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Extra resources for 2-Designs and a differential equation
198] of Part I ), and the solutions of inhomogeneous Helmholtz equation (1. 159), (1. 160) are transformed, in the three-dimensional case, to → − 1 1 − G3 → r,r = → − − 4π → r −r and, in the two-dimensional case, to → − i − G2 → r,r =− log 4π → − − → r −r . Let us consider the solution (1. 159) under the condition that k is nonzero, → − → − → → but − r − r → 0. Then if − r −r 1, G3 → − → 1 1 − − 1 → − r − r +O r,r = + ik − → → − − → 4π 2 r −r − → → − r −r 2 . (1. 161) This means that in the vicinity of the point source the electromagnetic field is of a static nature.
320) leads to the dual integral equations ∞ g (ν) J0 (νρ) dν = −1, ρ < a (1. 321) 0 ∞ ν 2 − k 2 g (ν) J0 (νρ) dν = 0, 0 © 2002 by Chapman & Hall/CRC ρ>a (1. 322) for the function g. The jump function ∂U ∂z ψ (ρ , φ ) = − z=+0 ∂U ∂z σD (ρ , φ ) , on S0 0, on S1 = z=−0 (1. 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.
255) that applies for both external and internal boundary value problems. 3 [right]). Both problems now are mixed and we refer to them as mixed boundary value problems for the Helmholtz equation. Let us formulate the mixed boundary conditions. When S0 (or Γ0 ) is acoustically hard the total fields Uet and Uit in the exterior and interior regions, respectively, satisfy ∂Uet ∂Uit = = 0 on S0 (Γ0 ) , ∂n ∂n Uet = Uit on S1 (Γ1 ) , © 2002 by Chapman & Hall/CRC (1. 256) (1. 4 The boundary separating two media with differing electromagnetic properties.