#### 2.1. Statement of the Problem and Decoupling of the Functional Difference Equations

[5] A wedge is placed in a circular cylindrical coordinate system (*r*, ϕ, *z*) in such a way that its edge coincides with the *z* axis and its faces are the half-planes ϕ = ±Φ with *π* > Φ > 0 (Figure 1). On the wedge faces the following boundary conditions must be met by the tangential components of the electric (*E*_{z}, *E*_{r}) and the magnetic (*H*_{z}, *H*_{r}) fields:

*Z*_{0} being the intrinsic impedance of the surrounding medium, *a*_{12}^{±} and *a*_{21}^{±} the normalized axially anisotropic surface impedances. The passivity requires that Re *a*_{12}^{±} ≥ 0 and Re *a*_{21}^{±} ≥ 0 hold good. Obviously, a scalar impedance wedge with *a*_{12}^{±} = *a*_{21}^{±} studied by *Lyalinov and Zhu* [2006] represents a special case of this work.

[6] Let a plane electromagnetic wave fall on this wedge. The *z* components of the incident field are given by (a time-dependence exp(−*iωt*) has been assumed, but is dropped in the remaining part of this work)

Here *k*_{0} stands for the wave number in the surrounding medium, the angles ϑ_{0} and ϕ_{0} characterize the incident direction of the plane wave, *U*_{10}/*Z*_{0} and *U*_{20} denote the amplitude of the magnetic and electric components along the edge, respectively.

[9] Equation (7) can be rewritten as

where

Auxiliary angles ϑ^{±} and *χ*^{±} have been introduced above according to

[10] Eliminating *f*_{2}(−*α*) from (8) one obtains a wholly decoupled FD equation for *f*_{1}(*α*),

After having obtained a solution for *f*_{1}(*α*) in the strip Π(−2Φ, 2Φ), *f*_{2}(*α*) in the same strip is given by, on use of (8),

The spectra (*α*) can be analytically extended to *α* outside of the basic strip Π(−2Φ, 2Φ) by virtue of (8).

#### 2.2. Simplified Difference Equation

[11] It is worth noting that there exists only one nonconstant coefficient at the left-hand side of (16), namely,

and this coefficient can be written in an equivalent form,

with the auxiliary function Ψ_{0}(*α*) given by

[13] In case of Φ > *π*/2, the two zeros of Ψ_{0}(*α*) inside the basic strip can be eliminated,

This new function *F*_{0}(*α*) is free of both zeros and poles in the basic strip.

#### 2.3. Fredholm Integral Equation of the Second Kind

[15] Expressing both the spectrum _{1}(*α*) and the right-hand side of (23) in their even and odd parts, _{1}(*α*) = _{1}^{e}(*α*) + _{1}^{o}(*α*), *H*_{1}(*α*) = *Q*_{1}(*α*)_{1}(−*α*) = *H*_{1}^{e}(*α*) + *H*_{1}^{o}(*α*), it is evident that (23) is identical to

To get the particular solution to the functional difference equation (25), which is equivalent to a system of equations,

we resort to the so-called S integral [*Tuzhlin*, 1973; *Buldyrev and Lyalinov*, 2001; *Babich et al.*, 2004] [see also *Lyalinov and Zhu*, 2003a, 2006].

[16] To find the particular solution to (24) by using the same procedure, it is advantageous to introduce a new odd function _{1}^{o}(*α*) via _{1}^{e}(*α*) = cot(*να*)_{1}^{o}(*α*). The respective difference equation follows from (24),

and can be dealt with in the same way as (25). Hence the particular solution is given by

The sought-for equivalent integral representation of (23) in the basic strip Π(−2Φ, 2Φ) reads

The nonintegral terms at the right-hand side recover the geometrical-optics poles and the asymptotic behavior of _{1}(*α*) at infinity.

[17] The constants *A*_{1}^{±} are determined by the radiation condition according to which the second spectrum *f*_{2}(*α*) given in the basic strip by (17) must be free of nonphysical poles. This leads to two additional conditions,

where

[18] As indicated clearly by (30), especially by the last term at its right-hand side (the particular solution), _{1}(*α*) in the basic strip depends upon its value along the imaginary axis of the complex plane. In line with (31), the same is true for the constants *A*_{1}^{±}. Therefore, for points on the imaginary axis of the complex *α* plane, (30), together with (31), amounts to a Fredholm integral equation of the second kind.

[19] The kernel of the integral equation (30), *Q*_{1}(−*t*)/cos[*ν*(*α* + *t*)], is free from singularity and does not contain the wave number *k*_{0} which may be a large parameter, as can be verified by recalling the definition of *Q*_{1}(*t*) and the properties of *F*_{0}(*t*) and *q*_{1}(*t*). Furthermore, this kernel decreases exponentially with ∣Im *t*∣, or more precisely,

as *t* ±i∞. It is precisely these properties that render the above described solution procedure a very efficient, that is, fast and accurate one; see section 3.

[20] From the asymptotic behavior of *Q*_{1}(*α*) and _{1}(*α*) it can be inferred that the kernel of (30) is square integrable. Hence its solvability follows from the uniqueness of solution to the present problem, which has been shown elsewhere [cf. *Lyalinov and Zhu*, 2003b, 2006].

#### 2.4. Uniform Asymptotic Solution

[21] Together with (31), (30) is solved by quadrature method, with the asymptotic behavior of _{1} accounted for explicitly [see *Lyalinov and Zhu*, 2003b, 2006]. Evaluating the Sommerfeld integrals (6) by the saddle-point method, a uniform asymptotic solution for the problem under study has been derived,

where the superscripts “go”, “sw” and “d” stand for the geometrical-optics part, the surface waves and diffracted waves, respectively. The formulae for these wave ingredients are given explicitly by *Lyalinov and Zhu* [2006], where now in the residues of the surface waves the following multiplication factors have to be used instead:

with

[23] On setting *a*_{12}^{±} = *a*_{21}^{±} = *η*^{±}, the above formulae reduce precisely to the ones for diffraction of a plane electromagnetic wave by a wedge with scalar impedance faces [*Lyalinov and Zhu*, 2006].