Pt. 1. Canonical constructions in capability thought -- pt. 2. Acoustic and electromagnetic diffraction by means of canonical constructions

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Extra info for Canonical Problems in Scattering and Potential Theory Part II

Example text

163) z=z −0 or ∞ ∞ cos ν (x − x ) 2 0 f (ν, µ) ν 2 + µ2 − k 2 cos µ (y − y ) dνdµ 0 = δ (x − x ) δ (y − y ) . (1. 164) Two applications of the Fourier cosine integral transform to equation (1. 164) show that 1 1 . (1. 165) f (ν, µ) = 2 2 2π ν + µ2 − k 2 → − → Thus, the function G3 − r , r is representable as → − − r,r = G3 → 1 2π ∞ ∞ cos ν (x − x ) 0 0 © 2002 by Chapman & Hall/CRC cos µ (y − y ) ν 2 + µ2 − k 2 √ 2 2 2 e− ν +µ −k |z−z | dνdµ (1. 166) where Im ν 2 + µ2 − k 2 ≤ 0. We may interpret this representation as an integrated spectrum of propagating and evanescent waves.

323) vanishes on S1 ; so we may extend the surface of integration in equation (1. 299) to the whole plane (z = 0) so that 2π π dφ 0 dρ ρ ψ (ρ , φ ) G3 (ρ, φ, 0; ρ , φ , 0) = 1, ρ < a (1. 324) 0 where the kernel G3 (ρ, φ, 0; ρ , φ , 0) is given by formula (1. 175) with z = z = 0, G3 (ρ, φ, 0; ρ , φ , 0) = 1 4π where Im series √ ∞ ∞ 0 2 − δm cos m (φ − φ ) 0 m=0 Jm (νρ ) Jm (νρ) √ νdν ν 2 − k2 (1. 325) ν 2 − k 2 ≤ 0. Now represent the function ψ (ρ , φ ) by the Fourier ∞ ∞ 2 − δs0 cos sφ ψ (ρ , φ ) = Gs (µ) Js (µρ ) dµ 0 s=0 ∞ +2 ∞ sin sφ Fs (µ) Js (µρ ) dµ.

2002 by Chapman & Hall/CRC When λ/l 1 low-frequency scattering that is a perturbation of the incident wave occurs and is referred to as Rayleigh scattering. When the wavelength λ is comparable to the characteristic dimension of the obstacle (λ ∼ l) one or several diffraction phenomena dominate; this region is also called the resonance region. The high-frequency region or quasi-optical region is characterised by λ l. To a greater or lesser extent, Rayleigh-scattering can be studied by various perturbation methods, and high-frequency scattering can be studied by well-developed high-frequency approximate techniques.