By Slavenas P.
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Extra resources for A Possible Way to Discuss the Fundamental Principles of Relativity
1. 231) and ∞ V = 2n + 1 p sin φ 4π dr n=1 n (n + 1) In a similar way the Debye potentials for the magnetic horizontal dipole are ∞ U (m) = 2n + 1 m cos φ 4π dr n=1 n (n + 1) ζn (kd) ψn (kr) , r < d ψn (kd) ζn (kr) , r > d Pn1 (cos θ) , (1. 232) © 2002 by Chapman & Hall/CRC ∞ V (m) = 2n + 1 im sin φ 4π dr n=1 n (n + 1) ζn (kd) ψn (kr) , r < d ψn (kd) ζn (kr) , r > d Pn1 (cos θ) . (1. 233) When the electric dipole is located at (0, 0, −d) in the lower half-space n+1 , and (z < 0) the general term in (1.
The electromagnetic field components radiated by the vertical electric or magnetic dipole may be calculated from the formulae (1. 55). An alternative way is to obtain the “generating” functions Hφ (for TM waves) and Eφ (for TE waves) and use the formulae (1. 86)–(1. 87). The “generating” functions are extracted from formulae (1. 55), (m) Hφ = ik ∂Πz ∂Πz , Eφ = −ik . ∂ρ ∂ρ From the representation (1. 197) we obtain the desired result ∞ 1 J (νρ) −√ν 2 −k2 |z−d| −ik Hφ √1 = e dν (1. 200) Eφ ik 4π 0 ν 2 − k2 √ where Im ν 2 − k 2 ≤ 0.
This representation of acoustical and electromagnetic point sources plays an important part in the construction of solutions to wave-scattering problems. Thus, we search for solutions to the inhomogeneous Helmholtz equation with the Dirac δ-function, → − − r −r ; ∇2 G + k 2 G = −δ → (1. 158) → − → the solution G − r , r is known as a Green’s function in free space. ) It can be readily verified that in three-dimensional space the solution of (1. 158) that → − → is radially symmetric about − r = r is given by − → − → G3 ik r − r → − 1 e − → r,r = → .