## 1. Introduction

[2] The geometrical theory of diffraction (GTD) is the most widely used theory for analysis of the edge diffraction. The GTD in the context of the time harmonic theory [*Keller*, 1962] deals with high-frequency asymptotic expansions of electromagnetic fields scattered from structures. It requires the solution of canonical problems on scattering of radio waves in order to recover transmitted, surface, and diffracted waves or, equivalently, to evaluate transmission, surface and diffraction coefficients. High fidelity analytical methods are essential for solving the canonical problems. Solution of these problems by pure numerical schemes is not always simple, and it is often costly in computational time. Considerable insight to the problem can be obtained if the solution is carried out analytically.

[3] Reliable detection of targets, specification of antenna ground plates and radar scattering evaluation requires a further investigation into electromagnetic scattering by plates and wedges. The analysis of half-plane diffraction problems is generally carried out using the Wiener-Hopf technique or the Clemmow dual integral equations method [*Senior and Volakis*, 1995, chapter 3]. Model problems for the exterior of wedges with impedance faces are effectively treated by the *Maliuzhinets* [1958] technique. This method is based on the use of the classical Sommerfeld integral representation of the solution to the Helmholtz equation and requires the solution of certain difference equations. When the electric and magnetic fields are not coupled by the boundary conditions and the corresponding difference equations are scalar, the solution can be derived in terms of special functions (known as the Maliuzhinets functions). For penetrative wedges, when the electromagnetic field has to be recovered in the interior and the exterior of the domain, in general, the difference equations are of the second order. For an arbitrary wedge angle, an analytical technique is not available in the literature. For a special case of the second-order difference equation, by using bilinear Riemann relations for abelian differentials, a partial solution was analyzed by *Senior and Legault* [2000] and *Legault and Senior* [2002]. A multi-valued non-physical solution for some canonical diffraction problems on right-angled scatterers was proposed by *Demetrescu et al.* [1998a, 1998b].

[4] Recently, *Antipov and Silvestrov* [2004a, 2004b] developed a novel method for second-order difference equations with meromorphic periodic coefficients and applied it to a diffraction problem for a right-angled wedge. One of the sheets of the scatterer was a conductive surface, and the second one was perfectly conductive. This method is a two-step-procedure which requires first to reduce the model problem to a scalar RH problem on a Riemann surface [*Antipov and Silvestrov*, 2004c] (in particular cases, this results in two scalar RH problems on a complex plane). The general single-valued meromorphic solution to the RH problems is derived in terms of 2*π*-periodic meromorphic functions with specified zeros and poles. These functions are arbitrary for the solution of the RH problem but not free for the main difference equation. The second step of the method is to find these functions from certain extra conditions. In the case considered by *Antipov and Silvestrov* [2004b], the coefficients *l*_{1}(*t*) and *l*_{2}(*t*) of the RH problems are continuous on the contour and do not vanish. It turns out that for some boundary conditions the coefficients of the associated RH problems may have poles and zeros on the contour and therefore the functions log *l*_{j}(*t*) are multi-valued. The procedure by *Antipov and Silvestrov* [2004b] if remains unchanged does not work for this multi-valued case. The main aims of the current paper are as follows: (1) to analyze the diffraction problem for a right-angled scatterer formed by an electrically resistive half-plane and a perfectly magnetically conductive half-plane; (2) to develop further the theory of second-order difference equations of diffraction theory focusing on the case when the coefficients of the associated RH problems have zeros and poles on the contour; (3) to determine the far field asymptotic expansion of the electric field.

[5] It is known [*Senior and Volakis*, 1995, p. 53] that for an electrically resistive half-plane, the *E*_{z}-component of the electric field is continuous, and the *H*_{x}-component of the magnetic field is discontinuous:

where *R*_{e} denotes the surface resistivity, and [*f*]_{−}^{+} = *f*∣_{y=+0} − *f*∣_{y=−0}. The electromagnetic dual of an electrically resistive screen is a magnetically conductive one. The magnetically conductive sheet boundary conditions stipulate the continuity of the *H*_{x}-component and the discontinuity of the *E*_{z}-component [*Senior and Volakis*, 1995, p. 74]:

where *R*_{m} is the conductivity. In the case of normal incidence (the incident wave *E*_{z}^{i} is orthogonal to the *z*-axis), the *E*_{z}- and *H*_{x}-components are linked by

where *k*_{0} is the wave number, and *Z*_{0} is the intrinsic impedance of the medium. Therefore, if *R*_{e} = 0 (a perfectly electrically conductive sheet), then the *E*_{z}-component is continuous and vanishes on the faces of the screen while its normal derivative is discontinuous. In the case *R*_{m} = 0 (a perfectly magnetically conductive sheet), the function ∂*E*_{z}/∂*y* is continuous and equal to zero, while the *E*_{z}-component is discontinuous. If *R*_{e} = ∞ in the case (1) or *R*_{m} = ∞ in the case (2), then the sheet ceases to exist.

[6] The paper is organized as follows. In section 2, the canonical problem of diffraction by a right-angled wedge with electrically resistive and perfectly magnetically conductive surfaces is formulated. It is reduced to a certain difference equation in section 3. In the next section, to solve this difference equation we analyze an auxiliary second-order difference equation and convert it into two RH problems. For the particular case *Z*_{0}/*R*_{e} = 4/, the general solution to the auxiliary difference equation is derived in terms of some unknown meromorphic periodic functions. In section 5, we establish under which conditions the auxiliary equation is equivalent to the main difference equation. These conditions are used to find the unknown meromorphic functions from section 4. An asymptotic expansion of the *E*_{z}-component of the electric field far away from the junction of the screens is derived in section 6. Numerical results for the diffraction coefficient are also reported.