534 BELL SYSTEM TECHNICAL JOURNAL 



force f{x) and the potential F(x) impressed at point x on the wave 

 antenna by the radio waves are given by the equations 



fix) = fix, d) cos 9, (107) 



Fix) = Fix, 6). (108) 



It is convenient to take one end, say x = 0, as a fixed reference point, 

 and then to express fix, 6) and Fix, 6) in terms of their values /(O, 6) 

 and F(0, 6) Sit X = 0. For this purpose, it will be assumed that the 

 radio waves are propagated in a simple exponential manner, so that 



fiO,d) FiO,d) ' ^'"^^^ 



r denoting the propagation constant of the radio waves, per unit length 

 measured horizontally along the direction of their propagation. Then 

 the equations (107) and (108) become 



fix) = /(O, 9) cos d e-r^cos9^ (110) 



Fix) = FiO, 0)g-i'^«°«^ (111) 



wherein /(O, 6) and 7^(0, d) may be supposed known. In this connec- 

 tion it should be remarked that /(O, 6) and F(0, 6) — and, more gen- 

 erally, fix, 6) and Fix, 0)— are not in phase.^* 



On substitution of (110) and (111), equations (49), ••• (53) now 

 become applicable for calculating the current /(x) at any point x of 

 the wave antenna; this current will be written /(x, 6) because it 

 depends on the incidence-angle 6, even when fix, 6) and Fix, 9) are 

 independent of 9. _ In the engineering of wave antennae, we are 

 usually concerned merely with the current lis, 9) received at the end 

 X = s. In general there will be four constituents of 7(5, 9), corre- 

 sponding to equations (50), (51), (52), (53) when x = 5. From the 

 discussion of the corresponding more general equations (41), (42), 

 (43), (44), it will be recalled that the current-constituent j(-^. ^) is due 

 to the impressed axial electric force acting throughout the length of the 

 wave antenna, Jois, 9) is due to the impressed voltage FiO, 9) acting 

 at the end x = 0, J sis, 9) is due to the corresponding impressed 

 voltage Fis, 9) = 7^(0, 0)g-i'*cos0 acting at the end x = s, and /o.(5, 9) 



dependence on 6. It may be noted that, in the calculation of the ordinary polar 

 diagram representing the directional selectivity of a wave antenna, the functions 

 f{x, 6) and F{x, 6) are regarded as independent of 6, in accordance with the very 

 definition of the directional selectivity. 



-'' The ratio of the horizontal electric force /(x, 0) to the vertical electric force 

 F{x, d)/H — where H here denotes the height of the wave antenna above the earth's 

 surface — is a complex number whose value depends on the conductivity, dielectric 

 constant, and permeability of the ground, and on the frequency. 



