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,* 



E 



2£o 



where 



PF 1 r 1 1 1 



1 + r-^ "^ 1 - T^ [ iTvidjX ^ {liridJXy J ' ^ ^ 



4- e-2i<r/f= e(l - i/Q). (18) 



T" 



In this equation IEq 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 Wfrom equation (16) is substituted in equation (17) 

 and terms which involve (1/^) to powers higher than the first are 

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



1 ^ + ^M r 1 __Z!_ 



1 - T'27rTHd/\\ I (1 + t2 



X 



) 



2Eo. (19) 



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, d, d) is 



/I a a ,1 a^ 52 4,r2\ _ 



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 



_ ^/^ . i^\ =_JL_4._J_r— L_ + ^1n 



47r2 \ a<f2 "^ d ad / 1 + t2 "^ 1 - T* L 2«d/x ^ {iiridixy} ° 



when the value of 3' = (1 + t^)TI/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 by £0 = — liOtTr^IIoIX 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 = lirfe'/g, 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 = /e/2<7. 



