Separation of the Helmholtz equation 1a2y(p,中, +k2y(p,p,z)=0(D.68) y(p,φ,3)=P(p)(φ)Z(z) (D.69) d-P(p) 1 dP(p) P(p)=0 dp a2Φ 2-+k中(p)=0 (D72) d-z(z) kaz dz2 z()=14:F1(k2x)+B2F2(k2),k≠0 k=0. Φ(φ) AφF1(k)+BF2(中),k≠0, φ+b ap Inp+be P(p) k=0 and ko≠0 Ap GI(ep)+ BeG2(kcp), otherwise F1(),F2()= 人() G1(),G2()= (D.78) Spherical coordinate system l=r,0≤r<∝ U=6,0≤6≤丌 (D.80) φ,-π≤φ≤丌 @2001 by CRC Press LLCSeparation of the Helmholtz equation 1 ρ ∂ ∂ρ  ρ ∂ψ(ρ, φ,z) ∂ρ  + 1 ρ2 ∂2ψ(ρ, φ,z) ∂φ2 + ∂2ψ(ρ, φ,z) ∂z2 + k2 ψ(ρ, φ,z) = 0 (D.68) ψ(ρ, φ,z) = P(ρ)(φ)Z(z) (D.69) k2 c = k2 − k2 z (D.70) d2P(ρ) dρ2 + 1 ρ d P(ρ) dρ + k2 c − k2 φ ρ2  P(ρ) = 0 (D.71) ∂2(φ) ∂φ2 + k2 φ(φ) = 0 (D.72) d2Z(z) dz2 + k2 z Z(z) = 0 (D.73) Z(z) = Az F1(kzz) + Bz F2(kzz), kz = 0, azz + bz, kz = 0. (D.74) (φ) = Aφ F1(kφφ) + Bφ F2(kφφ), kφ = 0, aφφ + bφ, kφ = 0. (D.75) P(ρ) =    aρ ln ρ + bρ, kc = kφ = 0, aρρ−kφ + bρρkφ , kc = 0 and kφ = 0, AρG1(kcρ) + BρG2(kcρ), otherwise. (D.76) F1(ξ), F2(ξ) =    e jξ e− jξ sin(ξ) cos(ξ) (D.77) G1(ξ), G2(ξ) =    Jkφ (ξ) Nkφ (ξ) H(1) kφ (ξ) H(2) kφ (ξ) (D.78) Spherical coordinate system Coordinate variables u = r, 0 ≤ r < ∞ (D.79) v = θ, 0 ≤ θ ≤ π (D.80) w = φ, −π ≤ φ ≤ π (D.81)