DIFFRACTION BY HILLS 



175- 



tious system of oscillating currents (or a system of 

 elementary doublets of proper strength) over the 

 surface of the plane. The field at the receiver is then 

 the superposition of all the fields produced by the 

 wavelets. 



Now let (Figure 6) >S be a plane perpendicular to 

 the line TR and let M be the point in which the line 

 TR intersects the plane S. Let Q be a point on the 

 plane S such that the difference in path between 

 TQR and TMR is just X/2. The locus of these 

 points is a circle about M. Similarly we can con- 

 struct other circles so that the corresponding path 

 differences are integral multiples of X/2. The area 

 Avithin the first circle is called the first Fresnel zone, 

 the subsequent ring-shaped areas are called the 

 second, third, etc., Fresnel zones. The secondary 

 wavelets originating in the first, third, fifth, etc., 

 Fresnel zones are in phase with each other and rein- 

 force each other by constructive interference at R, 

 while the secondary wavelets originating in the 

 second, fourth, etc., zones are in phase with each 

 other but out of phase with the former group and 

 tend to cancel the field produced by this group. 



Hence if the plane tS is opaciue except for a round 

 hole centered on M, the intensity of the radiation 

 field at R will depend on the number of Fresnel 

 zones that fall inside the hole. If we start out A\ith a 

 very small hole and progressively increase its size, 

 there mil be a maximum of intensity at R (nearly 

 twice the free-space field £'o) when the hole just 

 comprises the first Fresnel zone. If the size of the 

 hole is further increased, the destructive interference 

 of the second zone comes into play, decreasing the 

 intensity, and a minimum (very nearly zero) is 

 reached when the hole contains just the first two 

 zones. On continued increase of the hole size, 

 further maxima and minima appear. The amplitude 

 of these oscillations decreases very gradually until 

 eventually the field at R approaches the free-space 

 value. 



8.2.2 



Diffraction by a Slot 



The preceding considerations indicate that only a 

 comparatively small area of an opening, of the order 

 of one Fresnel zone, is required to produce an 

 illumination that is comparable in order of magni- 

 tude to the free-space field. It is also seen that the 

 simple geometrical construction of the Fresnel zones 

 is more suitable when dealing with the diifraction 



by round openings than with screens bounded by 

 straightedges. QuaUtatively, however, the conditions 

 are similar. 



As an example, consider the case of a slot bounded 

 by parallel edges at distances ht, and /)o' from the 

 point of mtersection M between the plane of the 

 slot and the direction from the observer to the 

 distant light source (see Figure 6). The diffracted 

 field E will obviously be equal to the free-space 

 field Eo if the slot is infinitely wide on both sides of 

 M, which corresponds to a vector joining the two 

 foci of the Cornu spiral. However, there is an infinite 

 number of other finite openings of the slot which 

 also will give the free-space field. Sujjpose, for 

 instance, that /lo = W in Figure 6 and that the slot 

 width is gradually increased from zero. A glance 

 at the Cornu spiral (Figure 3) shows that when 

 (' = 0.75 and v' = —0.75, the vector representing 

 the diffraction field is approximately equal to the 

 free-space field. This \\idth represents, for a slot, 

 the analogue of the fiist Fresnel zone for a circular 

 opening. 



S3 DIFFRACTION BY HILLS 



Introduction 



8.3.1 



The formula for diffraction by a straight edge may 

 be applied in radio practice to determine the diffrac- 

 tion field behind a ridge. The ridge need not be 

 perpendicular to the transmission path, but the 

 condition given in equation (1) must be approx- 

 imately fulfilled. The distance from transmitter 

 and recei^'er to the ridge should be large compared 

 to the height of the latter above the straight line 

 TR; and that height should be large compared to 

 the wavelength. 



Moreover, as pointed out in Section 8.1.3, the 

 diffraction formula applies in principle only to the 

 case where the effect of the currents induced on the 

 surface of the ridge upon the field at the receiver 

 can be neglected. This is the case (1) when the ridge 

 has the shape of a steep and narrow knife-edge 

 protruding from the surrounding countryside; or 

 (2) when the surface of the ridge is rough (see Sec- 

 tion 8.3.2). Experience shows that so long as the 

 profile of the ridge is reasonably compact and its 

 surface reasonably rough, the diffraction formula 

 will give the magnitude of the field behind the ridge 

 to within a few decibels. 



