356 



THEORY 



20.3 



FRESNEL ZONES 



In this section, equation (21) will be applied to 

 compute a general formula for the target strength of 

 a smooth and rigid target. Here, smooth means that 

 the radius of curvature of the target surface is large 

 compared to the wavelength of the sound striking it. 

 Moreover, the target is assumed to have a relatively 

 simple shape, always convex, with no marked bumps 

 or protuberances. While this ideal target hardly re- 

 sembles most actual targets, the consideration of this 

 simple problem gives some insight into the means by 

 which sound waves are actually reflected. Even under 

 these special assumptions, however, it will be shown 

 that the integral in equation (21) can be readily 

 evaluated only by an additional approximation, first 

 suggested by the French physicist, Fresnel. 



Consider only the case where the echo is observed 

 back at the sound source; this case corresponds to 

 the situation of chief practical interest, as pointed out 

 in Section 19.1, and in addition simplifies the com- 

 putations. Then r = r' and equation (21) reduces to 



V2 = — 



iBf^ r 



cos d 



-iiriT 



'^dS. 



(22) 



In the integral, however, both d and r vary over the 

 surface of the target. Therefore, the integral cannot 

 be evaluated by elementary methods, except for cer- 

 tain special cases illustrated in Section 20.4 where 

 an exact integration can be carried out. For most 

 practical purposes, however, an expression for the 

 reflected sound pressure p2 and, therefore, for the 

 target strength T can be derived by means of an 

 approximate method, which was originally developed 

 in optics and which is known as the method of 

 Fresnel zones. 



This method is based on the mathematical anal- 

 ysis developed in the preceding section. Physically, 

 according to equation (21), every point on the target 

 surface which is struck by the incident sound pres- 

 sure wave becomes in turn a center of outgoing wave- 

 lets so that the points on the target surface may be 

 considered "secondary sources" of soimd. In optics, 

 this is called Huyghens' principle. Simple addition of 

 the soimd pressure in each individual wavelet will 

 give the reflected sound pressure Pi. 



Now, every wavelet has a phase depending on the 

 total distance traveled by the sound out to the target 

 and back. In general, these wavelets interfere both 

 constructively and destructively. Destructive inter- 

 ference leads to .cancellations due to the phase dif- 



ferences. But a sharp maximum of amplitude — due 

 to constructive interference, where wavelets whose 

 amplitudes are all of the same sign are superimposed 

 — exists in the direction corresponding to specular 

 reflection. This is the direction in which the beam is 

 reflected according to ray acoustics. A quantitative 

 calculation of the amplitudes of the different wave- 

 lets will show exactly how much energy is reflected 

 in different directions. In this way, wave acoustics 

 can be shown to give the same results as ray acoustics 

 when the wavelength is very short. 



20.3.1 



Method 



To compute the amplitudes and phases of the 

 different wavelets, the surface may be divided into 

 successive areas from which all the wavelets emitted 

 are approximately in phase and thus do not interfere 

 destructively. This is the Fresnel method. According 

 to this method, consider a series of wave fronts pro- 

 ceeding outward from a source at the point P, 

 separated by a distance X/4 from each other, where 

 X is the wavelength of the projected sound. When 

 they strike the target, the surface of the target is 

 intersected by these wave fronts in a series of curves 

 which divide the surface into the so-called Fresnel 

 zones. The phase of each reflected wavelet, measured 

 back at P, is 2Tvjt — 47rr/X. Since if equals the product 

 of 4Tr/X times X/4, the distance between two adjacent 

 zones, the wavelets from each zone have an average 

 phase difference of ir from the wavelets of the ad- 

 jacent zones. But a change of phase by the amount t 

 results in multiplication of the amplitude by — 1 ; 

 hence the wavelets from each zone interfere destruc- 

 tively with those from the two adjacent zones. The 

 advantage of the Fresnel-zone approach, as will be 

 shown, is that most of the zones cancel each other, 

 leaving only the effects of the first and last zones to 

 be considered. 



While the analysis can be carried out for the Fres- 

 nel zones defined by the wave fronts at any one time, 

 it is simplest to take the zones resulting when one of 

 the particular wave fronts considered is just tangent 

 to the closest point on the target. Let R be the value 

 of r at this point, in other words, let R be the distance 

 from the sound source and receiver at P to the nearest 

 point on the target. The first zone is the area on the 

 surface of the target intercepted by the wave front 

 which is a distance X/4 from the wave front tangent 

 to the target. In general, the position of the nth zone 

 is then determined by the inequality of equation (23) 



