Series for the Wave Function of a Radiating Dipole at the 



Earth's Surface 



By S. O. RICE 



In this paper three series expansions are derived for the wave 

 function of a vertical dipole placed at the surface of a plane earth. 

 Two convergent series and one asymptotic series are obtained. A 

 remainder term for the latter series is given which enables one to set 

 an upper limit to the amount of error obtained by stopping at any 

 particular stage in the series. 



Introduction 



THE wave function above the earth of a vertical dipole placed at 

 the surface of a plane earth is ^ 



ni(r, Z) = [ki- + k2-) I TO,., , A , , , /T^r^ ^T^ ' ^^^ 



Jo ^2-V^^ - ki^ + ^i2V^2 _ ^^2 



where r and s are the horizontal and vertical distances from the dipole. 

 ki and k2 are constants depending upon the electrical properties of the 

 air and ground, respectively.^ We shall be concerned with the value 

 of this function at the surface of the earth. Setting 2 = gives us 

 an integral for Tli{r, 0) which is the function of r to be investigated 

 here. 



Although the electric and magnetic intensities are the properties of 

 an electromagnetic field which have the greatest physical significance, 

 writers on this subject often deal with the wave function because of 

 its simpler form and because in many cases of practical interest it is 

 nearly proportional to the electric intensity. However, the electro- 

 magnetic field may be obtained from the wave function by differ- 

 entiation. If the real parts of iJe~*"* and £e"'"' represent the electric 

 and magnetic intensities the field above the earth produced by the 

 dipole is 



dUi(r, z) 



Hr = Hz = 0, H^ = — 



dr 



ic' dm^jr, z) ic^^l d / dU,(r,z) 



drdz ' IP ' 0} r dr\ dr 



1 A. Sommerfeld, Ann. der Physik, vol. 28, pp. 665-736, No. 4 (1909). 



2 The symbols used here are defined in a list at the end of the paper. 



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