148 BELL SYSTEM TECHNICAL JOURNAL 



to the virtual image of the receiving antenna, R' , with a phase change 

 of 180° due to reflection between M and R. 



The received field is therefore propagated in four ways: (1) directly 

 from 2" to i? by diffraction at M represented by TMR, (2) by reflection 

 at Gi and diffraction at M represented by TGiMR, (3) by diffraction 

 at M and reflection at d represented by TMO^R, and (4) by reflection 

 at Gi, diffraction at M and a second reflection at G2 represented by 

 TG1AIG2R. The amplitudes and phases of these four components 

 can be calculated by usual methods of diffraction (see Appendix IV) by 

 assuming the components to travel from the real transmitting antenna 

 or its virtual image, to the real receiving antenna or its virtual image. 

 The ratio of the received field to the free space field may then be 

 calculated by combining the four components as follows: 



E/Eo = Cx exp \-j{rix + fi)] 



+ CiKx exp [-7(772 + U — ^\)\ 

 + dKi exp \-j{.-(]z + ^3 - <^2)] 

 + CJCxKi exp [-7(774 + u - (fi — ^2)], 



where the Cs are the ratios of the field strengths with and without 

 diffraction, the 77's are the phase lags introduced by diffraction and the 

 f's are the phase lags due to path lengths TR, T'R, TR' and T'R', 

 while the K's are magnitudes of the reflection coefficients and the cp's 

 are the phase advances at reflection. 



It is true that actual conditions will seldom be as simple as these. 

 The valleys will not be flat. There will often be more than one hill 

 and it may be impossible to represent the obstructions accurately by 

 the single straight edge, M, which we shall assume. It will often be 

 possible, however, to choose equivalent planes and straight edges in 

 such a way as to justify some confidence in the results. 



On these assumptions the frequency characteristic of the trans- 

 mission path from Deal to Lebanon has been calculated (Fig. 16). 

 By actual measurement it has been found that the attenuation over 

 this path at 17 mc. (17 meters) was more than that at 69 mc. (4 

 meters). This characteristic of poorer transmission on the longer 

 wave-lengths is the opposite of what would have been expected either 

 on the basis of diffraction alone or by analogy with the trend observed 

 on lower frequencies. The calculations show, however, that this is 

 the characteristic that we should expect on the theory outlined. In 

 view of uncertainties in the reflection coefficients and errors of measure- 

 ment, the agreement of the absolute values calculated and measured 

 is as good as should be expected. An improvement in this agreement 



