126 



SITING AND COVERAGE OF GROUND RADARS 



from the source to the diffracting edge is sufficiently 

 large. 



In Figure 19 is shown a cylindrical wavefront AB 



A 



Figure 19. Effect at point P of wavefront AB. 



with its axis at the line source S' (say an illuminated 

 narrow slit). The secondary wavelets from the 

 various line elements ds of the wavefront arrive at 

 P with different phases, having traveled different 

 distances MP. It is desired to find the resultant 

 field strength MP due to wavelets from any given 

 finite part of the front. 



Let the electric field strength at a point in the 

 wavefront be given by the expression 



E = E sin 2wft , (17) 



where t is the time, / the frequency, and E the 

 amplitude of E. The phase has been adjusted so as 

 to make E = when t = 0. 



Consider next the secondary wavelets spreading 

 from the front in the direction of P. The field 

 intensity at P due to the secondary wavelet emanat- 

 ing from the line element ds at the point M (see 

 Figure 19) is proportional to dsE and is inversely 

 proportional to the square root of the distance 

 MP = d (since this is a cylindrical wave). Further, 

 the field intensity must show a phase retardation 

 corresponding to the distance d, that is 2ird/\. Hence 

 the field strength of the wavelet at P is given by 

 an expression of the form 



dE = kE Q ds sin (2-irft 



2ird 



), 



(18) 



where k is a factor of proportionality which depends 

 to some degree on the angle MPM and the distance 

 d, but which will be considered constant here, as the 

 dependence of the phase on d is of much greater 

 importance. To obtain the intensity due to wavelets 

 emanating from a finite part of the front, equation 

 (18) must be integrated over the corresponding 

 region of s. For this purpose we need a relation 

 between d and s. This is obtained by applying the 



cosine law to the triangle MSP, which gives at once 

 d 2 = ( a + b) 2 + a- - 2a (a + b) cos - , (19) 



or after a simple reduction, using the identity 



s 



cos I s/a I = 1 



then 



2 sin 2 



l* = b 2 + 4a (a + b) sin 2 ^- 



(20) 



For the present purpose it is sufficient to consider 

 the case when angle s/a is so small that powers of 

 s/a above the square may be neglected in comparison 

 with unity. This means that 



-f 



+ 4a (a + b) sin 2 — ~ b 



+ 2a 



(a + b) 



sin J 



b 2a 



or again, on writing 



b + 



(a + b) 

 2ab 



V 



(a + b) 



2\ab S ~ 2 V ' 



the phase lag 2rd/\ assumes the form 

 27rd 2irb , t „ 



(21) 



(22) 



(23) 



Using equations (22) and (23), expanding the sine 

 expression of equation (18), it follows that 



b 



dE 



_ A \2(a 



ab\ 

 + b) 



E; 



cos 



(I") 



sin2ir[ ft — 



)Q|- V- 



cos 2ir 



m): 



dv . (24) 



This expression may now be integrated over a 

 certain region of the wavefront, say from v = v a to 

 v = v, corresponding to s = s to s = s, giving the 

 following expression for the electric field strength 

 at P: 



E 



where 



and 



^2l^4 /aV,0)Sin2 ' r ( /< -X-) 

 - g(v,v ) cos 2-kUI — -J 



f(v,v ) = I cos [%v-j dv , 

 g(v,v Q ) = / sin ( | v 2 j dv . 



(25) 

 (26) 

 (27) 



