172 



DIFFRACTION BY TERRAIN 



If these conditions are fulfilled, the field at the 

 receiver is given by 



E = En 



v 



2 7^00 



dv. 



(2) 



where E^ is the free-space field at the receiver in 

 absence of the screen and 



V = ±/li 



'^Ki-j) 



±^l?J?(a, + a=). (3) 



In the last formula, use is made of the fact that 

 ai and a2 are small angles 'so that approximately 

 ai = ho/disndao = ho/di. 



8.1.5 



The Fresnel Integrals 



An integral of the type appearing in equation (2) 

 is known as Fresnel's integral; its properties will 

 now be briefly discussed and numerical data given. 

 The standard Fresnel integral is usually defined as 



Civ)~jS(v)= / e' 



^( - '/i > 



(4) 



where 



C(«) = 



S(tO 



cos 



sm 



— I'- 1 dv. 

 2 / 



If this function is plotted in the complex plane, 

 with C and S as abscissa and ordinate, respectively, 

 for all values of v, a curve is obtained that is known 

 as Cornu's spiral (Figure .3). C - jS is represented, 

 in magnitude and phase, by a vector from the origin 

 to a point on this spiral. 



It may be shown that the length of arc along the 

 spiral, measured from the origin, is equal to v. 

 In the graph, values of v, counted positive in the first 

 quadrant and negative in the third quadrant, are 

 indicated along the spiral. As v approaches infinity, 

 the spii-al ^\■inds an infinity of tinies around two 

 points lying at the distance 1/V2 from the origin 

 on a 45-degree line. C and S for the end points are 



Cfico) = 



2 



Si±oo) = ± 



(5) 



" 0, 



Figure 3. Cornu spiral. 



