DIFFRACTION BY COASTS 



179 



the direct ray ■\\ith its full or nearly its full intensity. 

 The ray leaving the transmitter at an angle i^-o 

 (Figure 13), such that its reflected counterpart 

 undergoes reflection right at the shore line, divides 

 the coverage diagram into two parts. For angles of 



VERTICAL SECTION 



«— d,= do/cosr ^ 



■nn m -rf ti ' i'i ' r}ii 

 I LAND 



IWAtE 



OCEAN 



PLAN VIEW 



Figure 13. Diffraction by a coast line. 



elevation larger than ^ = -^n the field will be essen- 

 tially the free-space field ; for angles of elevation less 

 than \p = tpQ the familiar lobe pattern, for complete 

 reflection, will appear with maxima equal to twice 

 the free-space field. When diffraction bj^ the coast 

 line is taken into account, the discontinuity expressed 

 by this rough picture is replaced by a smooth transi- 

 tion of the field from one region to the other. 



The land surface may be considered as an opaque 

 screen for the image transmitter from which the 

 reflected rays seem to come (Figure 13). This prob- 

 lem is somewhat different from the diffiaction prob- 

 lem treated previously since the trace of the screen 

 in the vertical plane through T' and R is no longer 

 perpendicular to the line T'R as it was, for in.stance, 

 in Figure 2, upper part. In the present case, the 

 effective height ho of the diffracting edge for any 

 given ray is the perpendicular projection from the 

 coast line upon this ray, as shown in Figure 13. 

 The slant distance of the coast from the trans- 

 mitter is di. Assuming that the receiver (target) is 

 far distant, a condition usually fulfilled in radar 

 practice, d^ >> rfi and the angle tp between the 

 direct ray and the horizontal will be ecjual to the 

 angle between the image ray and the horizontal. 

 Then approximately, since the angles are small, 



7,„ = d,a, = d^{^Po - V'), (14) 



where di and ai have the significance given them in 

 Section 8.1.4. Here the signs have again been 

 chosen so that /)o is negative when the receiver 



(target) is in the shadow of the screen with regard 

 to the image transmitter. 



The distance from the transmitter to the diffract- 

 ing coast depends on the azimuth (Figure 13). 

 Therefore, with the designations of the figure, 



do 



di = 



cos y 



(15) 



8.4.3 



Equation for Field Strength 



The expression for the diffracted field of the 

 image transmitter is given by the straightedge 

 formula, equation (6), with v given by equation (3). 

 Since 1/(^2 is assumed negligibly small compared to 

 1/di, Ave find, on using equation (14), 



\2di 

 V = (\po- i)\ ^ ■ 



(16) 



This may be further simplified by introducing (as in 

 Section 8.3.2) a new variable, the lobe number 



_ 4/iii/' 

 X 



(17) 



{hi = transmitter height) . This quantity is equal to 

 1, 3, 5 • • • at the interference maxima and equal to 

 0, 2, 4, • • • at the interference minima but is here 

 taken as a continuous A'ariable, defined for any value 

 of tp. In particular for \p = \pa we put n = iio. Since 

 "Ao = h\/di, we have by equation (15) 



4Jii\pa 4/ii^ cos 7 ,. Qv 



no = = ; — • Uo) 



X Xdo 



Equation (16) may now be written 



Ho — n 



V2?2o 



The diffraction formula will again be written, in the 

 form of equation (7), as 



— =36 ^% 



where z and f are the functions of v shown in Figures 

 4 and 5. 



The total field obtained by the interference of the 

 direct and reflected ray is 



(19) 



E = £'o(l - 36" 



(20) 



where the negative sign in front of the second term 

 in parentheses accounts for the 180-degree phase 

 shift at reflection, and the phase lag irn corresponds 



