FRESNEL ZONES 



357 



R+ (n- \)^<r < R + n- (23) 



where r is the distance from the source to any point 

 in the zone. 



If S„ is the area of the nth zone, equation (22) can 

 be written as a sum of integrals in which each integral 

 extends over only one zone. Then 



p. = -?e--^'E r ^'e-'-'^'^dS; (24) 



the sum, denoted by the symbol S, extends over as 

 many values of n as there are zones. To evaluate the 

 integral in equation (2-4), define a new variable u„ for 

 the nth zone by 



r - ft - (n - 1) X/4 



Un = 



X/4 



(25) 



If this equation is substituted in inequaUty (23), u^ 

 satisfies the relationship 



<Un< 1. (26) 



Thus, M„ increases from at the near side of the 

 n zone to 1 at the far side. Equation (24) then be- 

 comes 



iB 



P2= — 



2n{/l-2R/ 



'^>Ze 



n-i)« e~""'dS. (27) 



Since e"-" = - 1, then e"'""'^" = (-1)""S and the 

 reflected soimd pressure pa may be written 



p, = _!^e2"(/'-2KA)|-p^ _ P^ + P3 . . . 



+ (-l)^-ip„], (28) 



where N is the total number of zones and P„ is 

 defined by 



cos 6 



'•-L 



"dS. 



(29) 



In each integral m„ hes between and 1 and obeys 

 inequality (26). 



For targets large compared to the wavelength, 

 whose surface is not too sharply curved, there will be 

 a large number of zones and the values of Pn found 

 in successive zones will not change very rapidly as n 

 is changed. The quantity r- will scarcely change at 

 all if the distance to the sound source and receiver 

 is much greater than the size of the target. The 

 quantity cos d may decrease from 1 in the first zone to 

 a small value for the higher zones, but if the target is 

 much larger than the wavelength and if its surface is 

 not cur^dng too sharply, the change in cos from one 

 zone to the next will not be large. Similarly the area 

 Sn of successive zones will not change very rapidly. 



The factor e~""" varies in the same way in all the 

 zones. Thus, on the average, the partitl pressure Pn 

 of the wavelets reflected from the nth zone may be 

 assumed to be equal to the average of the correspond- 

 ing partial pressure of the wavelets for the preceding 

 and following zones, or 



Pn = UPn-l + Pn+i). (30) 



Equation (30) forms the basis of the Fresnel approxi- 

 mation. It may be expected to become increasingly 

 accurate for a smooth surface as the wavelength X 

 decreases indefinitely and the order number n of the 

 particular zone increases indefinitely. 



With this approximation, the successive terms in 

 equation (28) cancel out, and the sum of all the Pn's 

 in equation (28) becomes simply 



Pi-P2 + P3--- -f-(-l)''P^ 



= K^i-f-(-l)^P^]. (31) 



In most practical cases, the value of cos d for the 

 last or iVth zone is zero, since the target surface at this 

 point is tangent to the sound rays. Thus P^r vanishes 

 and the sum of P„ over all the zones is simply one- 

 half the value of P for the first zone. This is a particu- 

 larly interesting and important result; if only half of 

 the first zone participates in the reflection, and the 

 entire target surface beyond it is neglected, the re- 

 flected sound wave is the same as if the entire target 

 were regarded as the reflecting surface. Only a small 

 part of the target surface perpendicular to the sound 

 rays produces the entire reflection; the reflection from 

 this small region is sometimes called a "highlight" as 

 it is in optics. It is this result, derived on the basis 

 of wave acoustics, which corresponds to the specular 

 reflection based on ray acoustics in Chapter 19. 



Therefore, set P^r equal to zero in equation (31), 

 substitute the result into equation (28), and p2 be- 

 comes 



2i\ 



(32) 



Now, since the value of r will be almost constant 

 throughout the first zone, unless the source is only 

 a few wavelengths away, r may be replaced by R, the 

 shortest distance from the source to the target. 

 Equation (29) may therefore be written as 



'^ = iJs'''' 



— I con e e-'"'"dS. 

 PVs, 



(33) 



Finally, by combining equations (32) and (33) the 

 pressure P2 in the reflected wave becomes 



Js, 



P2 



2nUt-2R/\) 



2XP2 



