Download Canonical Problems in Scattering and Potential Theory Part by S.S. Vinogradov, P. D. Smith, E.D. Vinogradova PDF

By S.S. Vinogradov, P. D. Smith, E.D. Vinogradova

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

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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.

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