Second‐order functional‐difference equations. II: Scattering from a right‐angled conductive wedge for E‐polarization

In part I, a new method for solving functional‐difference equations of the second order was proposed. The shift of the equation was assumed to coincide with the period of the coefficients. The method is based on the theory of the Riemann–Hilbert problem on a hyperelliptic surface and the Jacobi inversion problem. The procedure is applicable to any finite number of zeros of the discriminant of the equation in the strip. It yields the general single‐valued meromorphic solution. In the present paper, electromagnetic scattering by a right‐angled magnetically conductive wedge is analysed. The physical problem reduces to a second‐order difference equation with 2π‐periodic coefficients and with the shift π. A rigorous procedure for constructing the general solution is proposed. It consists of two steps. First, an auxiliary equation with the shift 2π and the period π is derived and solved by the method proposed in part I (the corresponding Riemann surface is a torus). Next, necessary and sufficient conditions for the solution of the auxiliary equation to satisfy the governing equation are derived. These conditions separate the general solution of the main equation from those solutions of the auxiliary equation which fail to satisfy the governing difference equation. In addition, the particular case of no branch points is analysed by the machinery of the Riemann–Hilbert problem for a segment on the complex plane. A high‐frequency asymptotic expression for the electric field is presented. Numerical results for the backscattering coefficient are reported.