PHYSICAL REALITY OF ZENNECK'S SURFACE WAVE 37 



The left hand side of (8) has been altered to correspond with the 

 facts as now known. 



P is the wave-function for a surface wave which at great distances 

 behaves like Zenneck's plane surface wave. 



The series (5) and (6) constitute the complete wave-function for 

 a unit vertical dipole centered on the interface between air and ground. 



The series (8) and (9) are the asymptotic expansions of (5) + P/2 

 and (6) - P/2. 



The series (5), (6), (8) and (9) are exact and it is from them that the 

 attenuation charts in a paper by C. R. Burrows in this issue of the Bell 

 System Technical Journal were computed. 



Since interchanging k\ and k^ in (5) gives (6) and interchanging ki 

 and ^2 in (8) gives (9) but interchanging ki and k^ in P changes its 

 sign it follows that if (6) ~ (9) + P/2 then (5) ~ (8) - P/2. Hence 

 the complete wave-function H, = (5) + (6) ~ [(8) - P/2] + [(9) 

 -\- P/2] = (8) -f- (9) and P does not appear in the asymptotic expan- 

 sion of the wave-function. 



The series (5) and (6) have been computed and found to be respec- 

 tively equal to (8) — P/2 and (9) -f P/2.* These computations show 

 again that Hz = (5) -+- (6) ~ (8) + (9) or putting it in words, that 

 there is no surface wave wave-function P in the asymptotic expansion 

 of the complete wave-function. 



As a further check S. O. Rice has derived the series (5) and (6) in 

 an entirely different manner and verified that their asymptotic expan- 

 sions are indeed ^o — P/2 and Q^ + P/2. 



In order to get a direct numerical check on the series the wave- 

 function integral was computed by mechanical quadrature for two 

 cases. Van der Pol's transformation of the wave-function integral 

 with the path of integration deformed upward along the lines Im{ihru) 

 constant was used.^ 



1. With r/X = l/47r and e — i2c\<T = 12.5 — i 12.5 mechanical 

 quadrature gave IIj = (.800 — i .578)/r while the series (5) and (6) 

 gave (.9247 - i .4334)/f and (- .1242 - i .1438)/r respectively which 

 add up to (.8005 — i .5772)/r. This is a good check on the series 

 (5) and (6). 



2. With r/X = 50 and e - i2c\a = 80 - i .7512 mechanical quad- 

 rature gave n^ = (.094 — i .178) jr while the serie s (8) and (9) 

 gave <3o ~ (.086 - i A87)/r and Qz « 1.2 X 10"" [u^/r. Since 

 P = (4.47 — i 1.92)/r there can be no doubt that it must be omitted 

 in computing Hz asymptotically. This is a good check on the above 

 stated relation U, = (5) + (6) ~ (8) -f (9) or H^ ~ (2o + Q2. Be- 

 cause the asymptotic series ^0 here starts to diverge at the third term 



* Eq. (1) in paper 4 says that (5) ~ (8) - P/2. 



