INTRODUCTION 



65- 



If the earth were flat and perfectly reflecting, the 

 envelope of the maxima of the curve in Figure 4 

 would coincide with the line 2Eo, twice the free- 

 space field, corresponding to the in-phase addition 

 of the direct and reflected waves. An envelope of the 

 minimum points would be £ = 0, corresponding 

 to the destructive interference of the direct and 

 reflected waves . The curvature of the earth, resulting 

 in increased divergence of the waves (see Section 

 5.2.5), and the lack of perfect reflection (see Section 

 5.2.4) cause the maximal and minimal envelopes to 

 differ from 2Eo and 0, respectively. In the neighbor- 

 hood of the first maximum in Figure 4 (i.e., when the 

 direct ray makes small angles with the earth), the 

 reflection coefficient tends to be unity in magnitude 

 for both polarizations except for the increase in diver- 

 gence which results in the deviation of the maxi- 

 mal and minimal lines from 2E and 0, respectively. 

 At a smaller distance, for vertical polarization, as 

 shown in Figure 4, the deviation is caused principally 

 by the smaller magnitude of the reflection coeffi- 

 cient. The virtual meeting of the maximal and 

 minimal lines corresponds to the minimum value 

 of the reflection coefficient at the pseudo-Bi-ewster 

 angle. (For horizontal polarization at small dis- 

 tances, the envelope of maxima would virtually 

 coincide with 2i?o and the minima would be closer to 

 zero. As the distance is increased, the difference 

 between the envelopes for vertical and horizontal 

 polarization gradually decreases.) 



2. Both antennas low; h < he- In a discussion of 

 the height function, it is convenient to distinguish 

 between high and low antennas. The critical height 

 separating the two cases for horizontal polarization 

 or ultra-short waves is given by 



(21) 



SOX-''^ meters 



where X is expressed in meters. For X = 0.1 meter, 

 he = 6.46 metei's, and for X = 10 meters, he = 139.5 

 meters. If both antennas are at elevations less than 

 he, the height-gain functions f(h), to a first approx- 

 imation, are the same for all the modes, so that the 

 complete solution 



/i(/u) • Mh) • Fr(d) +/o(/iO • Mh.) • F,{d) + + + 



can be written in the form 



/(/u)•/(/^2)■(Fl + F=-f...), 



(22) 



or 



/(Ai) • m) ■ Fid), (23) 



where /(/ii) replaces fiQii), fQh) replaces fiih), etc., 

 while F{d) stands for the sum Fi + F2 + • • • . 



The distance function F(d) can be calculated for 

 particular cases. This has been done for high fre- 

 quencies and is represented graphicallj^ in Section 

 5.7, the results being valid for low antennas for all 

 distances in the optical as well as in the diffraction 

 region such that 2hihz < < \d (see Figure 3). The 

 condition 2hihi < < \d assures that the antennas are 

 below the interference pattern. 



At the ground, f{h) = 1, so that if both antennas, 

 are close to the ground, the distance dependence is 

 given by F(d) only. 



3. One or both aMennas elevated; h> hc= 30X~'^. 

 For elevated antennas, h > h^ and the height-gain 

 functions of f{h) vary with the modes. Conse- 

 quently, it is not possible to separate the height and 

 distance effects as in the previous paragraph. 



In the optical-interference region, it is more ad- 

 vantageous to use the method of combining the 

 direct and reflected waves. This is equivalent to 

 the rigorous solution which is illustrated by the dots 

 in Figure 4. 



Simple graphical aids can be given for points well 

 within the diffraction region where the first mode 

 predominates. The range of usefulness of the first 

 mode can be extended by plotting the field strength 

 given by the first mode as a function of height (or 

 distance) and plotting a similar curve by using the 

 ray method as far as the lowest (or first) maximum 

 (see Figure 7). Then by joining these partial curves 

 into a smooth overall curve, a fairly good value of 

 the field can be obtained for intermediate points. 



There is a further possibility occurring with the 

 transmitting antenna elevated, the receiver low and 

 lying below the interference pattern, and the dis- 

 tance short. In this event, none of the pi'evious 

 methods apply. However, the reciprocity principle 

 (see Chapter 2) can be applied to find the radio 

 gain at the receiver by interchanging the role of 

 receiver and transmitter. Suppose the original 

 transmitter height is 100 meters, the original re- 

 ceiver height 15 meters, and the wavelength 1 meter. 

 Now let the transmitter height be 15 meters. If the 

 receiver height is low (/i2 < 30 meters), values of 

 the gain can be found (Section 5.7) ; if the receiver 

 height is in the interference region, the gain can be 

 found by the ray method. Now suppose a curve be 

 drawn for these results, giving the attenuation 

 versus receiver height. From this graph, the value 

 of the gain factor A at /jj = 100 meters can be read. 

 This value of A by the principle of reciprocity is the 

 gain factor for the original heights. 



