448 BELL SYSTEM TECHNICAL JOURNAL 



We wish to acknowledge the helpful suggestions which have been 

 received in the course of this work from Messrs. H. T. Friis, J. C. 

 Schelleng, and M. E. Strieby. Valuable advice on some of the mathe- 

 matical questions encountered has been received from Mr. T. C. Fry. 



Appendix 

 The following formulas for the power radiated by transmission 

 lines were obtained by the conventional method of postulating the 

 current distribution, calculating the electromagnetic fields and from the 

 fields, the associated radiation by means of Poyn ting's theorem. As 

 an independent check the same current distribution was postulated and 

 the radiated power calculated following the methods of Pistolkors^^ and 

 Bechman.^^ 



Case I 



An approximation for a line terminated in its characteristic im- 

 pedance is a balanced two-wire line carrying a non-attenuated traveling 

 wave. For this case the power radiated is: 



Pi = 120/2 



log, (2L) - Ci{2D + ^^^2Zr ^ ^'^^^^^ ~ ^ 



,,..,,, , sin A sin (VL^ -\- A^ - U ^ sin (VL^ + A"" + L) 

 2Ct{A) -\- 



A 2VL2 -f A^ 



watts, (Ij 



-f- Cii^U -\- A'~ - L) -{- Ci{<U + ^2 ^ L) 



in which 



. lira 



_27rZ 



^ "X' 



- = line spacing in wave-lengths, 



A 



- = line length in wave-lengths, 



A 



/ = r.m.s. value of current in each wire, and 

 Ci{ ) = cosine integral. ^^ 



The equation simplifies considerably if it is assumed that a/\ is 



small so that: 



sin ^ ^ A and L> y4, 



" A. A. Pistolkors, Proc. L R. E., p. 562, March, 1929. 

 22 R. Bechman, Proc. L R. E., p. 461, March, 1931. 

 ^^ See Jahnke und Emde, "Funktionentafehi." 



