Diffraction by a Wedge or by a Cone with Impedance-Type Boundary Conditions and Second-Order Functional Difference Equations

This work reports some recent advances in diffraction theory by canonical shapes like wedges or cones with impedancetype boundary conditions. Our basic aim in the present paper is to demonstrate that functional difference equations of the second order deliver a very natural and efficient tool to study such a kind of problems. † To this end we consider two problems: diffraction of a normally incident plane electromagnetic wave by an impedance wedge whose exterior is divided into two parts by a semi-infinite impedance sheet and diffraction of a plane acoustic wave by a right-circular impedance cone. In both cases the problems can be formulated in a traditional fashion as boundary-value problems of the scattering theory. For the first problem the Sommerfeld-Malyuzhinets technique enables one to reduce it to a problem for a vectorial system of functional Malyuzhinets equations. Then the system is transformed to uncoupled second-order functional difference-equations (SOFDE) for each of the unknown spectra. In the second problem the incomplete separation of variables leads directly to a functional difference-equation of the second order. Hence, it is remarkable that in both cases the key † For a thorough and up-to-date overview of the scattering and diffraction in general the readers are referred to a special section of the journal “Radio Science” edited by Uslenghi [1]. 240 Zhu and Lyalinov mathematical tool is an SOFDE which is an analog of a second-order differential equation with variable coefficients. The latter is reducible to an integral equation which is known to be the most traditional tool for its solution. It has recently been recognised that reducing SOFDEs to integral equations is also one of the most efficient approaches for their study. The integral equations which are developed for the problems at hand are both of the second kind and obey Fredholm property. In the problem of diffraction by a wedge the generalised Malyuzhinets function is exploited on the preliminary step then “inversion” of a simple difference operator with constant coefficients leads to an integral equation of the second kind. The corresponding integral operator is represented as a sum of the identical operator and a compact one [2]. However, in the second problem the situation is slightly different: the integral operator can be represented by a sum of the boundedlyinvertible (Dixon’s operator) and compact operators. This situation was earlier considered by Bernard in his study of diffraction by an impedance cone, and important advances have been made (see [3–6]). The Fredholm property is crucial for the elaboration of different numerical schemes. In our cases we exploited direct numerical approaches based on the quadrature formulae and computed the farfield asymptotics for the problems at hand. Various numerical results are demonstrated and discussed. 1. DIFFRACTION BY AN IMPEDANCE WEDGE WITH A SEMI-INFINITE IMPEDANCE SHEET ATTACHED TO ITS EDGE This section deals with diffraction of a normally incident plane wave in a wedge-shaped region which consists of an impedance wedge and a semi-infinite impedance sheet joint at the edge of the wedge. The looked-for fields on both sides of the impedance sheet are expressed in terms of the Sommerfeld integrals. Inserting the Sommerfeld integrals into the boundary conditions on the faces of the wedge and across the impedance sheet, and inverting the resultant expressions, one obtains a system of difference equations for the spectra. Eliminating one of the spectra leads to a functional difference-equation of higher order. The latter can be converted by means of a recently developed technique for second-order functional difference-equations into an equivalent integral expression. For points located on the imaginary axis of the complex angle α the integral expression turns out to be an integral equation of the second kind which permits an efficient solution by use of quadrature Progress In Electromagnetics Research B, Vol. 6, 2008 241 method. From evaluation of the Sommerfeld integrals by virtue of the saddle-point method a first-order uniform asymptotic solution follows. The results to be given below follow the same line as [2, 7]. Therefore, the following exposition is confined to main steps and the readers are referred to [2, 7] for details. 1.1. Statement of the Problem Figure 1 depicts the scattering obstacle. For convenience, a cylindrical co-ordinate system (r, φ, z) is chosen in such a way that the edge of the wedge and one rim of the semi-infinite impedance sheet coincide with the z-axis, the wedge faces and the impedance sheet are half-planes given by φ = ±Φ and φ = Φ0 with 0 < Φ ≤ π. In the following, it is assumed that 0 ≥ Φ0 ≥ −Φ. Figure 1. Diffraction of a normally incident plane wave in a wedgeshaped region. A plane wave impinges perpendicularly on the edge from the direction φ = φ0 with Φ0 < φ0 < Φ assumed in this study. The alternative case can be dealt with analogously. The incident electric field oscillates along the edge of the wedge-shaped region and is given by (a time-dependence e−iωt is assumed and suppresed in this section) E z (r, φ) = E0 exp [−ikr cos(φ− φ0)] , (1) where E0 is the amplitude of the incident wave. 242 Zhu and Lyalinov This incident wave will be scattered by the obstacle. Owing to the translational symmetry of both the incident wave and the wedgeshaped region with respect to the z-axis, the boundary value problem is a two-dimensional one and the total electric field has only one component Ez(r, φ). In the region surrounding the wedge and the sheet, Ez obeys the two-dimensional Helmholtz equation. On the faces of the wedge φ = ±Φ, the boundary conditions to be met by Ez read 1 r ∂ ∂φ Ez ∓ i k η± Ez = 0, (2) η±Z0 being the surface impedance of the upper (lower) wedge face and Z0 the intrinsic impedance of the surrounding medium. The electric property of the impedance sheet is given by ys/Z0, the shunt admittance. On the impedance sheet at φ = Φ0, Ez is subjected to the semi-transparency conditions [8]: [ 1 r ∂ ∂φ Ez ]+ − [ 1 r ∂ ∂φ Ez ]− + i k ys 2 [E z + E − z ] = 0, [E z − E− z ] = 0. (3) Furthermore, Ez must satisfy the Meixner’s edge condition Ez = O(1) as r → 0 and the radiation conditions [9] (see also [10]). 1.2. Higher-Order Functional Difference-Equation According to Sommerfeld, the solution can be constructed through a superposition of plane waves