By S.S. Vinogradov, P. D. Smith, E.D. Vinogradova
Even supposing the research of scattering for closed our bodies of straightforward geometric form is easily constructed, constructions with edges, cavities, or inclusions have appeared, beforehand, intractable to analytical equipment. This two-volume set describes a step forward in analytical suggestions for appropriately making a choice on diffraction from periods of canonical scatterers with comprising edges and different advanced hollow space gains. it really is an authoritative account of mathematical advancements during the last 20 years that gives benchmarks opposed to which ideas bought by way of numerical equipment might be verified.The first quantity, Canonical buildings in power idea, develops the maths, fixing combined boundary capability difficulties for constructions with cavities and edges. the second one quantity, Acoustic and Electromagnetic Diffraction via Canonical buildings, examines the diffraction of acoustic and electromagnetic waves from a number of periods of open constructions with edges or cavities. jointly those volumes current an authoritative and unified therapy of capability idea and diffraction-the first entire description quantifying the scattering mechanisms in advanced buildings.
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Additional info for Canonical Problems in Scattering and Potential Theory Part 2: Acoustic and Electromagnetic Diffraction by Canonical Structures
To avoid confusion with the formulae in , note that we use the harmonic time dependence exp (−iωt) instead of exp (+iωt)). In terms of the “generating” functions Hφ or Eφ the electromagnetic field of the vertical electric (magnetic) dipole uses formulae (1. 201) and the represen→ − tation of the electric (magnetic) z-component of Hertz vector Π in spherical coordinates, where → − − → → − Π = ir G3 cos θ + iθ (−G3 sin θ) ; (1. 203) © 2002 by Chapman & Hall/CRC recall that G3 = 1 eikR , 4π R (1.
Analogously, when the magnetic horizontal dipole is located at the same point in the lower n space, the general term in (1. 232) is multiplied by the factor (−1) and the n+1 . general term in (1. 233) is multiplied by the factor (−1) To obtain the Debye potentials UP HS and VP HS , describing the electromagnetic field of a point Huygens source, first set the dipole strengths to be equal: p = m = α. The potentials are derived by linear superposition of the representations (1. 230)–(1. 233): UP HS = αk cos φ αk sin φ Φ (r, θ) , VP HS = Φ (r, θ) , 4πr 4πr (1.
108]), → − − r,r = G3 → ∞ √ 2 2 2 f (ν, µ) cos µ (y − y ) e− ν +µ −k |z−z | dνdµ ∞ cos ν (x − x ) 0 0 (1. 5). Furthermore, f (ν, µ) is an unknown function of two spectral parameters ν and µ, to be determined. Substitute this form into equation (1. 158), and integrate with respect to z over a small interval (z − ε, z + ε) about z . Remembering the continuity of the solution at z = z and passing to the limit ε → 0, we obtain ∂ G3 (x, y, z; x , y , z ) ∂z z=z +0 = −δ (x − x ) δ (y − y ) (1. 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 ) .