By S.S. Vinogradov, P. D. Smith, E.D. Vinogradova
Pt. 1. Canonical buildings in power thought -- pt. 2. Acoustic and electromagnetic diffraction through canonical constructions
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Extra info for Canonical Problems in Scattering and Potential Theory Part II
Example text
1) (2) The constants Cm and Cm are known as polarisation constants. The argument above is quite general and is employed in the formulation and analysis of vectorial scattering problems for other structures such as the circular disc, the hollow finite cylinder, etc. In order to find physically reasonable solutions the so-called Sommerfeld radiation conditions must be imposed. , |rψ| < K (1. 280) for some constant K, and r ∂ψ − ikψ ∂r →0 (1. 281) uniformly with respect to direction as r → ∞. In two-dimensional problems conditions (1.
143) n=m © 2002 by Chapman & Hall/CRC For the prolate spheroidal system (ξ, η, φ), introduced in equation (1. 92) the scalar wave equation (1. 144) ∆ + k2 U = 0 takes the form ∂ ∂ξ ξ2 − 1 ∂ ∂U + ∂ξ ∂η + 1 − η2 ∂U ∂η ξ 2 − η2 ∂2U + c2 ξ 2 − η 2 U = 0 (1. 145) (ξ 2 − 1) (1 − η 2 ) ∂φ2 where c is a dimensionless parameter proportional to the ratio of focal distance d to the wavelength λ : c = k d2 = π λd . The method of separation of variables provides solutions of the form Umn = Rmn (c, ξ) Smn (c, η) cos mφ sin mφ where m = 0, 1, 2, .
152) The radial prolate spheroidal functions of first, second, third and fourth kind are defined as the solutions of the equation (1. 146) at λ = λmn (c), over the range 1 ≤ ξ < ∞, that possess the following asymptotics 1 cξ 1 (2) Rmn (c, ξ) = cξ 1 (3) Rmn (c, ξ) = cξ 1 (4) Rmn (c, ξ) = cξ (1) Rmn (c, ξ) = π −2 (n + 1) + O (cξ) 2 π −2 sin cξ − (n + 1) + O (cξ) 2 π −2 exp i cξ − (n + 1) + O (cξ) 2 π −2 exp −i cξ − (n + 1) + O (cξ) 2 cos cξ − (3) (1) (1. 153) (2) as cξ → ∞, respectively. It is obvious that Rmn (c, ξ) = Rmn (c, ξ)+iRmn (c, ξ) (4) (1) (2) and Rmn (c, ξ) = Rmn (c, ξ)−iRmn (c, ξ).