By S.S. Vinogradov, P. D. Smith, E.D. Vinogradova
Pt. 1. Canonical buildings in strength idea -- pt. 2. Acoustic and electromagnetic diffraction by means of canonical constructions
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Extra resources for Canonical Problems in Scattering and Potential Theory Part II
1. 309) n=0 Our aim is to show that the Fredholm equation of the first kind (1. 299) is equivalent to the dual series equations (1. 307)–(1. 308). Because the jump function ∂U ∂U σD (θ , φ ) on S0 ψ (θ , φ ) = − = (1. 310) 0 on S1 ∂r r=a−0 ∂r r=a+0 vanishes on S1 , the surface of integration in equation (1. 299) may be extended to the whole of the spherical surface S = S0 ∪ S1 (given by r = a), so that 2π a2 π dθ sin θ ψ (θ , φ ) G3 (a, θ, φ; a, θ , φ ) = eika cos θ , dφ 0 0 θ ∈ (0, θ0 ) (1. 311) where the kernel G3 (a, θ, φ; a, θ , φ ) is given by formula (1.
285) uniformly with respect to direction as r → ∞; in the two-dimensional case √ the factor r occurring in (1. 284) and (1. 285) is replaced by r. For smooth closed scatterers, the enforcement of these boundary and radiation conditions is adequate to ensure that a unique solution to the scattering problem at hand exists (see , ). However if the scattering surface has singular points, or is open, further conditions to guarantee uniqueness must be imposed. © 2002 by Chapman & Hall/CRC The distinctive feature of open thin-walled cavities is the presence of sharp edges.
These conditions mean that in two- (respectively, three-) dimensional space the scattered field must behave as an outgoing cylindrical (respectively, spherical) wave at very large distances from the scatterer. The minus sign in both formulae is replaced by a plus sign if the time harmonic dependence is changed from exp (−iωt) to exp (+iωt) . The corresponding conditions for the three-dimensional electromagnetic case are → − → − r E < K, r H < K (1. 284) and → − → − − → → − → → r E + Z0 − r × H → 0, r H − Z0−1 − r ×E →0 (1.