This chapter presents an exact solution to diffraction of a skew-incident plane electromagnetic wave by a wedge with axially anisotropic impedance faces. Applying the Sommerfeld- Malyuzhinets technique to the boundary-value problem under study yields a coupled system of difference equations for the spectra. On elimination, a difference equation of higher order for one spectrum arises. After simplification in terms of a generalized Malyuzhinets function and accounting for Meixner's edge condition as well as the poles and residues of the spectrum in the basic strip of the complex plane, the functional difference equation is converted, via the S-integrals, to an integral equivalent. For points on the imaginary axis which is inside the basic strip the integral equivalent becomes a Fredholm equation of the second kind with a nonsingular, wavenumber-free and exponentially decreasing kernel. Solving this integral equation by the quadrature method the spectrum can be determined by integral extrapolation and by analytical continuation. A first-order uniform asymptotic solution follows from evaluating the Sommerfeld integrals with the saddle-point method. Comparison with available exact solutions in several special cases shows that this approach leads to a fast and accurate solution of the problem under study.

Chapter Contents:

• 2.1 Introduction
• 2.2 Statement of the problem and uniqueness
• 2.2.1 Statement of the problem
• 2.2.2 On uniqueness of a solution
• 2.3 Sommerfeld integral and functional equations
• 2.4 A functional difference equation of higher order
• 2.4.1 A difference equation for one spectrum
• 2.4.2 The generalized Malyuzhinets function χΦ(α)
• 2.4.3 Simplifying the functional difference equation of higher order
• 2.5 Second-order functional difference equation and Fredholm integral equation of the second kind
• 2.5.1 An integral equivalent to the difference equation
• 2.5.2 Determining the constants C ± 1 l
• 2.5.3 Fredholm integral equation of the second kind
• 2.6 Uniform asymptotic solution
• 2.6.1 Poles and residues
• 2.6.2 First-order uniform asymptotics
• 2.7 Numerical results
• 2.7.1 Numerical computation of the spectra
• 2.7.2 Examples
• 2.8 Appendix: Computation of the generalized Malyuzhinets function
• 2.8.1 Numerical integration
• 2.8.2 Series representation

Inspec keywords: electromagnetic wave diffraction; boundary-value problems; difference equations

Other keywords: Sommerfeld-Malyuzhinets technique; saddle-point method; Meixner edge condition; axially anisotropic impedance; boundary-value problem; S-integrals; Fredholm equation; coupled system; skew-incident plane electromagnetic wave diffraction; functional difference equation

Subjects: Function theory, analysis; Electromagnetic wave propagation; Electromagnetic waves: theory; Mathematical analysis