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Electromagnetic wave scattering by a circular impedance cone

Electromagnetic wave scattering by a circular impedance cone

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This chapter is devoted to the study of electromagnetic scattering of a plane wave by a circular cone with impedance boundary conditions on its surface. The technique developed in the previous chapter is extended and applied to the electromagnetic diffraction problem with the aim to compute the far field. By means of the Kontorovich-Lebedev integral representations for the Debye potentials and a 'partial' separation of variables, the problem is reduced to coupled functional-difference equations for the relevant spectral functions. For a circular cone the functional-difference equations are then further reduced to integral equations, which are shown to be Fredholm-type equations. We then solve numerically the integral equations. The radar cross section in the domain which is free from both the reflected and the surface waves has been computed numerically. Certain useful further integral representations for the solution of 'Watson-Bessel' and Sommerfeld types are developed, which give, as in acoustic case, a theoretical basis for subsequent calculation of the far-field (high frequency) asymptotics for the diffracted field. We also discuss the asymptotic expressions for the surface waves propagating from the conical vertex to infinity. While the problem we are considering is physically sound, from the mathematical viewpoint it is particularly attractive as containing various earlier studied models, both with ideal and nonideal boundaries, as particular or limiting cases. At the same time one has to deal with some novel effects of coupling, which provides in total a good test ground for advancing appropriate technical tools, both analytical and numerical, as we are aiming in this chapter.

Chapter Contents:

  • 6.1 Formulation and reduction to the problem for the Debye potentials
  • 6.1.1 The far-field pattern
  • 6.1.2 The Debye potentials
  • 6.1.3 Boundary conditions for the Debye potentials
  • 6.2 Kontorovich-Lebedev (KL) integrals and spectral functions
  • 6.2.1 KL integral representations
  • 6.2.2 Properties of the spectral functions
  • 6.2.3 Boundary conditions for the spectral functions
  • 6.2.4 Verification of the boundary and other conditions
  • 6.2.5 Diffraction coefficients
  • 6.3 Separation of angular variables and reduction to functional-difference (FD) equations
  • 6.4 Fredholm integral equations for the Fourier coefficients
  • 6.4.1 Reduction to integral equations
  • 6.4.2 Comments on the Fredholm property and unique solvability of the integral equations
  • 6.5 Electromagnetic diffraction coefficients in M' and numerical results
  • 6.5.1 Numerical Solution
  • 6.5.2 Numerical examples
  • 6.6 Sommerfeld and Watson-Bessel (WB) integral representations
  • 6.6.1 Sommerfeld integral representations
  • 6.6.2 Regularity domains for the Sommerfeld transformants
  • 6.6.3 Diffraction coefficients and Sommerefeld transformants
  • 6.7 The diffraction coefficients outside the oasis as ω ∊ M''
  • 6.8 Problems for the Sommerfeld transformants and some complex singularities
  • 6.8.1 Problems for the Sommerfeld transformants
  • 6.8.2 Local behavior of the Sommerfeld transformants near complex singularities
  • 6.9 Asymptotics of the Sommerfeld integrals and the electromagnetic surface waves
  • 6.9.1 Derivation of the functionals C0u(n)
  • 6.9.2 Some comments on the asymptotics uniform with respect to the direction of observation

Inspec keywords: Fredholm integral equations; functional equations; electromagnetic wave scattering; electromagnetic wave propagation; difference equations; radar cross-sections; electromagnetic wave diffraction; electromagnetic wave reflection; electromagnetic coupling; electric impedance

Other keywords: electromagnetic diffraction problem; coupling effect; spectral function; impedance boundary condition; reflected wave; plane wave; radar cross section; integral equation; asymptotic expression; conical vertex; Sommerfeld solution; Debye potential; circular impedance cone; functional-difference equation; surface wave propagation; Watson-Bessel solution; electromagnetic wave scattering; Kontorovich-Lebedev integral representation; Fredholm type equation

Subjects: Differential equations (numerical analysis); Electromagnetic wave propagation; Numerical approximation and analysis; Integral equations (numerical analysis); Electromagnetic waves: theory; Nonlinear and functional equations (numerical analysis)

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