex of the sector   \begin{align} U(r,\vartheta,\varphi)=D(\omega,\omega_{0})\frac{\exp(\mathrm{i}kr)}{-\mathrm{i}kr}\left(1+ O\left(\frac{1}{kr}\right)\right),\,\,\, kr\to \infty. \end{align} (2.8) In the asymptotics (2.8), which is non-uniform with respect to $$\omega \in \overline {\varOmega }_{0}$$, the diffraction coefficient D(ω, ω0) is one of the most important characteristics of the scattered field.4 In the exterior of the oasis the structure of asymptotics is more complex and contains also other wave components in the far field. Consider the directions from $$(S^{2}\setminus (\varOmega _{r}^{*}\cup \varOmega _{r}\cup \varOmega _{B}))\cap \varOmega _{A}$$ in which the spherical wave, the cylindrical wave from the edge A as well as surface waves (possibly also other multiply diffracted) are observed in the far field   \begin{align} U(r,\vartheta,\varphi)&=D(\omega,\omega_{0})\frac{\exp(\mathrm{i}kr)}{-\mathrm{i}kr}\left(1+ O\left(\frac{1}{kr}\right)\right)\nonumber \\[10pt] &\quad+ d_{A}(\omega,\omega_{0})\,\frac{\exp\left(-\mathrm{i}kr\cos\theta_{A}(\omega,\omega_{0})\right)} {\sqrt{-\mathrm{i}kr\sin\psi_{A}}}\left(1+ O\left(\frac{1}{kr\sin\psi_{A}}\right)\right)+V_{s}(r,\vartheta,\varphi) +\dots, \end{align} (2.9) where dots denote the waves multiply diffracted from the edges, provided the corresponding directions belong also to $$\varOmega _{BA}, \varOmega _{ABA},\dots $$ .5 The yet unknown function dA(ω, ω0) in (2.9) is connected with the diffraction coefficient of the cylindrical wave from the edge A. The summand Vs(r, ϑ, φ) is the sum of the surface waves generated by the interaction of the incident waves with the vertex or edges. It is only essential provided the observation point is in the close vicinity of the sector, $$\omega \in \varOmega _{\delta _{0}}$$ and as ε± = 0, χ± < 0,   \begin{align}{}\displaystyle V_{s}(r,\vartheta,\varphi)&= \nonumber D_{s}(\omega,\omega_{0})\,\frac{\exp(-\mathrm{i}kr\cos\theta_{s}(\omega,\omega_{0}))}{\sqrt{-\mathrm{i}kr}} \left(1+ O\left(\frac{1}{kr}\right)\right)\\ &\quad+ {D_{s}^{A}}(\omega,\omega_{0})\,\exp\left(-\mathrm{i}kr{\cos\theta^{A}_{s}}(\omega,\omega_{0})\right) + {D_{s}^{B}}(\omega,\omega_{0})\,\exp\left(-\mathrm{i}kr{\cos\theta^{B}_{s}}(\omega,\omega_{0})\right) +\dots\, \end{align} (2.10) with complex valued eikonals $$\theta _{s}(\omega ,\omega _{0}),\theta _{s}^{A,B}(\omega ,\omega _{0})$$. These eikonals solve the equations   \begin{align} (\nabla_{\omega} \theta_{s})^{2}\,=\,1,\,\,\,\,\,\left(\nabla_{\omega} \theta_{s}^{A,B}\right)^{2}\,=\,1, \end{align} (2.11) satisfying the conditions   \begin{align*} \Im(\theta_{s})|_{\sigma}&=0,\,\,\,\Im(\theta_{s})>0,\,\,\,0\leqslant\Re(\theta_{s}) <\pi\!,\,\,\\ \Im(\theta_{s}^{A,B})|_{\sigma}&=0,\,\,\,\Im(\theta_{s}^{A,B})>0\,\,\,0\leqslant\Re(\theta_{s}^{A,B}) <\pi\,\, \end{align*} as ε± = 0, χ± < 0. In the domain $$(S^{2}\setminus (\varOmega _{r}^{*}\cup \varOmega _{r}\cup \varOmega _{A}))\cap \varOmega _{B}$$ the asymptotics has the same form as in (2.9) with the change of the subscript A on to B in the second summand which describes the cylindrical wave from the edge B. In the directions ω from Ωr ∩ (ΩB ∪ΩA) the leading terms consist of the reflected, spherical and diffracted (from the edges A and B) waves (as well as surfaces ones)   \begin{align} U(r,\vartheta,\varphi)=&\ R(\omega,\omega_{0}) \exp\left(-\mathrm{i}kr\cos\theta_{r}(\omega,\omega_{0})\right)+ D(\omega,\omega_{0})\frac{\exp(\mathrm{i}kr)}{-\mathrm{i}kr}\left(1+ O\left(\frac{1}{kr}\right)\right)\nonumber\\&+\ d_{A}(\omega,\omega_{0})\,\frac{\exp\left(-\mathrm{i}kr\cos\theta_{A}(\omega,\omega_{0})\right)}{\sqrt{-\mathrm{i}kr\sin\psi_{A}}} \left(1+ O\left(\frac{1}{kr\sin\psi_{A}}\right)\right)\\\nonumber&+\ d_{B}(\omega,\omega_{0})\,\frac{\exp\left(-\mathrm{i}kr\cos\theta_{B}(\omega,\omega_{0})\right)}{\sqrt{-\mathrm{i}kr\sin\psi_{B}}} \left(1+ O\left(\frac{1}{kr\sin\psi_{B}}\right)\right)+V_{s}(r,\vartheta,\varphi) \,+ \dots, \end{align} (2.12) where R(ω, ω0) is the reflection coefficient in the first summand $$U_{r}(\omega ,\omega _{0})=R \exp (-\mathrm {i}kr\cos \theta _{r}(\omega ,\omega _{0}))$$ of (2.12) which is the reflected wave. The summand Vs(r, ϑ, φ) has the same meaning as in the formula (2.9). The wave field U + Ui in the shadow of the incident wave, i.e. as $$\omega \in \varOmega _{r}^{*}\cap (\varOmega _{B}\cup \varOmega _{A})$$, reads   \begin{align} U(r,\vartheta,\varphi)+U_{i}(r,\vartheta,\varphi)=&\ D(\omega,\omega_{0})\frac{\exp(\mathrm{i}kr)}{-\mathrm{i}kr}\left(1+ O\left(\frac{1}{kr}\right)\right)\nonumber\\&+\ d_{A}(\omega,\omega_{0})\,\frac{\exp\left(-\mathrm{i}kr\cos\theta_{A}(\omega,\omega_{0})\right)} {\sqrt{-\mathrm{i}kr\sin\psi_{A}}}\left(1+ O\left(\frac{1}{kr\sin\psi_{A}}\right)\right)\nonumber\\ &+\ d_{B}(\omega,\omega_{0})\,\frac{\exp\left(-\mathrm{i}kr\cos\theta_{B}(\omega,\omega_{0})\right)} {\sqrt{-\mathrm{i}kr\sin\psi_{B}}}\left(1+ O\left(\frac{1}{kr\sin\psi_{B}}\right)\right)\nonumber\\ &+\ V_{s}(r,\vartheta,\varphi) + \dots, \end{align} (2.13) $$kr\to \infty $$. We do not describe the asymptotics of the far field in the transition domains, where the expressions depend on special transition functions, see e.g. Lyalinov (2013), Sections 6 and 7. It is worth mentioning the work of Shanin (2011), where similar asymptotics for diffraction by an ideal circular cone were established. 3. Watson–Bessel integral representation of the solution and reduction to the problem for the spectral function We turn to the separation of the radial variable of the solution which exploits the Watson–Bessel integral representation. We begin with that for the incident wave. 3.1. Watson–Bessel integral representation for the incident wave We make use of the known Watson–Bessel integral representation for the incident wave (2.1)   \begin{align} U_{i}(r,\vartheta,\varphi)= 4\mathrm{i}\sqrt{\frac{\pi}{2}}\int\limits_{C_{\phi}}\nu \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}u_{\nu}^{i}(\omega,\omega_{0}) \frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\,d \nu, \end{align} (3.1) where $$u_{\nu }^{i}(\omega ,\omega _{0})=-\frac {P_{\nu -1/2}(-\cos \theta _{i}(\omega ,\omega _{0}))}{4\cos \pi \nu },$$  \begin{align} \left(\triangle_{\omega}+(\nu^{2}-1/4)\right)u_{\nu}^{i}(\omega,\omega_{0})=\delta(\omega-\omega_{0}), \end{align} (3.2) Pν−1/2(x) is the Legendre function and Cϕ with ϕ ∈ [0, π/2) is shown in Fig. 6. The contour Cϕ is traditionally taken for ϕ = 0; however, for some reductions it is possible also to use Cϕ with ϕ ∈ (0, π/2). The ‘spectral’ function $$u_{\nu }^{i}(\omega ,\omega _{0})$$ corresponding to the incident plane wave admits the estimate  \begin{align}\displaystyle \left|u_{\nu}^{i}(\omega,\omega_{0})\right|\,<\,C\,\frac{1}{\sqrt{|\nu|}}\exp\left\{-|\nu|\,|\sin\phi|\,\theta(\omega,\omega_{0})\right\}\!, \end{align} (3.3) Fig. 6. View largeDownload slide The contours C0 and Cϕ. Fig. 6. View largeDownload slide The contours C0 and Cϕ. as $$|\nu |\to \infty $$ and ν ∈ Cϕ. Remark that $$J_{\nu }(kr)\sim \frac {[kr/2]^{\nu }}{\varGamma (\nu +1)},\,\,\,\Re \,\nu \to +\infty $$, $$|\arg \nu |\,<\,\pi /2$$. The estimate of the integrand in (3.1) on Cϕ as $$|\nu |\to \infty $$, ν = |ν|eiϕ, $$0\leqslant \phi < \pi /2$$ is given by6  \begin{align} \left|\nu\,\mathrm{e}^{-\mathrm{i}\nu\pi/2}u_{\nu}^{i}(\omega,\omega_{0}) {J_{\nu}(kr)}\right|\,<\,& C\, \exp\left\{-|\nu|(\log|\nu|-1)\cos\phi- |\nu|(\sin\phi[{\arg(kr/2)-\pi/2-\phi}]\right. \nonumber\\[-5pt] &\qquad\quad\left.+\,|\sin\phi|\theta(\omega,\omega_{0})-\cos\phi\log|kr/2|)\right\}. \end{align} (3.4) It is assumed in (3.4) that k may be complex with |argk| < π/2 although we consider argk = 0 in this work. Actually the constants in the estimates (3.4) and (3.3) are different; however, for convenience we take the maximal one and denote it by C. It is also useful to have these estimates not only for real k > 0 but for $$|\arg k|<\pi /2$$ as well. The integral in (3.1) then rapidly converges. It is worth noting that the function $$u_{\nu }^{i}(\omega ,\omega _{0})$$ is even with respect to ν and is holomorphic in the strip Π1/2,   \begin{align*} \varPi_{\delta}=\left\{\nu\in\mathrm{C}:\,\,\,|\Im(\nu)|<\delta\right\}\!.\end{align*} Its simple poles are located at zeros of $$\cos \pi \nu $$. 3.2. Watson–Bessel integral representation for the scattered field and separation of the radial variable In order to separate the radial variable for the problem at hand we look for the solution in the integral form   \begin{align} U(r,\vartheta,\varphi)= 4\mathrm{i}\sqrt{\frac{\pi}{2}}\int\limits_{C_{0}}\nu \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}\,u_{\nu}(\omega,\omega_{0}) \frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\,d \nu \end{align} (3.5) with unknown ‘spectral’ function uν(ω, ω0). We begin with some formal reductions enabling us to motivate our approach and to make reasonable assumptions about the basic properties of the spectral function to be constructed. Let uν(ω, ω0) satisfy the equation   \begin{align} \left(\triangle_{\omega}+(\nu^{2}-1/4)\right)u_{\nu}(\omega,\omega_{0})=\,0\, \end{align} (3.6) in the classical sense, i.e. uν(⋅, ω0) ∈ C2(Σ), Σ = S2 ∖ σ then, as is well known (see e.g. Smyshlyaev, 1991 and others), the scattered field U(r, ϑ, φ) in (3.5) solves the Helmholtz equation (2.2). Now we turn to the boundary condition (2.3)   \begin{align*} &\left.\frac{1}{r}\frac{\partial (U_{i}+U)}{\partial\mathscr{N}_{\pm}}\right|_{\omega\in\sigma_{\pm}}\,- \, \mathrm{i}k\eta_{\pm}\,(U_{i}+U)|_{\omega\in\sigma_{\pm}}\\ &\quad= 4\mathrm{i}\sqrt{\frac{\pi}{2}}\int\limits_{{C_{0}^{b}}}\nu \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}\left(\left. \frac{\partial \widehat{u}_{\nu}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\, \frac{(-\mathrm{i})J_{\nu}(kr)}{(-\mathrm{i}kr)^{3/2}}\,-\,\mathrm{i}\eta_{\pm}\, \widehat{u}_{\nu}(\omega,\omega_{0})|_{\sigma_{\pm}}\,\frac{J_{\nu}(kr)}{(-\mathrm{i}kr)^{1/2}}\right)\,d \nu\,=\,0, \end{align*} where $${C_{0}^{b}}$$ is shown in Fig. 7, $$\widehat {u}_{\nu }(\omega ,\omega _{0})={u}_{\nu }(\omega ,\omega _{0})+{u}_{\nu }^{i}(\omega ,\omega _{0}) $$. We assumed that uν is holomorphic7 and even w.r.t. ν in some vicinity of the imaginary axis, meromorphic in C and some of its possible sigularities are located inside the contour $${C_{0}^{b}}$$. Fig. 7. View largeDownload slide Integration contour $${C_{0}^{b}}$$ and its deformation. Fig. 7. View largeDownload slide Integration contour $${C_{0}^{b}}$$ and its deformation. We take into account the identity zJν−1(z) + zJν+1(z) = 2νJν(z). After some reductions one has   \begin{align*} 4\sqrt{\frac{\pi}{2}}\int\limits_{{C_{0}^{b}}}\,d \nu\, \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}\left(\left. \frac{\partial \widehat{u}_{\nu}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\, \frac{\mathrm{i}(J_{\nu-1}(kr)+J_{\nu+1}(kr))}{2\sqrt{-\mathrm{i}kr}}\,+\,\eta_{\pm}\,\nu\, \widehat{u}_{\nu}(\omega,\omega_{0})\Bigg|_{\sigma_{\pm}}\,\frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\right)\,=\,0, \end{align*} then appropriately change the variables of integration so that   \begin{align*} &2\mathrm{i}\sqrt{\frac{\pi}{2}}\left(\int\limits_{{C_{0}^{b}}-1}\!d \nu\, \,\mathrm{e}^{-\mathrm{i}(\nu+1)\pi/2}\left. \frac{\partial \widehat{u}_{\nu+1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\, \frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\!+\! \int\limits_{{C_{0}^{b}}+1}\!d \nu\, \,\mathrm{e}^{-\mathrm{i}(\nu-1)\pi/2}\left. \frac{\partial \widehat{u}_{\nu-1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\, \frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\,\right.\\ &\qquad\qquad\left. -\, 2\mathrm{i}\eta_{\pm}\,\int\limits_{{C_{0}^{b}}}\,d \nu\,\nu\,\mathrm{e}^{-\mathrm{i}\nu\pi/2} \widehat{u}_{\nu}(\omega,\omega_{0})\bigg|_{\sigma_{\pm}}\,\frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\,\right)\,=\,0, \end{align*} where the contours of integration $${C_{0}^{b}}+1$$ and $${C_{0}^{b}}-1$$ are those in Fig. 7 obtained by shifting $${C_{0}^{b}}$$ along the real axis to the right- or left-hand side accordingly and $${C_{0}^{b}}$$ consists of the parts $$(\infty -\mathrm {i}b,-\mathrm {i}b],$$ [−ib, ib] and $$[\mathrm {i}b, \infty +\mathrm {i}b)$$. The contours $${C_{0}^{b}}+1$$ and $${C_{0}^{b}}-1$$ are explicitly described by $$\{{C_{0}^{b}}+1\}=(\infty -\mathrm {i}b+1,-\mathrm {i}b+1]\cup [-\mathrm {i}b+1,\mathrm {i}b+1]\cup [\mathrm {i}b+1, \infty +\mathrm {i}b+1)$$ and $$\{{C_{0}^{b}}-1\}=(\infty -\mathrm {i}b-1,-\mathrm {i}b-1]\cup [-\mathrm {i}b-1,\mathrm {i}b-1]\cup [\mathrm {i}b-1, \infty +\mathrm {i}b-1)$$. Let us now assume that uν(ω, ω0) is taken meromorphic w.r.t. ν ∈ C with the poles inside some strip |I(ν)| < b, holomorphic in Πδ for some small δ > 0 and such that $$\left . \frac {\partial \widehat {u}_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}$$ is regular (holomorphic) in the strip Πδ+1. We are now able to deform the contours $${C_{0}^{b}}+1$$ and $${C_{0}^{b}}-1$$ into that $${C_{0}^{b}}$$ in Fig. 7 thus having   \begin{align*} 2\sqrt{\frac{\pi}{2}}\,\int\limits_{{C_{0}^{b}}}\,d \nu\,\mathrm{e}^{-\mathrm{i}\nu\pi/2}\left( \left.\frac{\partial \widehat{u}_{\nu+1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\,-\, \left.\frac{\partial \widehat{u}_{\nu-1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}} +2\eta_{\pm}\nu\, \widehat{u}_{\nu}(\omega,\omega_{0})|_{\sigma_{\pm}}\,\right)\frac{J_{\nu}(kr)}{\sqrt{-\mathrm{i}kr}}\,\,=\,0. \end{align*} We conclude that, provided $$\widehat {u}_{\nu } (\omega ,\omega _{0})=u_{\nu }(\omega ,\omega _{0})+u_{\nu }^{i}(\omega ,\omega _{0})$$ satisfies the boundary condition on σ = σ+ ∪ σ−  \begin{align} \left.\frac{\partial \widehat{u}_{\nu+1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\,-\, \left.\frac{\partial \widehat{u}_{\nu-1}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}} +\,2\nu\,\eta_{\pm}\, \widehat{u}_{\nu}(\omega,\omega_{0})|_{\sigma_{\pm}}\,=\,0, \end{align} (3.7) the wave field U + Ui fulfils the boundary condition (2.3). Remark The condition (3.7) is nonlocal with respect to the spectral variable ν, which means that, contrary to the Helmholtz equation, the radial variable in the mixed boundary condition (2.3) is not separable in a traditional meaning. In order to transform the boundary condition to an alternative form we take into account a simple Lemma which actually follows from the known technique developed for a class of functional equations, Babic et al. (2008), Chapter 7. Lemma 3.1 Let H(ν) be holomorphic as ν ∈ Πδ and $$|H(\nu )|\leqslant C\mathrm {e}^{-\kappa |\nu |}\,,\,|\nu |\to \infty \,\,, \kappa>0$$ in this strip, H(ν) = −H(−ν). Then an even solution s(ν) of the equation   \begin{align*} s(\nu+1)-s(\nu-1)= -2\mathrm{i}H(\nu), \end{align*} which is regular (holomorphic) in the strip ν ∈Π1+δ and exponentially vanishes as $$|\nu |\to \infty $$ there, is given by   \begin{align*} s(\nu)&=\frac{1}{4}\int\limits_{{-\textrm i}\infty}^{\mathrm{i}\infty}\, d\tau\, H(\tau)\,\left( \frac{\sin(\pi\tau/2)}{\cos(\pi\tau/2)-\sin(\pi\nu/2)}\,+\, \frac{\sin(\pi\tau/2)}{\cos(\pi\tau/2)+\sin(\pi\nu/2)}\, \right)\\&= \frac{1}{4}\int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\, d\tau\, H(\tau)\, \frac{\sin\pi\tau}{\cos\pi\tau+\cos\pi\nu}\,,\,\,\,\,\,\,\nu\in \varPi_{1+\delta}. \end{align*} Assuming exponential decreasing of $$\widehat {u}_{\nu }$$ as $$|\nu |\to \infty ,$$ν ∈ Πδ and making use of this Lemma, we arrive at a new form of the boundary condition (3.7)   \begin{align} \left.\frac{\partial \widehat{u}_{\nu}(\omega,\omega_{0})}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\,= \frac{\eta_{\pm}\,}{2\mathrm{i}} \int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\, d\tau\, \frac{\tau\,\sin\pi\tau\,\widehat{u}_{\tau}(\omega,\omega_{0})|_{\sigma_{\pm}}}{\cos\pi\tau+\cos\pi\nu}\, \,,\,\,\,\,\nu\in \varPi_{\delta}. \end{align} (3.8) Remark Provided $$\widehat {u}_{\nu }(\omega ,\omega _{0})|_{\sigma _{\pm }}$$ is holomorphic as ν ∈ Πδ, then from the Lemma and condition (3.8) $$\left .\frac {\partial \widehat {u}_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$ is holomorphic as ν ∈ Π1+δ. In order to formulate a basic statement of this Section we first postulate a set of conditions specifying a desired class for solution of the problem (3.6), (3.8). 3.3. Conditions specifying a class of solutions for spectral function uν(ω, ω0) We make use of some known definitions (Dieudonne, 1960; Chapter 7) dealing with a meromorphic in the complex plane C (or holomorphic in a domain D ⊂ C) mapping fν from C (or from D) into a Banach space B, i.e. with such meromorphic (or holomorphic) function of ν that fν : ν↦fν(⋅), where fν(⋅) ∈ B. Let the spectral function uν(ω, ω0) satisfy the conditions The spectral function is such a mapping that uν : ν↦uν(⋅, ω0) , ($$u_{\nu }(\cdot , \omega _{0})\in \,C^{2}_{loc}(\varSigma ) \,$$) is meromorphic in the complex plane C for all $$\omega _{0}\notin \varOmega _{\delta _{0}}$$, where ω0 is a parameter. For all regular ν it admits the estimate (Meixner’s condition on the unit sphere) $$u_{\nu }(\omega ,\omega _{0})= c_{0}\,+\,c_{1}(\chi _{A})\,\psi _{A}^{1/2}+\dots $$, ψA → 0, where ψA is the geodesic distance from A to ω = (ψA, χA). A similar condition is valid for the point B. The trace of the spectral function on σ is holomorphic in Πδ, i.e. the mapping uν|σ : ν↦uν(⋅|σ, ω0) is holomorphic in ν ∈ Πδuν(⋅|σ, ω0) ∈ C(σ) for all $$\omega _{0}\notin \varOmega _{\delta _{0}}$$. The set Q of poles of the meromorphic function uν is contained in the strip |I(ν)| < b for some b > 0. The spectral function is even with respect to ν, uν(ω, ω0) = u−ν(ω, ω0). There exist the traces $$ \left .\frac {\partial \widehat {u }_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$ on σ which are holomorphic functions of ν ∈ Π1+δ with the values in C(σ), $$\omega _{0}\notin \varOmega _{\delta _{0}}$$. Notice that, in view of the boundary conditions (3.7), (3.8), these traces admit meromorphic continuation on the complex plane. The spectral function satisfies the estimate ($$|\nu |\to \infty $$, ν = |ν|eiϕ, − π/2 < ϕ < π/2)   \begin{align} |u_{\nu}(\omega,\omega_{0})|\,\leqslant\,C\, \frac{1}{\sqrt{|\nu|}}\exp\left\{ -|\nu|\,\left(|\sin\phi|\,\tau_{0}(\omega,\omega_{0})-\cos\phi \,|\tau_{1}(\omega,\omega_{0})|\right.\right\}\, \end{align} (3.9) for some τ0, τ1, where τ0 > 0. This estimate is valid on the contour Cϕ in Fig. 6. Remark that analogous properties may be verified for $$u_{\nu }^{i}(\omega ,\omega _{0})$$. It is worth commenting on the origin of the estimate (3.9). The equation (3.6) has high-frequency (or quasi-classical) structure as $$\nu \to \infty $$. The solution of the equation in this case may be determined as a sum of the ‘ray’ expansions   \begin{align*} u_{\nu}(\omega)\asymp&\, \sum_{j} U^{\,j}(\nu,\omega)\,,\\ U^{\,j}(\nu,\omega)=&\,\mathrm{e}^{\mathrm{i}\nu\tau_{j}(\omega)}\sum_{m=1}^{\infty} v_{m}(\omega)/\nu^{1/2+m}. \end{align*} The boundary condition for uν(ω) also depends on the large parameter and is not self-adjoint because, in particular, it depends on complex η±. As a result, amongst solutions τj(ω) of the eikonal equations (∇ωτj(ω))2 = 1 there are those complex valued having the structure $$\tau _{j}(\omega )={\tau ^{j}_{0}}(\omega )+\mathrm {i}{\tau ^{j}_{1}}(\omega )$$ with the real valued $${\tau ^{j}_{0}}(\omega )>0\,$$ and $$\,{\tau ^{j}_{1}}(\omega )$$. Assuming that the ray solutions admit continuation to ν = |ν|eiϕ, $$|\nu |\to \infty $$, 0 < ϕ < π/2, we may expect   \begin{align*} |u_{\nu}(\omega)|\leqslant\text{Const}\,|\nu|^{-1/2} \left|\mathrm{e}^{\mathrm{i}\nu(\tau_{0}(\omega)+\mathrm{i}\tau_{1}(\omega))}\right|, \end{align*} where $$\tau _{0}(\omega )=\min _{j}({\tau _{0}^{j}}(\omega ))$$ and $$|\tau _{1}(\omega )|=\max _{j}(|{\tau _{1}^{j}}(\omega )|)$$. This accounts for the condition 6. It is worth noting that rapid convergence of the Watson–Bessel integral representation (3.5) follows from the estimate (see also (3.4)) given by   \begin{align} \left|\nu\,\mathrm{e}^{-\mathrm{i}\nu\pi/2}u_{\nu}(\omega,\omega_{0}) {J_{\nu}(kr)}\right|<&\, \,C\, \exp\left\{-|\nu|(\log|\nu|-1)\cos\phi--|\nu|(\sin\phi[{\arg(kr/2)-\pi/2-\phi}]\right.\nonumber\\ &\left.\quad\qquad+\ |\sin\phi|\tau_{0}(\omega,\omega_{0}))+|\nu|\cos\phi[ |\tau_{1}(\omega,\omega_{0})|+\log|kr/2|]\right\} \end{align} (3.10) on Cϕ as $$|\nu |\to \infty $$, ν = |ν|eiϕ, − π/2 < ϕ < π/2. Taking into account the discussion in this section, we arrive at Theorem 3.1 Let uν(ω, ω0) be a solution of the problem (3.6), (3.8) from the class of functions described by the conditions 1–6. Then Watson–Bessel integral representation (3.5) for U(r, θ, φ) is the desired classical solution of the problem. We mentioned that the radiation condition (2.6) is not valid for the solution in the case of the plane wave incidence. On the other hand, we should emphasize that description of the far-field asymptotics for the diffraction problem at hand will not be exhaustively considered in this work, however, we shall discuss the far-field expression in the oasis Ω0. Nevertheless, we shall develop an efficient formalism based on the Sommerfeld integral representation which is well adapted for such a description (Lyalinov, 2013). In the following section we shall consider an approach that enables one to show a way of construction of the spectral function solving the problem (3.6), (3.8). 4. Study of the problem for the spectral function In this section we consider the problem (3.6), (3.8) and restrict it on $$\nu \in [0,\mathrm {i}\infty )$$ exploiting also that uν(ω, ω0) is even. Then it is useful to study the inhomogeneous equation instead of (3.6) and, vice versa, homogeneous boundary condition so that   \begin{align} \left(\triangle_{\omega}+(\nu^{2}-1/4)\right)w_{\nu}(\omega)=&\,F(\nu,\omega)\,,\,\,\,\,\omega\in \varSigma\nonumber\\ \left.\frac{\partial w_{\nu}(\omega)}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}=&\, \frac{\eta_{\pm}\,}{\mathrm{i}} \int\limits_{0}^{\mathrm{i}\infty}\, d\tau\, \frac{\tau\,\sin\pi\tau\,w_{\tau}(\omega)|_{\sigma_{\pm}}}{\cos\pi\tau+\cos\pi\nu}\,. \end{align} (4.1) This problem is not traditional because of the non-local dependence on the parameter of separation ν. It should be noticed that the problems (3.6), (3.8) and (4.1) are actually connected by a simple change of the unknown function uν(ω, ω0) = wν(ω, ω0) + v(ν, ω, ω0), where v is uniquely defined. We shall write wν(ω) instead of wν(ω, ω0) omitting dependence on the parameter ω0. We are looking for solution wν(ω) of (4.1) which is from $$C([0,\mathrm {i}\infty ),C^{2}(\varSigma ))$$, having also continuous value of $$\frac {\partial w_{\nu }(\omega )}{\partial \mathscr {N}_{\pm }}$$ on σ such that the boundary condition in (4.1) is satisfied. Having such a solution, we shall continue it appropriately onto the whole complex plane w.r.t ν and also ensure the conditions 1–6. for the spectral function uν(ω, ω0). It is convenient to make use of the variable x instead of ν and the new unknown function $${\mathscr {U}}(x,\omega )$$ defined by the expressions   \begin{align*} \,x=\,1/\cos\pi\nu\,,\,\,\, \nu=\,\mathrm{i} d(x),\,\,\,d(x):=&\,\frac{1}{\pi}\,\text{arccosh}(1/x)= \frac{1}{\pi}\, \log\left(\frac{1}{x} +\sqrt{\frac{1}{x^{2}}-1}\right)\,,\,\,\,\,x\in [0,1]\,,\\ {\mathscr{U}}(x,\omega)\,=&\,\sqrt{d(x)}\,\cos\left(\mathrm{i}\pi d(x)\right)\,w_{\mathrm{i}d(x)}(\omega). \end{align*} Remark that   \begin{align*} 0\leqslant\,\frac{1}{\pi}\,\log(1/x)\leqslant d(x)\,\leqslant\,d_{0}(x)\end{align*} with $$d_{0}(x)=\frac {1}{\pi }\,\log (2/x)$$. As a result, we write the problem (4.1) in the form   \begin{align} \left(-\triangle_{\omega}+d^{2}(x)+1/4\right){\mathscr{U}}(x,\omega)\,=&\,f(x,\omega)\,,\,\,\,\,\omega\in \varSigma\nonumber\\[15pt] \left.\frac{\partial{\mathscr{U}}(x,\omega)\,}{\partial\mathscr{N}_{\pm}}\right|_{\sigma_{\pm}}\,+ \eta_{\pm}\,\mathscr {A} \mathscr {U}(x,\omega|_{\sigma_{\pm}})\,=&\,0, \end{align} (4.2) where $$f(x,\omega )\,=\, -\sqrt {d(x)}\cosh (\pi d(x) F(\mathrm {i}d(x),\omega )$$ and the operator-impedance $${\mathscr {A}}$$ is given by   \begin{align}\displaystyle \mathscr {A} \mathscr {U}(x,\omega|_{\sigma_{\pm}})\,=\,\frac{1}{\pi}\,{\int\limits_{0}^{1}}\,\frac{\sqrt{d(x)d(y)}}{x+y}\, {\mathscr{U}}(y,\omega|_{\sigma_{\pm}})\,\mathrm{d}y. \end{align} (4.3) The operator $${\mathscr {A}}$$ in (4.3) is formally symmetric. It is worth commenting on the reduction of the problem (4.1) to that (4.2). We are looking for the solution of the problem (4.1) which meromorphically depends on the variable ν. However, the study of the meromorphic operator-function   \begin{align*} {\mathscr{M}}_{\nu} \,=\,\left\{\triangle_{\omega}+(\nu^{2}-1/4)\,,\,\,\, \left.\left(\frac{\partial\,\,\,\,}{\partial\,\mathscr{N}_{\pm}}\,-\, \eta_{\pm}\,{\mathscr{A}}_{0}\,\right)\right|_{\sigma_{\pm}}\,\right\},\end{align*} and of the equation   \begin{align*} {\mathscr{M}}_{\nu}\, w_{\nu}(\omega) \,= \,{\mathscr{F}}_{\nu} (\omega) \end{align*} is not a simple task. In the latter problem the operator $${\mathscr {A}}_{0}$$ is defined by the right-hand side of the boundary condition in (4.1) and $${\mathscr {F}}_{\nu } (\omega )=\{F_{\nu }(\omega ),0\}$$. Instead, we consider the reduced problem (4.2). The solution of the latter problem $${\mathscr {U}}(x,\omega )$$ is defined on the segment x ∈ (0, 1) and is then analytically continued as a meromorphic function wν(ω) onto the complex plane ν ∈ C taking into account the change of the variable x↦ν and of $${\mathscr {U}}(x,\omega )\mapsto w_{\nu }(\omega )$$. Our further goal is to study unique solvability of the problem (4.2) in an appropriate functional space. To this end, we introduce a Hilbert space and attribute an operator, acting in this space, with the problem (4.2). It is performed by use of the traditional technique based on a sesquilinear form connected with the problem (4.2). The corresponding sectorial form is taken densely defined and proved to be closable, see Kato (1972), Chapter 6. The latter circumstance enables one to define an m-sectorial operator A attributed to (4.2), see also Assier et al. (2016). It turns out that this operator is boundedly invertible, A−1 = (A−Λ)−1|Λ=0 is correctly defined because Λ = 0 belongs to the resolvent set of the operator A, where Λ is the spectral parameter. 4.1. Definition of the sectorial form attributed to the problem (4.2) Consider the Hilbert space H = L2((0, 1);L2(Σ)) of such functions u of x ∈ (0, 1) with the values in L2(Σ) denoting them u = u(x, ω). The Hilbert norm $$\|v\|_{H}=\sqrt {<v,v>}$$ is specified by the scalar product   \begin{align*} \displaystyle <u,v>\,=\,{\int\limits_{0}^{1}}\,d\,x\,\left(\int\limits_{\varSigma}\, u(x,\omega)\overline{v(x,\omega)} \,\mathrm{d}\,\omega\right)\,. \end{align*} In order to give a motivated expression for the sesquilinear form tA attributed to the problem (4.2) we apply the Green’s identity to the differential operator and make use of the boundary condition for u  \begin{align*}<(-\triangle_{\omega}+d^{2}+1/4) u,v>\,=&\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma}\,(-\triangle_{\omega}+d^{2}(x)+1/4) u(x,\omega)\overline{v(x,\omega)}\mathrm{d}\,\omega\right)\\[6pt] =&\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\!\left(\int\limits_{\varSigma}\,(\nabla_{\omega} u(x,\omega)\cdot \nabla_{\omega} \overline{v(x,\omega)}+(d^{2}(x)+1/4) u(x,\omega)\overline{v(x,\omega)})\mathrm{d}\,\omega \right)\!-\,\\[6pt] &-{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{+}}\mathrm{d}\sigma\, \frac{\partial u(x,\sigma)\,}{\partial\mathscr{N}_{+}}\overline{v(x,\sigma)} \,-\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{-}}\mathrm{d}\sigma\, \frac{\partial u(x,\sigma)\,}{\partial\mathscr{N}_{-}}\overline{v(x,\sigma)}\\[6pt] =&\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma}\,\mathrm{d}\omega( \nabla_{\omega} u(x,\omega)\cdot \nabla_{\omega} \overline{v(x,\omega)}+ (d^{2}(x)+1/4) u(x,\omega)\overline{v(x,\omega)}) \right)\\[6pt] &+\, \frac{\eta_{+}}{\pi}{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{+}}\mathrm{d}\sigma\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{\sqrt{d(x)d(y)}}{x+y}\, u(y,\sigma) \overline{v(x,\sigma)}\\[6pt] &+\, \frac{\eta_{-}}{\pi}{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{-}}\mathrm{d}\sigma\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{\sqrt{d(x)d(y)}}{x+y}\, u(y,\sigma) \overline{v(x,\sigma)}\,, \end{align*} where u and v are taken such that the reductions above are justified. Let us now define a sesquilinear form   \begin{align} t_{A}[u,v]\,=\,{t^{1}_{A}}[u,v]+{t_{A}^{2}}[u,v]+{t_{A}^{3}}[u,v]\,, \end{align} (4.4) where   \begin{align*} {t^{1}_{A}}[u,v]\,&=\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int_{\varSigma}\, \left(\nabla_{\omega} u(x,\omega)\cdot \nabla_{\omega} \overline{v(x,\omega)} \,+\, \frac{1}{4}u(x,\omega)\overline{v(x,\omega)}\right)\mathrm{d}\omega \right)\!,\\ {t_{A}^{2}}[u,v]&=\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma} d^{2}(x) u(x,\omega)\overline{v(x,\omega)}\mathrm{d}\,\omega \right)\!,\\ {t_{A}^{3}}[u,v]\,&=\,\mathop{\sum}_{\pm}\,\frac{\eta_{\pm}}{\pi}{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{\sqrt{d(x)d(y)}}{x+y}\, u(y,\sigma) \overline{v(x,\sigma)} , \end{align*} with the domains   \begin{align*} D\left({t_{A}^{1}}\right)&=\left\{v:\,\,v\in L_{2}\left((0,1);H^{1}(\varSigma)\right)\subset H\right\}\!,\\ D\left({t_{A}^{2}}\right)&=\big\{v:\,\,v\in H=L_{2}((0,1);L_{2}(\varSigma))\,\,\text{such that}\,\,\,d\,v\in L_{2}((0,1);L_{2}(\varSigma))\, \big\}\!,\\ D\left({t_{A}^{3}}\right)&=\left\{w:\,w\in H,\,\,\,w|_{\sigma}\in L_{2}((0,1);L_{2}(\sigma))\,\, \text{such that}\,\,\,\sqrt{d}\,w|_{\sigma}\in L_{2}((0,1);L_{2}(\sigma))\, \right\}\, \end{align*} correspondingly. Then the domain of tA is   \begin{align}\displaystyle D(t_{A})\,=\, \bigcap_{i=1}^{3} D\left({t_{A}^{i}}\right) \end{align} (4.5) and tA is densely defined. It is worth mentioning that H1(Σ) is the usual Sobolev space and is boundedly embedded into L2(σ). The norm in L2((0, 1);H1(Σ)) is given by   \begin{align*} \|v\|^{2}_{1}\,=\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma}\,(|\nabla_{\omega} u(x,\omega)|^{2}+ |u(x,\omega)|^{2}\mathrm{d}\,\omega\right)\, \end{align*} and this space is boundedly embedded into L2((0, 1);L2(σ)). The form $${t_{A}^{3}}$$ may be also written as   \begin{align*} {t_{A}^{3}}[u,v]\,=\,\mathop{\sum}_ {\pm}\,\eta_{\pm}{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, \, ({\mathscr{A}} u)(x,\sigma) \overline{v(x,\sigma)} \,=\,\mathop{\sum}_{\pm}\,\eta_{\pm}({\mathscr{A}} u,v)_{\pm}, \end{align*} where the operator-impedance $${\mathscr {A}}\geqslant 0$$ is an integral operator, has the kernel $$\frac {\sqrt {d(x)d(y)}}{x+y}\,$$ and is formally symmetric. We aim to show that the form tA is sectorial (Kato, 1972, Chapter 6) with the vertex Γ which is a real number and with the half-angle Θ. We verify that the range R(tA) of the form, which is the set of values of the quadratic form tA[u, u] provided u ∈ D(tA) and ∥u∥H = 1, is located in the sector $$|\text {arg}(\varLambda -\varGamma )|\leqslant \,\Theta ,$$$$0\leqslant \Theta <\pi /2$$ on the complex plane of the variable Λ. To that end, we represent the form as   \begin{align*} t_{A}\,=\,\Re(t_{A})\,+\,\mathrm{i}\Im(t_{A}), \end{align*} where (see Kato, 1972, Chapter 6) real part $$\Re (t_{A})$$ of the form is defined by   \begin{align*} \Re(t_{A})[u,v]=\frac{t_{A}[u,v]+ \overline{t_{A}[v,u]}}{2}\end{align*} and similarly the imaginary part   \begin{align*} \Im(t_{A})[u,v]=\frac{t_{A}[u,v]- \overline{t_{A}[v,u]}}{2\mathrm{i}}\end{align*} for any u, v ∈ D(tA). Simple calculations lead to the expressions   \begin{align*} \Re(t_{A})[u,v]&= {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma}\, \nabla_{\omega} u(x,\omega)\cdot \nabla_{\omega} \overline{v(x,\omega)} \,+\, \left(d^{2}(x)+\frac{1}{4}\right)\,u(x,\omega)\overline{v(x,\omega)} \mathrm{d}\omega \right)\,\\&\quad+ \,\mathop{\sum}_{\pm}\,\varepsilon_{\pm}({\mathscr{A}} u,v)_{\pm}, \Im(t_{A})[u,v]=\,\mathop{\sum}_{\pm}\,\chi_{\pm}({\mathscr{A}} u,v)_{\pm}\, \end{align*} recalling that η± = ε± + iχ±,   \begin{align*} ({\mathscr{A}} u,v)_{\pm}\,=\,\frac{1}{\pi}\,{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{\sqrt{d(x)d(y)}}{x+y}\, u(y,\sigma) \overline{v(x,\sigma)} \,. \end{align*} It is obvious that   \begin{align*} \Re(t_{A})[u,u]\,\geqslant\,\frac{1}{4}\|u\|_{H}^{2} \end{align*} so that the vertex $$\varGamma \geqslant \frac {1}{4}$$ and the form $$\Re (t_{A})$$ is positive definite. For the imaginary part we find   \begin{align*} |\Im(t_{A})[u,u]|\,&\leqslant\,\mathop{\sum}_{\pm}\,|\chi_{\pm}|({\mathscr{A}} u,u)_{\pm}\,\leqslant\, \mathop{\sum}_{\pm}\,\frac{|\chi_{\pm}|}{\varepsilon_{\pm}}\, \left(\varepsilon_{\pm}({\mathscr{A}} u,u)_{\pm}\,+\,\|\nabla_{\omega} u\|_{H}^{2}+\,\|d\,u\|_{H}^{2}\right)\,\\&\leqslant\, \left(\frac{|\chi_{+}|}{\varepsilon_{+}}+\frac{|\chi_{-}|}{\varepsilon_{-}}\right)\, \left(\Re(t_{A})[u,u]\,-\,\frac{1}{4}\|u\|_{H}^{2}\right) \end{align*} for any u ∈ D(tA). As a result, we have the estimate for the half-angle Θ of the sectorial form tA  \begin{align*} 0\,\leqslant\,\tan\Theta\,\leqslant\, \left(\frac{|\chi_{+}|}{\varepsilon_{+}}+\frac{|\chi_{-}|}{\varepsilon_{-}}\right)\!. \end{align*} It is crucial to prove that the sesquilinear sectorial form tA is closable, i.e. admits a closed extension, which is equivalent to that for $$\Re (t_{A})$$. 4.2. The form tA is closable In order to prove that tA admits a closure it is sufficient to show that its summands $${t_{A}^{i}},\,\,i=1,2,3$$ are closable (i.e. admit closed extensions, see Theorem 1.31 in Chapter 6 of Kato, 1972). To do this we shall use the following simple statement (see, Birman & Solomyak, 1987, Section 10.1(4)) Lemma 4.1 Let h be a densely defined sesquilinear form and G be an h −dense in D(h) set. Also let the conditions h[un − um, un − um] → 0 and ∥un∥H → 0 (as $$n,m\to \infty $$) be followed by   \begin{align*} h[u_{n},g]\,\to\, 0\,,\,\,\,n\to\infty \end{align*} for any g ∈ G then the form h is closable. Introduce the set G by   \begin{align*} G\,=\,\left\{\,g\in D(t_{A})\,:\, \,\,\,\,{d_{0}^{2}} g\in H=L_{2}((0,1);L_{2}(\varSigma)),\,\,\, {d_{0}^{2}} g\big|_{\sigma}\in L_{2}((0,1);L_{2}(\sigma))\,\right\}\!, \end{align*} $$d_{0}(x)=\frac {1}{\pi }\log (2/x)$$. Remark that the form $${t_{A}^{1}}$$ is closable because it is directly connected with the form   \begin{align*} \tau^{1}[u,v]\,=\, \,\int\limits_{\varSigma}\, \left(\nabla_{\omega} u(\omega)\cdot \nabla_{\omega} \overline{v(\omega)} \,+\, \frac{1}{4}u(\omega)\overline{v(\omega)}\right)\mathrm{d}\omega \, \end{align*} which is known to be closable in H1(Σ), see e.g. Assier et al. (2016). That the form $${t^{1}_{A}}$$ is closable implies the following: from ∥un∥→ 0 and $${t_{A}^{1}}[u_{n}-u_{m},u_{n}-u_{m}]\to 0$$ ($$n,m\to \infty $$) it follows that (see Theorem 1.17 in Chapter 6, Kato, 1972)   \begin{align} {t_{A}^{1}}[u_{n},u_{n}]\to 0\,,\,\,\,\text{as}\,\,\, n\to \infty. \end{align} (4.6) The form $${t_{A}^{2}}$$ is closable because   \begin{align*} \left|{t_{A}^{2}}[u_{n},g]\right|\,&\leqslant\,{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma} \big|d^{2}(x) u(x,\omega)\big|\,|g(x,\omega)|\mathrm{d}\,\omega \right)\,\\[15pt] &\leqslant\, {\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\left(\int\limits_{\varSigma} |u(x,\omega)|\,\left|{d_{0}^{2}}(x) g(x,\omega)\right|\mathrm{d}\,\omega \right)\, \leqslant\,\|u_{n}\|_{H}\,\|{d_{0}^{2}} g\|_{H}\,\to\,0\,, \end{align*} as ∥un∥H , → 0 , $$n\to \infty $$ for any g ∈ G. The corresponding estimate for $${t_{A}^{3}}$$ requires a bit more work   \begin{align*} \left|{t_{A}^{3}}[u_{n},g]\right|\,&\leqslant\, \,\mathop{\sum}_{\pm}\,\frac{|\eta_{\pm}|}{\pi}{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{\sqrt{d(x)d(y)}}{{d_{0}^{2}}(x)(x+y)}\, |u_{n}(y,\sigma)|\, \left|{d_{0}^{2}}(x)g(x,\sigma)\right| \,\\&\leqslant\, \,\mathop{\sum}_{\pm}\,\frac{|\eta_{\pm}|}{\pi}\left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, \left[\int\limits_{0}^{1}\,\mathrm{d}y\,\frac{\sqrt{d(y)}}{\sqrt{d_{0}(x)}^{3}(x+y)}\, |u_{n}(y,\sigma)|\right]^{2}\,\right\}^{1/2}\\ &\quad\ \ \left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, |{d_{0}^{2}}(x)g(x,\sigma)|^{2} \right\}^{1/2}\,\\&\leqslant\, \,\mathop{\sum}_{\pm}\,C\,\frac{|\eta_{\pm}|}{\pi}\, \left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{d(y)}{d_{0}(x)^{3}(x+y)^{2}}\, \right\}^{1/2}\,\\&\quad\times \left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, | u_{n}(x,\sigma)|^{2} \right\}^{1/2}\, \,\left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, \left|{d_{0}^{2}}(x)g(x,\sigma)\right|^{2} \right\}^{1/2}\,, \end{align*} where we have multiply used the Cauchy–Schwarz inequality. From (4.6) and that L2((0, 1), H1(Σ)) is boundedly embedded into L2((0, 1), L2(σ)) (un ∈ L2((0, 1), H1(Σ)) we have   \begin{align*} \left\{{\int\limits_{0}^{1}}\,\mathrm{d}\,x\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, | u_{n}(x,\sigma)|^{2} \right\}^{1/2}\,\leqslant\,C\|u_{n}\|_{1}\,\,\leqslant\,C_{1}\, {t_{A}^{1}}[u_{n},u_{n}]\to 0\,,\,\,\,\text{as}\,\,\, n\to \infty. \end{align*} It remains to verify that   \begin{align*} {\int\limits_{0}^{1}}\,\mathrm{d}\,x\, {\int\limits_{0}^{1}}\,\mathrm{d}y\,\frac{d(y)}{d_{0}(x)^{3}(x+y)^{2}}\,\leqslant \,\text{Const}, \, \end{align*} i.e. it is bounded. The latter double integral over the square S = (0, 1) × (0, 1) is represented as a sum of two integrals, one is over $$S_{1}=\{(x,y):\,\,\sqrt {x^{2}+y^{2}}<1\}\cap S $$ and that over S ∖ S1. The integral over $$\overline {S\setminus S_{1}}$$ is obviously bounded because the integrand is continuous in this domain. The integral over the sector S1 is estimated by making use of the polar coordinates ρ, ϕ in its evaluation and is estimated, which is a simple exercise, by a constant C1,   \begin{align*} \int\limits_{0}^{\pi/2}\,\mathrm{d}\phi\, {\int\limits_{0}^{1}}\,\frac{\mathrm{d}\rho}{\rho}\,\frac{\left|\log\rho+\log(\sin\phi)\right|}{ \left|\log(2/\rho)-\log(\cos\phi)\right|{}^{3}(\cos\phi+\sin\phi)^{2}}\,\leqslant \,\text{C}_{1}. \, \end{align*} Taking into account the estimates above we find that $$|{t_{A}^{3}}[u_{n},g]|\,\to \,0$$ as ∥un∥H → 0 , $$n\to \infty $$ for any g ∈ G. As a result, we can assert that the form tA is closable. 4.3. An m-sectorial operator A corresponding to the problem (4.2). Solvability of the problem Let us denote TA the closure of tA with D(tA) ⊂ D(TA).8 In accordance with the theorem on representation for the closed sectorial forms (see Kato, 1972, Section 2, Chapter 6, Theorem 2.1) we obtain Theorem 4.1 There exists a unique m-sectorial operator A such that Dom(A) ⊂ D(TA) and   \begin{align*} <Au,v>= T_{A}[u,v] \end{align*} with u ∈ Dom(A), v ∈ D(TA). It is important to notice that this operator is the desired operator attributed to the problem (4.2). Let u ∈ D(TA) and for any v ∈ D(TA)   \begin{align*} T_{A}[u,v]=<f,v>\,,\,\,\,\,f\in H, \end{align*} then u is called weak solution of the equation   \begin{align} Au=f. \end{align} (4.7) This solution is also a weak solution of the problem (4.2) by definition. From the theorem on representation it also follows that, provided u ∈ D(TA), f ∈ H and the equality   \begin{align*} \displaystyle T_{A}[u,v]=<f,v>\end{align*} is valid for any v from the core of the form TA (i.e. from Dom(A)) then u ∈ Dom(A) and u is a solution of the equation (4.7). The range Θ(A) of the m-sectorial operator A is a dense subset of the range Θ(TA), which implies that Λ = 0 belongs to the resolvent set of the operator A. The inverse operator A−1 = (A−Λ)−1|Λ=0 is bounded and the problem (4.2) has a weak solution $${\mathscr {U}}$$. Recall that   \begin{align*} F(\nu,\omega)|_{x=1/\cos\pi\nu}\,=\,-\,\frac{f(x,\omega)}{\sqrt{d(x)}\cosh\left(\pi d(x)\right)} \end{align*} in (4.1) and, provided f is from C([0, 1];C(σ)) ⊂ H the function F(ν, ω) is continuous with respect to ν and is from C(σ) w.r.t. ω, exponentially vanishing as $$\nu \to \mathrm {i}\infty $$. Similar simple arguments enable us to conclude that solution of the problem (3.6), (3.8) exists and is unique, uν(ω, ω0) is from $$C([0,\mathrm {i}\infty );C^{2}(\varSigma ))$$ which admits an estimate   \begin{align*} \displaystyle |u_{\nu}(\omega,\omega_{0})|\,\leqslant\, C\,\left|\frac{\text{e}^{\mathrm{i}\nu\tau_{*}}}{ \sqrt\nu}\right|\,,\,\,\,\,\nu\to\mathrm{i}\infty, \end{align*} $$0<\tau _{*}\leqslant \pi /2$$ as dist(ω0, σ) > π/2. This estimate follows from the fact that $${\mathscr {U}}\in L_{2}((0.1);H^{1}(\varSigma ))$$. Remark that $${\mathscr {U}}_{i}(x,\omega )\,=\,\sqrt {d(x)}\,\cos (\mathrm {i}\pi d(x))\,u_{\mathrm {i}d(x)}^{i}(\omega ,\omega _{0})\,$$ is from H = L2((0, 1);L2(Σ)) provided dist(ω0, σ) > π/2, which is implied. 5. Meromorphic continuation of uν(ω, ω0) In order to continue uν(ω, ω0) specified on $$\nu \in [0,\mathrm {i}\infty ) $$, as demonstrated in the previous section, onto the complex plane we need to develop some technical tools. First, we continue uν(ω, ω0) from $$[0,\mathrm {i}\infty )$$ onto iR making use of evenness w.r.t. ν. Then, consider gν(ω, ω′) the Green’s function of the operator △ω + (ν2 − 1/4) on the unit sphere   \begin{align*} g_{\nu}(\omega,\omega^{\prime})=-\frac{P_{\nu-1/2}\left(-\cos\theta(\omega,\omega^{\prime})\right)}{4\cos\pi\nu}, \end{align*} satisfying   \begin{align*} \left(\triangle_{\omega}+(\nu^{2}-1/4)\right)g_{\nu}(\omega,\omega^{\prime})=\delta(\omega-\omega^{\prime}). \end{align*} We apply Green’s identity to uν(ω, ω0) and gν(ω, ω′) and, omitting some technical details, obtain   \begin{align*} &\int\limits_{\varSigma}\,\mathrm{d}\omega^{\prime} \,\left( \triangle_{\omega^{\prime}}g_{\nu}(\omega,\omega^{\prime})\,u_{\nu}(\omega^{\prime},\omega_{0})\right)\,-\, g_{\nu}(\omega,\omega^{\prime})\,\triangle_{\omega^{\prime}}u_{\nu}(\omega^{\prime},\omega_{0}))\\&\quad=\, \mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\left(\frac{\partial g_{\nu}(\omega,\sigma)}{\partial{\mathscr{N}}_{\pm}} u_{\nu}(\sigma,\omega_{0})\,-\, \frac{\partial u_{\nu}(\sigma,\omega_{0})}{\partial{\mathscr{N}}_{\pm}} g_{\nu}(\omega,\sigma)\right)\,. \end{align*} Making use of the equations for gν(ω, ω′) and uν(ω, ω′) and the boundary condition (see (3.8)), we arrive at the representation   \begin{align} u_{\nu}(\omega,\omega_{0})\,=\, \mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\left(\frac{\partial g_{\nu}(\omega,\sigma)}{\partial{\mathscr{N}}_{\pm}} u_{\nu}(\sigma,\omega_{0})\,-\,\eta_{\pm} \,{\mathscr{A}}_{0}u_{\nu}(\sigma,\omega_{0})\,g_{\nu}(\omega,\sigma)\right)\,+\, \varPsi_{\nu}(\omega,\omega_{0}), \end{align} (5.1) where   \begin{align*} {\mathscr{A}}_{0}u_{\nu}(\sigma,\omega_{0})\,:&=\,\frac{ 1\,}{2\mathrm{i}} \int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\, d\tau\, \frac{\tau\,\sin\pi\tau\,u_{\tau}(\sigma,\omega_{0})}{\cos\pi\tau+\cos\pi\nu}\,,\\ \varPsi_{\nu}(\omega,\omega_{0})&=\,\mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\,\mathrm{d}\sigma \left(\frac{\partial u^{i}_{\nu}(\omega,\sigma)}{\partial{\mathscr{N}}_{\pm}} g_{\nu}(\sigma,\omega_{0})\,-\,\eta_{\pm } \,{\mathscr{A}}_{0}u_{\nu}^{i}(\sigma,\omega_{0})\,g_{\nu}(\omega,\sigma)\right)\!. \end{align*} The integral equation for uν(s, ω0) as s belongs to the boundary σ+ ∪ σ− is then derived and takes the form   \begin{align} \alpha_{s} u_{\nu}(s,\omega_{0})\,=\, \mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\left(\frac{\partial g_{\nu}(s,\sigma)}{\partial{\mathscr{N}}_{\pm}} u_{\nu}(\sigma,\omega_{0})\,-\,\eta_{\pm} \,{\mathscr{A}}_{0}u_{\nu}(\sigma,\omega_{0})\,g_{\nu}(s,\sigma)\right)\,+\, \varPsi_{\nu}(s,\omega_{0}), \end{align} (5.2) where $$\alpha _{s} =\frac {1}{2}$$ as s ∈ (σ+ ∪ σ−), s ∉ {A, B} and αs = 1 as s ∈ {A, B}. The properties of the single and double layer potentials have been used in (5.2). It is not difficult to show that Ψν(ω, ω0) in (5.2), (5.1) admits meromorphic continuation as ν ∈ C such that it is holomorphic as ν ∈ Πδ for some δ > 0. Remark that the integral equation really has solution, because it is derived from the uniquely solvable problem (3.6), (3.8). It is easily verified that Ψν is a holomorphic mapping ν ↦ L2(σ) as ν ∈ Πδ, Ψν(⋅, ω0) ∈ L2(σ). Now we explain that the solution uν(s, ω0) of the equation (5.2) obeys the same property. To that end, we write the equation (5.2) in an equivalent form   \begin{align} (I-{\mathscr{K}}_{0})u_{\nu} = 2\chi_{\nu,0}, \end{align} (5.3) where   \begin{align*} {\mathscr{K}}_{0} u_{\nu}(s,\omega_{0})\,&=\, \mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\frac{\partial g_{\nu}(s,\sigma)}{\partial{\mathscr{N}}_{\pm}}\, u_{\nu}(\sigma,\omega_{0})\,,\,\,s\in \sigma\setminus\{A,B\} \\2\chi_{\nu,0}(s,\omega_{0})&=\, 2\mathop{\sum}_{\pm} \eta_{\pm}\,\int\limits_{\sigma_{\pm}}\mathrm{d}\sigma\, \,g_{\nu}(s,\sigma) {\mathscr{A}}_{0}u_{\nu}(\sigma,\omega_{0})\,+\,2\varPsi_{\nu}(s,\omega_{0}), \end{align*} so that the operator $$(I-{\mathscr {K}}_{0})$$ is boundedly invertible for any ν ∈ Πδ and $$(I-{\mathscr {K}}_{0})^{-1}$$ is holomorphic with respect to ν ∈ Πδ. Because the right-hand side χν, 0 in (5.3) is a holomorphic mapping ν ↦ L2(σ) as ν ∈ Πδ , χν, 0(⋅, ω0) ∈ L2(σ), we conclude that the solution uν of the equation (5.2) is a holomorphic mapping ν ↦ L2(σ) as ν ∈ Πδ , uν(⋅, ω0)|σ ∈ L2(σ) (see condition 2 in Section 3.3). The representation (5.1) enables one to assert that uν is also a holomorphic mapping ν ↦ L2(Σ) as ν ∈ Πδ , uν(⋅, ω0) ∈ L2(Σ). Indeed, the integrand in the right-hand side of (5.1) is holomorphic as ν ∈ Πδ for all ω ∈ Σ because gν(ω, ⋅)|σ obeys this property as well as uν(⋅, ω0)|σ in the integrand, gν(ω, ⋅) is continuous w.r.t. ω ∈ Σ. uν(s, ω0)|σ is continuous w.r.t. s assuming that $$\omega _{0}\notin \varOmega _{\delta _{0}}$$. The same regularity is valid for $${\mathscr {A}}_{0}u_{\nu }(\sigma ,\omega _{0})$$, which follows from the explicit formula for $${\mathscr {A}}_{0}u_{\nu }(\sigma ,\omega _{0})$$. After integration w.r.t. σ we obtain the desired, where we also make use of the regularity of Ψν(ω, ω0) declared above. Making use of the regularity of the mapping $$(u_{\nu }+u_{\nu }^{i})|_{\sigma }$$ as ν ∈ Πδ, from the boundary condition (3.8) we have that $$ \left .\frac {\partial \widehat {u}_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$ specifies a holomorphic mapping as ν ∈ Π1+δ. The latter directly follows from the fact that the denominator $$\cos \pi \tau +\cos \pi \nu \ne 0 $$ as τ ∈iR and ν ∈ Π1 in the integrand of (3.8), whereas $$(u_{\nu }+u_{\nu }^{i})|_{\sigma }$$ is holomorphic as ν ∈ Πδ. Further use of the integral equation (5.2) enables one to continue analytically uν onto the strip ν ∈ Π1+δ as a meromorphic mapping with the poles located in the strip |Iν| < b, b > 0. Indeed, we make use of the same argumentation, exploiting the equation written in the form (5.3) and the fact that $$(I-{\mathscr {K}}_{0})$$ is boundedly invertible, in order to verify that uν(s, ω0)|σ has only polar singularities as ν ∈ Π1+δ because $${\mathscr {A}}_{0}u_{\nu }(\sigma ,\omega _{0})$$ is regular and gν(s, σ), Ψν(s, ω0) are meromorphic in this strip with the poles located on the real axis. The condition (3.7) shows that having the meromorphic functions in the strip Π1+δ one can continue $$ \left .\frac {\partial \widehat {u}_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$ from this strip onto a neighbouring strip on the left- or right-hand sides. Because the shift of the argument ν is performed along the real axis, the corresponding poles appear only in some strip |Iν| < b, b > 0. On this way we obtain that $$ \left .\frac {\partial \widehat {u}_{\nu }(\omega ,\omega _{0})}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$ and uν(s, ω0)|σ are meromorphic in the strip Π2+δ. The corresponding procedure can be iterated and one can prove meromorphic continuation of the mapping uν (as well as of $$ \left .\frac {\partial \widehat {u}_{\nu }}{\partial \mathscr {N}_{\pm }}\right |_{\sigma _{\pm }}\,$$) from the strip Π2+δ onto the complex plane with the singularities located in some strip |Iν| < b, b > 0. The representation (5.1) enables one to assert that uν(s, ω0)|σ is a meromorphic mapping ν ↦ L2(Σ) as ν ∈ C , with the values uν(⋅, ω0) ∈ L2(Σ). 6. Sommerfeld integral representation for the wave field In order to study the far-field asymptotics of the scattered wave field U(r, φ) it is profitable to reduce the Watson-Bessel representation to the Sommerfeld integral because, contrary to that of the Watson-Bessel, it is well adapted to this goal. 6.1. Sommerfeld representation for the incident wave We transform the Watson–Bessel representation (3.1) for the incident plane wave making use of the Sommerfeld formula for the Bessel function   \begin{align*} J_{\nu}(kr)=\frac{1}{2\pi}\int\limits_{\gamma_{-}}\mathrm{e}^{-\mathrm{i}kr\cos\alpha}\, \mathrm{e}^{\mathrm{i}\nu\pi/2-\mathrm{i}\nu\alpha}\,d \alpha, \end{align*} where γ− is the lower part of the double-loop Sommerfeld contour (Fig. 8). Fig. 8. View largeDownload slide Sommerfeld double-loop contour, γ = γ+ ∪ γ−. Fig. 8. View largeDownload slide Sommerfeld double-loop contour, γ = γ+ ∪ γ−. Substitute the latter into (3.1) and change the orders of integration, which is justified,   \begin{align} U_{i}(r,\omega,\omega_{0})&= 4\mathrm{i}\sqrt{\frac{\pi}{2}}\int\limits_{C_{0}}\, d \nu\,u_{\nu}^{i}(\omega,\omega_{0}) \frac{\nu \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}}{\sqrt{-\mathrm{i}kr}}\left( \frac{1}{2\pi}\int\limits_{\gamma_{-}}\mathrm{e}^{-\mathrm{i}kr\cos\alpha}\,\mathrm{e}^{\mathrm{i}\nu\pi/2-\mathrm{i}\nu\alpha}\,d \alpha\, \right)\,\nonumber\\ &=\frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma_{-}} d \alpha\,\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}} \left( -2\sqrt{2\pi}\int\limits_{C_{0}} \,\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha} u_{\nu}^{i}(\omega,\omega_{0})\,d \nu\, \right)\, \nonumber\\ &=\frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma_{-}}\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}} \varPsi_{i}(\alpha,\omega,\omega_{0})\, d \alpha, \end{align} (6.1) where we introduced notation   \begin{align*} \varPsi_{i}(\alpha,\omega,\omega_{0})\,=\,-2\sqrt{2\pi}\int\limits_{C_{0}} \,\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha} u_{\nu}^{i}(\omega,\omega_{0})\,d \nu, \end{align*} Ψi(α, ω, ω0) is regular in D− = {α ∈ C : ℑα < 0} because $$u_{\nu }^{i}(\omega ,\omega _{0})$$ is bounded on C0 for any ω ∈ S2. In view of the estimate (3.3) on the contour Cϕ one can verify that Ψi(⋅, ω, ω0) is regular in the domain   \begin{align*} D_{\theta}^{\phi} =\,\left\{\alpha\in C:\,\,\sin\phi\,\Re\alpha+\cos\phi\,\Im\alpha < \theta|\sin\phi|\right\}\cap \{\alpha\in C:\,\,-\sin\phi\,\Re\alpha+\cos\phi\,\Im\alpha < \theta|\sin\phi|\} \end{align*} because   \begin{align*} \left|\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha}u_{\nu}^{i}(\omega,\omega_{0}) \right|\,< \,C\,|\nu|^{1/2} \exp\big\{|\nu|\big(|\sin\phi|\,\Re(\alpha)+\cos\phi\,\Im(\alpha)-\theta|\sin\phi|\big)\big\}, \end{align*} θ = θ(ω, ω0). It is useful to notice that the regularity domain varies from the halfplane $$D_{\theta }^{0}$$ to the strip $$D_{\theta }^{\pi /2}$$ (see also Fig. 7 in Lyalinov, 2013). Consider Ψi(⋅, ω, ω0) in $$D_{\theta }^{\pi /2}$$ and introduce a regular function Φi(⋅, ω, ω0) by the equality   \begin{align*} \varPhi_{i}(\alpha,\omega,\omega_{0})\,:=\,\frac{1}{2} \varPsi_{i}(\alpha,\omega,\omega_{0})\,= \mathrm{i}\sqrt{2\pi}\int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty} \,\nu\, \sin(\nu\alpha) u_{\nu}^{i}(\omega,\omega_{0})\,d \nu, \end{align*} which is obviously odd in α, Φi(α, ω, ω0) = −Φi(−α, ω, ω0). We easily verify that Φi(⋅, ω, ω0) is analytically continued as a holomorphic function into C∖l±, $$l_{\pm }\!=\!\{\alpha \in C:\,\pm \Re (\alpha )\geqslant \theta (\omega ,\omega _{0}),\,\Im (\alpha )=0\}$$, i.e. it is regular in the complex plane with the cuts along the lines l±. Remark that Φi(α, ω, ω0) can be computed in a closed form Lyalinov (2013)  \begin{align*} \varPhi_{i}(\alpha,\omega,\omega_{0})=\frac{\partial \widetilde\varPhi_{i}(\alpha,\omega,\omega_{0})}{\partial \alpha} \end{align*} with   \begin{align*} \widetilde\varPhi_{i}(\alpha,\omega,\omega_{0})=-\frac{\sqrt{\pi}}{2}\,\big[\cos\alpha-\cos\theta(\omega,\omega_{0})\big]^{-1/2}, \end{align*} the branch is fixed by the condition $$\sqrt {\cos \alpha -\cos \theta (\omega ,\omega _{0})}>0$$ as − θ < α < θ. This expression is obtained by means of the formula (see Gradstein & Ryzhik, 1980, 7.216)   \begin{align*} \frac{1}{\sqrt{2}}\,[\cos\alpha-\cos\theta]^{-1/2}\,=\, \int\limits_{0}^{\infty}\mathrm{d}\tau \frac{P_{\mathrm{i}\tau-1/2}\big(\cos[\pi-\theta]\big)}{\cos\mathrm{i}\pi\tau} \,\cos\mathrm{i}\alpha\tau\,,\,\,\,\,\,\,(\mathrm{i}\alpha>0)\,. \end{align*} The Sommerfeld representation for the incident wave takes the form   \begin{align} U_{i}(r,\vartheta,\varphi)= \frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma} \,\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}}\, \varPhi_{i}(\alpha,\omega,\omega_{0}) \,d \alpha, \, \end{align} (6.2) and also   \begin{align} U_{i}(r,\vartheta,\varphi)= \frac{{\sqrt{-\mathrm{i}kr}}}{2\pi\mathrm{i}} \int\limits_{\gamma} \,{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}\, \sin\alpha\, \widetilde\varPhi_{i}(\alpha,\omega,\omega_{0}) \,d \alpha, \, \end{align} (6.3) where γ = γ+ ∪ γ− is shown in Fig. 8. 6.2. Sommerfeld representation for the scattered field The derivations given in the previous section motivate appearance of the analogous representations for the scattered field. We have   \begin{align} U(r,\vartheta,\varphi)&= 4\mathrm{i}\sqrt{\frac{\pi}{2}}\int\limits_{C_{0}}\, d \nu\,u_{\nu}(\omega,\omega_{0}) \frac{\nu \,\mathrm{e}^{-\mathrm{i}\nu\pi/2}}{\sqrt{-\mathrm{i}kr}}\left( \frac{1}{2\pi}\int\limits_{\gamma_{-}}\mathrm{e}^{-\mathrm{i}kr\cos\alpha}\,\mathrm{e}^{\mathrm{i}\nu\pi/2-\mathrm{i}\nu\alpha}\,d \alpha\, \right)\,\nonumber\\ &=\frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma_{-}} d \alpha\,\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}} \left( -2\sqrt{2\pi}\int\limits_{C_{0}} \,\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha} u_{\nu}(\omega,\omega_{0})\,d \nu\, \right)\, \nonumber\\&=\frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma_{-}}\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}} \varPsi(\alpha,\omega,\omega_{0})\, d \alpha, \end{align} (6.4) where we introduced   \begin{align*} \varPsi(\alpha,\omega,\omega_{0})\,=\,-2\sqrt{2\pi}\int\limits_{C_{0}} \,\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha} u_{\nu}(\omega,\omega_{0})\,d \nu, \end{align*} Ψ(⋅, ω, ω0) is regular in $$D_{\tau _{1}}=\{\alpha \in C:\, \,\Im ( \alpha )< -|\tau _{1}(\omega ,\omega _{0})|\}$$ because uν(ω, ω0) satisfies the estimate (3.9) on C0 (see Fig. 7) for any ω ∈ S2. We take into account the estimate   \begin{align*} \left|\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha}u_{\nu}(\omega,\omega_{0}) \right|\,< \,C\,|\nu|^{1/2} \exp&\left\{|\nu|\left(|\sin\phi|\Re(\alpha)+\cos\phi(\Im\alpha)\right.\right.\\&\quad\left.-\ \tau_{0}(\omega,\omega_{0})|\sin\phi|\,+ \,\mathrm{sign}(\phi)\,\cos\phi|\tau_{1}(\omega,\omega_{0})|\right\}\!, \end{align*} and conclude that the regularity domain for Ψ(⋅, ω, ω0) is   \begin{align*} D_{\tau_{0}\tau_{1}}^{\phi} =\,&\left\{\alpha\in C:\,\,\sin\phi\Re(\alpha)+\cos\phi\Im(\alpha) < \tau_{0}|\sin\phi|-\cos\phi\,\tau_{1}\right\}\cap\\ &\left\{\alpha\in C:\,\,-\sin\phi\Re(\alpha)+\cos\phi\Im(\alpha) < \tau_{0}|\sin\phi|+\cos\phi\,\tau_{1}\right\}\!, \end{align*} which is verified from   \begin{align*} \varPsi(\alpha,\omega,\omega_{0})\,=\,-2\sqrt{2\pi}\int\limits_{C_{\phi}} \,\nu\,\mathrm{e}^{-\mathrm{i}\nu\alpha} u_{\nu}(\omega,\omega_{0})\,d \nu, \end{align*} where ϕ ∈ (0, π/2). Remark Provided the contour Cϕ varies from C0 into Cπ/2 the domain of regularity $$D_{\tau _{0}\tau _{1}}^{\phi }$$ of Ψ(⋅, ω, ω0) deforms from the halfplane ℑ(α) < −|τ1(ω, ω0)| onto the strip |I(α)| < τ0(ω, ω0). We consider Ψ(⋅, ω, ω0) in the strip $$D_{\tau _{0}\tau _{1}}^{\pi /2}$$ and introduce there a regular function   \begin{align*} \varPhi(\alpha,\omega,\omega_{0})\,:=\,\frac{1}{2} \varPsi(\alpha,\omega,\omega_{0})\,= \mathrm{i}\sqrt{2\pi}\int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty} \,\nu\, \sin(\nu\alpha) u_{\nu}(\omega,\omega_{0})\,d \nu, \end{align*} which is odd in α, Φ(α, ω, ω0) = −Φ(−α, ω, ω0). Simple analysis enables us to conclude that Φ(⋅, ω, ω0) is continued as a regular function into the domain   \begin{align*} D\,=\,D_{\tau_{0}\tau_{1}}^{\pi/2}\cup D_{\tau_{0}\tau_{1}}^{0}\cup \left(D_{\tau_{0}\tau_{1}}^{\pi/2}\right)^{*}\!, \end{align*} where $$\left (D_{\tau _{0}\tau _{1}}^{\pi /2}\right )^{*}$$ is symmetric to $$D_{\tau _{0}\tau _{1}}^{\pi /2}$$ with respect to the origin, (Fig. 9). In other words, we can assert that all singularities of Φ(⋅, ω, ω0) are located in C ∖ D. For the diffraction by a sector with Dirichlet boundary conditions such singularities are symmetrically distributed on the real axis Lyalinov, 2013. For the sector with the impedance boundary conditions there are some additional complex singularities which are responsible for the surface waves propagating along the surface of the sector. Fig. 9. View largeDownload slide Domain of regularity D, singularities are in C ∖ D. Fig. 9. View largeDownload slide Domain of regularity D, singularities are in C ∖ D. The Sommerfeld representations for the scattered field take the form   \begin{align} U(r,\vartheta,\varphi)= \frac{1}{2\pi\mathrm{i}} \int\limits_{\gamma} \,\frac{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}{\sqrt{-\mathrm{i}kr}}\, \varPhi(\alpha,\omega,\omega_{0}) \,d \alpha, \, \end{align} (6.5) or   \begin{align} U(r,\vartheta,\varphi)= \frac{{\sqrt{-\mathrm{i}kr}}}{2\pi\mathrm{i}}1 \int\limits_{\gamma} \,{\mathrm{e}^{-\mathrm{i}kr\cos\alpha}}\, \sin\alpha\, \widetilde\varPhi(\alpha,\omega,\omega_{0}) \,d \alpha\, \, \end{align} (6.6) with   \begin{align*} \displaystyle \varPhi(\alpha,\omega,\omega_{0})=\frac{\partial \widetilde\varPhi(\alpha,\omega,\omega_{0})}{\partial \alpha}\,. \end{align*} As we remarked the singularities of $$\varPhi (\alpha ,\omega ,\omega _{0}), \widetilde \varPhi (\alpha ,\omega ,\omega _{0})$$ play a crucial role in studying the far-field asymptotics by use of the Sommerfeld integral representations. In order to describe the singularities of the Sommerfeld transformants $$\varPhi (\alpha ,\omega ,\omega _{0}), \widetilde \varPhi (\alpha ,\omega ,\omega _{0})$$ we turn to formulation of the problems for them, although the singularities themselves will be studied elsewhere. 6.3. Problems for the Sommerfeld transformants $$\varPhi (\alpha ,\omega ,\omega _{0}),\widetilde {\varPhi }(\alpha ,\omega ,\omega _{0})$$ It is obvious that the Fourier transform of (3.6) with integration along the imaginary axis, enables us to write down the equation for the Sommerfeld transformant (Lyalinov, 2013),   \begin{align} \left(\triangle_{\omega}-\partial^{2}_{\alpha}-1/4\right)\varPhi(\alpha,\omega,\omega_{0})=0 \end{align} (6.7) as ω ∈ S2 ∖ σ. Let us turn to the boundary conditions   \begin{align} \left. r^{-1}\frac{\partial\left(U + U^{\mathrm{i}}\right)}{\partial{\mathscr{N}}_{\pm}} \right|_{\sigma_{\pm}} &=\frac{1}{\sqrt{-\mathrm{i} kr}}\frac{1}{2\mathrm{i}\pi}\int\limits_{\gamma}\mathrm{e}^{-\mathrm{i} kr\cos\alpha}(-\mathrm{i} k\sin\alpha) \left.\frac{\partial}{\partial{\mathscr{N}}_{\pm}}\left(\widetilde{\varPhi}+\widetilde{\varPhi}_{i}\right)(\alpha,\omega,\omega_{0})\right|_{\sigma_{\pm}} \mathrm{d} \alpha \nonumber\\ &=\,\left.\mathrm{i} k \eta_{\pm} \left(U + U^{\mathrm{i}}\right)\right|_{S}\,=\,\,\frac{\mathrm{i} k \eta_{\pm}}{\sqrt{-\mathrm{i} kr}} \frac{1}{2\mathrm{i}\pi}\int\limits_{\gamma}\mathrm{e}^{-\mathrm{i} kr\cos\alpha} \left.\frac{\partial}{\partial \alpha}\left(\widetilde{\varPhi}+\widetilde{\varPhi}_{ i}\right)(\alpha,\omega,\omega_{0})\right|_{\sigma_{\pm}}\mathrm{d} \alpha, \end{align} (6.8) then, exploiting the Malyuzhinets theorem (see Lyalinov & Zhu, 2012, Chapter 1), we obtain   \begin{align*} \sin\alpha \left.\frac{\partial \left(\widetilde{\varPhi}+\widetilde{\varPhi}_{\mathrm{i}}\right)}{\partial{\mathscr{N}}_{\pm}}(\alpha,\omega,\omega_{0})\right|_{\sigma_{\pm}} = \left.-\eta_{\pm}\frac{\partial \left(\widetilde{\varPhi}+\widetilde{\varPhi}_{ i}\right)} {\partial \alpha}(\alpha,\omega,\omega_{0})\right|_{\sigma_{\pm}}\!, \end{align*} which is followed by   \begin{align} \left.\frac{\partial (\varPhi+\varPhi_{ i})}{\partial{\mathscr{N}}_{\pm}}(\alpha,\omega,\omega_{0})\right|_{\sigma_{\pm}} = \left.-\eta_{\pm}\frac{\partial}{\partial \alpha}\frac{(\varPhi+\varPhi_{ i})(\alpha,\omega,\omega_{0})}{\sin\alpha}\right|_{\sigma_{\pm}}\!, \end{align} (6.9) where   \begin{align*} \varPhi_{ i}(\alpha,\omega,\omega_{0})=\frac{\partial} {\partial \alpha}\widetilde{\varPhi}_{ i}(\alpha,\omega,\omega_{0}). \end{align*} The equation (6.7) and the condition (6.9) are proved to be valid in the domain D by means of the analytic continuation. For real α from this strip the equation (6.7) is of hyperbolic type. In order to have a correct problem for this equation on S2 ∖ σ with the boundary condition (6.9) as α > 0 we should add initial conditions at α = 0. For any α ∈ [0, τ0) we have   \begin{align} \varPhi(\alpha,\omega,\omega_{0})+\varPhi_{i}(\alpha,\omega,\omega_{0})= \,\mathrm{i}\sqrt{2\pi}\int\limits_{\mathrm{i} R} \,\nu\, \sin({\nu\alpha}) u_{\nu}(\omega,\omega_{0})\,d \nu\, \,-\frac{\sqrt{\pi}}{4} \frac{\sin\alpha}{\left(\cos\alpha-\cos\theta(\omega,\omega_{0})\right)^{3/2}} \end{align} (6.10) which is a regular function. Therefore, assuming that uν(ω, ω0) is known in the integrand, we arrive at the initial conditions   \begin{align} \varPhi(0,\omega,\omega_{0})= 0,\,\,\,\, \left.\frac{\partial \varPhi(\alpha,\omega,\omega_{0})}{\partial\alpha}\right|_{\alpha=0}= \mathrm{i}\sqrt{2\pi}\int\limits_{\mathrm{i} R} \,\nu^{2}\, u_{\nu}(\omega,\omega_{0})\,d \nu.\, \end{align} (6.11) The Cauchy boundary value problem (6.7), (6.9), (6.11) can be used in order to determine singularities of Φ(α, ω, ω0) as α ∈ C ∖ D. However, instead of the conditions (6.11) one can exploit the equality (6.10) for any point α ∈ [0, τ0) together with (6.7), (6.9). The analogous problem is valid for $$\widetilde \varPhi (\alpha ,\omega ,\omega _{0})$$  \begin{align} \left(\triangle_{\omega}-\partial^{2}_{\alpha}-1/4\right)\widetilde\varPhi(\alpha,\omega,\omega_{0})=0 \end{align} (6.12) as ω ∈ S2 ∖ σ and   \begin{align} \sin\alpha \left.\frac{\partial \left(\widetilde{\varPhi}+\widetilde{\varPhi}_{\mathrm{i}}\right)} {\partial N_{\pm}}\right|_{\sigma_{\pm}} = \left.-\eta_{\pm}\frac{\partial \left(\widetilde{\varPhi}+\widetilde{\varPhi}_{ i}\right)}{\partial \alpha}\right|_{\sigma_{\pm}}\!, \end{align} (6.13) and   \begin{align}\displaystyle \widetilde\varPhi(0,\omega,\omega_{0})= -\mathrm{i}\sqrt{2\pi}\int\limits_{\mathrm{i} R} \, u_{\nu}(\omega,\omega_{0})\,d \nu,\,\,\,\, \left.\frac{\partial \widetilde\varPhi(\alpha,\omega,\omega_{0})}{\partial\alpha}\right|_{\alpha=0}= 0 .\, \end{align} (6.14) The discussion above enables us to prove Theorem 6.1 There exists an analytic function Φ(α, ω, ω0) which is holomorphic in the domain D. For real α ∈ [0, τ0) this function solves the problem (6.7), (6.9), (6.11). For the strip $$|\Re (\alpha )|<\tau _{0}(\omega ,\omega _{0})$$ it is specified by the equality (see (6.10))   \begin{align*} \varPhi(\alpha,\omega,\omega_{0})\,= \,\mathrm{i}\sqrt{2\pi}\int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty} \,\nu\, \sin({\nu\alpha}) u_{\nu}(\omega,\omega_{0})\, \mathrm{d} \nu. \end{align*} A similar statement is valid for $$\widetilde \varPhi (\alpha ,\omega ,\omega _{0})$$. 7. Comments on the derivations of the far-field asymptotics. Diffraction coefficient for narrow sector in ‘oasis’ In this section we give some comments dealing with the derivations of the far-field asymptotics described in Section 2.2 although the detailed exposition of the results will appear in a future work. Fig. 10. View largeDownload slide Deformation of the Sommerfeld contour into the steepest descent paths γ+ ∪ γ−: (left) singularities are not captured, ω ∈ Ω0; (right) singularities are captured ω ∈ S2 ∖ Ω0. Fig. 10. View largeDownload slide Deformation of the Sommerfeld contour into the steepest descent paths γ+ ∪ γ−: (left) singularities are not captured, ω ∈ Ω0; (right) singularities are captured ω ∈ S2 ∖ Ω0. Solution Φ(α, ω, ω0) of the problems for the hyperbolic equation (6.7) as α ∈ [0, τ0) is unique. This solution is represented by the Fourier type integral (see, Theorem 6.1) as $$|\Re (\alpha )|<\tau _{0}(\omega ,\omega _{0})$$. However, one needs to have analytic continuation of the Sommerfeld transformant into C ∖ D, in particular, into vicinities of the singularities9 located in C ∖ D. As we mentioned, these singularities are of crucial importance for studying the far-field asymptotics (Lyalinov, 2013). Indeed, in order to evaluate the Sommerfeld integral representation (6.5) asymptotically as $$kr \to \infty $$ we ought to deform the Sommerfeld double-loop contour γ into the steepest descent paths γ±π (Fig. 10). In the process of such deformation some singularities of the transformant can be captured. Contributions of these singularities (Fig. 10, right) give rise to the corresponding components of the far field such as reflected wave, edge waves (see Lyalinov, 2013 for the Dirichlet boundary conditions) as well as the surface waves which may be excited near the impedance surface of the sector (2.9), (2.10). These surface waves are governed by complex singularities which do not exist in the case of the ideal boundary conditions. It is worth mentioning that the singularities migrate (Fig. 10) when the observation point varies its position. The saddle point ±π are responsible for the spherical wave from the vertex of the sector. The diffraction coefficient of this wave is easily calculated by use of the direct application of the saddle point technique provided the observation point (r, ω) is located in the oasis, ω ∈ Ω0. In this case the singularities are not captured (Fig. 10, left) and the asymptotics of the scattered far-field is given by (see also (2.8))  \begin{align*} \displaystyle U(r,\vartheta,\varphi)=D(\omega,\omega_{0})\frac{\exp(\mathrm{i}kr)}{-\mathrm{i}kr}\left(1+ O\left(\frac{1}{kr}\right)\right),\,\,\, kr\to \infty. \end{align*} with   \begin{align} D(\omega,\omega_{0})\,=\,-\sqrt{\frac{2}{\pi}}\,\varPhi(\pi,\omega,\omega_{0})\,=\, \,\frac{2}{\mathrm{i}}\int\limits_{\mathrm{i} R} \,\nu\, \sin(\pi\nu) u_{\nu}(\omega,\omega_{0})\, \mathrm{d} \nu\, \end{align} (7.1) or10  \begin{align*} D(\omega,\omega_{0})\,= 2\int\limits_{\mathrm{i} R} \,\nu\, \exp(-\mathrm{i}\pi\nu) u_{\nu}(\omega,\omega_{0})\, \mathrm{d} \nu, \end{align*} where the integrals converge exponentially as ω ∈ Ω0. It is obvious that the numerical determination of the spectral function uν(ω, ω0) is necessary for the calculation of the diffraction coefficient. In order to demonstrate also some practical issues from the results developed above we consider derivation of the closed form of the asymptotic expression for the diffraction coefficient in the case of a narrow impedance sector, β = 2a ≪ 1. In this case a closed asymptotic formula for the spectral function uν(ω, ω0) can be given and the integral for the diffraction coefficient (7.1) can be computed explicitly leading to a simple expression in the high order approximation as β ≪ 1. 7.1. Expression for D(ω, ω0) as β = 2a ≪ 1 There are, at least, two ways to determine such an expression. The first one is in asymptotic solution of the boundary value problem (3.6), (3.8) for the spectral function uν(ω, ω0). It is based on matching local asymptotic expansions (see e.g. Babich, 1997; Lyalinov & Zhu, 2012, pp. 122–124) for the problem on the unit sphere S2 with the narrow cut σ = σ+ ∪ σ−, mes(σ) = 2mes(σ±) = 2β ≪ 1. In this work we exploit an alternative approach, based on approximate solution of the integral equation (5.2) for uν(σ, ω0) as mes(σ±) = β ≪ 1, Bernard & Lyalinov (2001). It leads to the same desired result and also makes use of the integral representation (5.1) of the solution uν(ω, ω0), ω ∈ Ω0. Simple analysis of the integral equation (5.2) enables us to assert that $$ u_{\nu }(\omega |_{\sigma _{\pm }},\omega _{0})$$ is of O(β), therefore, taking into account that mes(σ+) = β and mes(σ−) = β, from the integral representation (5.1) we find   \begin{align} u_{\nu}(\omega,\omega_{0})&=\, \, \varPsi_{\nu}(\omega,\omega_{0})(1+O(\beta\log\beta)\nonumber\\&=\, \,- \,\frac{ 1\,}{2\mathrm{i}}\mathop{\sum}_{\pm} \eta_{\pm} \int\limits_{\sigma_{\pm}}\!\mathrm{d}\sigma\, g_{\nu}(\omega,\sigma)\!\left(\ \int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\,\mathrm{d}\tau\, \frac{\tau\,\sin\pi\tau\,u_{\tau}^{i}(\sigma,\omega_{0})}{\cos\pi\tau+\cos\pi\nu}\,\right) \left(1+O(\beta\log\beta)\right)\!,\,\,\,\,\,\omega\in\varOmega_{0}, \end{align} (7.2) where we took into account the estimate   \begin{align*} \mathop{\sum}_{\pm} \int\limits_{\sigma_{\pm}}\,\mathrm{d}\sigma\,g_{\nu}(\omega,\sigma) \,\left.\frac{\partial u_{\nu}^{i}}{ \partial{\mathscr{N}}_{\pm}}\right|_{\sigma}\, =\,O(\beta^{2}\log\beta). \end{align*} Substituting the integral expression from (7.2) into the formula for the diffraction coefficient (7.1) in oasis, in the leading approximation we arrive at   \begin{align*} D(\omega,\omega_{0})\,&=\mathop{\sum}_{\pm} \frac{\eta_{\pm}}{16} \int\limits_{\sigma_{\pm}}\,\mathrm{d}\sigma\,\\&\quad \left(\ \int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\,\mathrm{d} \tau\, \int\limits_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\,\mathrm{d} \nu\, \frac{\tau\,\tan\pi\tau\,\,\nu\,\tan\pi\nu\,}{\cos\pi\tau+\cos\pi\nu}\, P_{\nu-1/2}(\cos\theta(\omega,\sigma))P_{\tau-1/2}(-\cos\theta(\sigma,\omega_{0})) \right) \end{align*} The iterated integral in the brackets was computed in the work (Bernard & Lyalinov (1999) so that one has   \begin{align*} D(\omega,\omega_{0})\,= \,-\, \mathop{\sum}_{\pm} \frac{\eta_{\pm}}{4\pi} \int\limits_{\sigma_{\pm}}\, \frac{\mathrm{d}\sigma\,} {\left[\cos\theta(\omega,\sigma)+\cos\theta\left.(\sigma,\omega_{0}\right)\right]^{2}}\left(1+\,O(\beta \log\beta)\right)\, \end{align*} as θ(ω, σ) + θ(σ, ω0) > π. Integration in the latter formula is conducted along asymptotically small arcs σ± then asymptotically equivalent version of this formula takes the form   \begin{align*} D(\omega,\omega_{0})\,= \,-\, \frac{\eta_{+}+\eta_{-}}{2\pi} \frac{2a\,} {\left[\cos\theta(\omega,M)+\cos\theta\left.(M,\omega_{0}\right)\right]^{2}}\,\left(1+\,O(\beta \log\beta)\right)\!, \end{align*} θ(ω, M) + θ(M, ω0) > π and ω ∈ Ω0, M is the middle point of the arc AB. The advantage of the latter formula is that it is quite elementary and can be easily used in the engineering applications.11 It is instructive to compare this result with that obtained by Babich (1997), where he made use of matching asymptotic series in the case of Dirichlet boundary condition on the surface of a cone,   \begin{align*} D_{d}(\omega,\omega_{0})\,= \, -4\pi\, \int\limits_{-\infty}^{\infty}\,\mathrm{d} \tau\,\,\tau\,\mathrm{e}^{\pi\tau} \frac{g_{\tau} (\omega,M)\,g_{\tau}(M,\omega_{0})}{W_{\beta}-\Re{\psi(\mathrm{i}\tau-1/2)-{\mathscr{C}}- \log 2}}\, \left(1+\,O(\beta \log\beta)\right), \end{align*} θ(ω, M) + θ(M, ω0) > π, where $$W_{\beta }= - \log (a/2)=\mathrm {O}(\log \beta )$$, Wβ is the Wiener capacity, an integral characteristic of the segment of the length 2a, $${\mathscr {C}}$$ is Euler constant, $$g_{\tau } (\omega ,M)= -\frac {P_{\mathrm {i}\tau -1/2}(-\cos \theta (\omega ,M))}{4 \cosh \pi \tau }$$. As mentioned in Babich (1997) the latter expression for the diffraction coefficient Dd(ω, ω0) integral may be simplified by formally neglecting $$-\Re \psi (\mathrm {i}\tau -1/2)-{\mathscr {C}}- \log 2$$ in comparison with Wβ then computed, which leads to   \begin{align*} D_{d}(\omega,\omega_{0})\,= \,\,-\, \frac{1}{2\log\frac{a}{2}} \frac{1} {\left[\cos\theta(\omega,M)+\left.\cos\theta(M,\omega_{0}\right))\right]}\,\left(1+\,O(1/ \log\beta)\right)\!, \end{align*} as θ(ω, M) + θ(M, ω0) > π. It is obvious that the diffraction coefficient Dd(ω, ω0) for the Dirichlet case is of $$O(1/\log (a))$$ in the leading approximation, whereas D(ω, ω0) = O(a) as a → 0 for a narrow sector with the impedance boundary conditions. 8. Conclusion In this work we developed a motivated procedure to study the problem of diffraction by a semi-infinite sector with impedance boundary conditions. Such kind of the boundary conditions thwarts complete separation of variables. The separation of the radial variable in the boundary conditions leads to a condition which is non-local with respect to the parameter of separation. Nevertheless, after separation of the radial variable the problem on the unit sphere with the non-local condition on the cut AB admits an efficient study. To that end, the traditional theory of extension of sectorial sesquilinear forms has been exploited. The Watson–Bessel integral representation for the solution is not efficient for the derivation of the far-field asymptotics that is why reduction to the Sommerfeld integral representations was used. Analytic properties of the Sommerfeld transformants were considered. In particular, domains, where singularities of the transformants are localized, were indicated. Although a complete study of the singularities and of their contributions to the far-field asymptotics is postponed to a future work, we obtained a practically useful simple formula for the diffraction coefficient of the spherical wave from the vertex in ‘oasis’ in the case of scattering by a narrow impedance sector. One of the goals for the further studies is to develop a reliable numerical procedure in order to compute the spectral function uν(ω, ω0). Such a development could be based on our study of the problem for the spectral function given in this work. The situation with the impedance boundary conditions is more complex than that for the ideal ones. In particular, there is no reason to assert that the spectrum of the operator A from the Section 4.3 is discrete. Contrary to the case of the Laplace–Beltrami operator on the unit sphere with usual mixed (Robin) boundary conditions with complex impedance parameter (see Assier et al., 2016, Theorem 4.10) on the cut the argumentation based on the compact resolvent does not work in our spectral problem. However, there are some hopes that the study of the spectral properties of the non-local operator $${\mathscr {A}}$$ in the boundary condition of the problem (4.2) may be exploited in order to create an efficient numerical procedure to compute the spectral function uν(ω, ω0). This might be used in order to reduce the spectral problem with non-local operator in the boundary condition to that with simpler conditions like those complex Robin in Assier et al. (2016). It is also important to give a complete description of the far-field asymptotics. Some natural extensions of the ideas proposed in Lyalinov (2013) might be useful in this case. Acknowledgements The author is grateful to Dr Ning Yan Zhu (Stuttgart University) for his important comments on the subject of the work. It is also a pleasure to thank Prof. V.P. Smyshlyaev (University College of London) for his hospitality and fruitful discussions on diffraction by sectors during the visit of the author to UCL. The latter visit has been supported by the travel grant N 11.42.678.2017 of Saint Petersburg State University (SPbGU) for scientific collaboration. Funding The work was supported in part by the Russian Foundation of Basic Research, RFBR 17-01-00668a. Footnotes 1 It is worth remarking that in our study the use of closable sectorial forms and m-sectorial operators looks natural for the problem on the unit sphere with the cut. It has been partly inspired by a very useful work by Assier et al. (2016), where similar but simpler spectral problems for the Laplace–Beltrami operator with ideal boundary conditions on the cut have been considered. 2 The harmonic time-dependence $$\mathrm {e}^{-\mathrm {i}\widehat \omega t}$$ is assumed and suppressed throughout the paper. 3 The sign of χ± is not fixed; however, in the case χ± < 0, ε± = 0 surface waves are excited on the impedance surfaces of the sector and propagate to infinity without attenuation. 4  The asymptotics (2.8) fails provided the observation point approaches the boundary of Ω0, near the boundaries of the domains $$\varOmega _{i},\,\, i=0,r,A,B,\dots $$ some special transition functions apply to match the local asymptotics. 5 Usually these waves are neglected in comparison with the first two terms because they have higher order with respect to (kr)−1. 6 The Stirling asymptotics for the complex argument of the gamma-function is also exploited. 7 More accurate formulations will be given in the following section. 8 It is possible to describe the domain of TA more efficiently. 9 These singularities are of the branch point type, see also Lyalinov (2013). 10 For the first time this kind of formulae for the ideal circular cones has been obtained by Smyshlyaev, see e.g. Smyshlyaev (1990). 11 It is worth remarking that the identical result can be obtained by use of the matching the local asymptotic series, which is shown in Lyalinov & Zhu (2012), Chapter 5, for a convex impedance cone. 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Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com

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