50 BELL SYSTEM TECHNICAL JOURNAL 



by using equation (15) to calculate the field strength. This is for- 

 tunate since series D requires laborious calculations. 



The attenuation factor may be obtained from W by means of the 

 relation,* 



/here 



E ^ W 1_ 



2£:o ~ 1 + T-^ 1 - 



1 



' +74yKp]- (") 



lirid/X {lirifj 



e - Half = e{\ - i/Q). (18) 



In this equation 2Eo is the inverse distance, or radiation, component 

 of the field that would result from transmission over a perfectly con- 

 ducting plane, and Q is the ratio of the imaginary component to the 

 real component of the admittance of the ground. In other words, 

 Q is the ratio of the dielectric current to the conduction current.f 

 The parameter e occurring in equation (18) is the relative dielectric 

 constant (with respect to vacuum), a pure numeric that is numerically 

 equal to the dielectric constant measured in electrostatic units. 



If the value of W f rom equation (16) is substituted in equation (17) 

 and terms which involve (l/d) to powers higher than the first are 

 neglected as may be done at the greater distances, we have 



1 - r' lirrHd/X J [ 



1 ... (-5) 



2Trld f , 1\ -, 

 X 



2E^. (19) 



(1 + r^) 



The magnitude of the second factor on the right differs from unity 



* This expression may be obtained as follows. II satisfies the wave equation 

 which in cylindrical coordinates (z, Q, d) is 



(I d d I d^ d^ iir^ \ 



ddd'^dd'^d'^dd^'^di^'^1^)^^^- 



Because of symmetry the second term is zero. Solving for the value of the last two 

 terms and substituting it in equation (11) yields 



The differential equation given by Wise " for II becomes 



_ xi/a^ , i3n\ _ n 2 r 1 ^ In 



in'' \ dd'^ ^ d dd J 1 + t2 "^ 1 - T* L 2Tridl\ ^ (iTid/xyj " 



when the value of y = (1 + t2)II/2 is substituted in his equation (7), and the result 

 multiplied by 2/(1 + t^). Substitution of this relation in the preceding equation and 

 division hy Eo = ~ liOiir^UoIX gives equation (17) of the text. Since Eo is the 

 inverse distance component of the free space field, this relation follows from equa- 

 tion (11). 



t In practical units Q = Iwfe'jg, where «' is the dielectric constant in farads per 

 meter and g is the conductivity in mhos per meter. On frequent occasions, the 

 constants of the dielectric are expressed in electrostatic units; then Q = fe/2<7. 



