Asymptotic Dipole Radiation Formulas 



By W, HOWARD WISE 



THE analysis of the radiation from dipoles as given by Sommerfeld 

 and by von Hoerschelmann is deficient in one respect: it does 

 not give the true ^ asymptotic expressions for the radiation leaving at 

 a considerable angle from the horizontal. The correct asymptotic 

 formulas have already been easily supplied by an appeal to the 

 Reciprocal Theorem; lately M. J. O. Strutt ^ has got them directly 

 from the boundary conditions and H. Weyl ^ has derived the correct 

 asymptotic formula for a vertical dipole at the surface of the earth 

 by a method quite different from Sommerfeld's. In the present 

 paper it is shown how they can be got by merely improving the rigor 

 of Sommerfeld's analysis. 



The present analysis begins with the formulas of von Hoerschelmann 

 for the wave potentials of vertical and horizontal dipoles at a finite 

 distance above the surface of the earth and generally follows Sommer- 

 feld. The derivation of an asymptotic approximation for the wave 

 potential of a vertical dipole is considerably different from Sommer- 

 feld's and results in the simpler and more precise formulas deduced 

 from the reciprocal theorem. 



Most of the analysis is somewhat simplified by taking the permea- 

 bility of the earth to be unity. 



The notation used is chiefly that of Bateman.'* 



T = variable of integration, throughout the paper. 



I — Vr^ — ^l^ nt = Vr^ — ki^. 



The subscripts 1 and 2 refer to air and ground respectively. 

 Ri, R2, a, p, (p, w, X, y and z are adequately defined by Fig. 1. 



cos d^ = x/R, cos By = yJR, cos ^^ = z/R, R^ = x^ -j- y^ -j- z 

 The wave potential of a horizontal dipole is ^' * 



1 See paragraph following equation (8). 



2 M. J. O. Strutt, Ann. d. Phys., Bd. 1, p. 721, 1929. 



3 H. Weyl, Ann. d. Phys., Bd. 60, p. 481, 1919. 



* "Electrical and Optical Wave Motion," pp. 73-75. 

 8H. V. Hoerschelmann, Jahrb. d. draht. Teleg., Bd. 5, 1912, pp. 14-188. 



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