358 



THEORY 



20.;;. 2 Application 



To find the target strength corresponding to the 

 pressure of the reflected wave in equation (34), equa- 

 tion (6) in Chapter 19 may be written in the form 



T = 201og|p2| - 201og|B| -l-401ogi2 (35) 

 where the vertical bars mean that absolute values of 

 the complex quantities involved must be taken. The 

 term 20 log |p2| is the rms echo level E where ps is the 

 actual echo pressure; 20 log |S| is the rms source 

 le\'el S at 1 yd ; and 40 log R is twice the transmission 

 loss from 1 yd out to the target at a range R, ex- 

 cluding attenuation losses. Strictly speaking, the rms 

 level is the average value of the square of the real 

 part of the complex quantity rather than the abso- 

 lute value; however, a more elaborate computation 

 along these lines leads to exactly equation (35). If 

 equation (34) is substituted into equation (35) the 

 target strength T becomes 



20.4.1 



Sphere 



T = 20 log ^ cos e e~ 



"dS 



(3G) 



where the bars again denote that an absolute value 

 must be taken. The quantity 7^1 in the exponent is 

 defined by 



u, = -(r ~ R). (37) 



A 



(S'l in equation (36) is the area of the target in which 

 Ml is less than 1; the integral is evaluated only over 

 those surface elements lying within Si. 



The evaluation of equation (36) provides the solu- 

 tion of the problem presented at the beginning of this 

 section. 



20.1 TARGET STRENGTH OF SIMPLE 

 TARGETS 



In this section, equation (36) will be used to com- 

 pute the target strength of relatively simple surfaces, 

 such as spheres, cylinders, and other objects, which 

 have a single highlight. The results obtained may 

 also be applied to more complicated surfaces, as long 

 as the radius of curvature is greater than the wave- 

 length. Whenever several highlights are present, the 

 reflected wave is the sum of the waves reflected from 

 each one separately. In general, they will interfere. 

 However, if an average is taken over a considerable 

 spread of target aspects, and if the highlights are 

 spaced much further apart than the wavelength, the 

 interference will tend to be random; in this situation, 

 the intensity of the echo is simply the sum of the 

 intensities computed for each highlight individually. 



The target strength of a sphere, on the basis of 

 wave acoustics, may be easily derived from equation 

 (36). The results of this analysis may be used not 

 only for a perfect sphere but also for any target sur- 

 face whose first Fresnel zone is essentially spherical. 



Consider a wave from a source P striking a sphere 



Figure 2. Reflection from a sphere. 



of radius A, illustrated in Figure 2, whose nearest 

 point is a distance R away from the source. If </> is the 

 angle subtended at the center of the sphere by Q, 

 which bounds an element dS of area, dS is simply 



dS = 2tA~ sin <t>d<t>. (38) 



By the law of cosines, the distance r from the source 

 P to the point Q is given by 



r^ = (R + Ay + .42 - 2A(R + A) cos <i> 



= K^ -I- 4.4(4 + R)sin' 



(39) 



When R is much greater than A, r is approximately 



r = R + 



2A(A +R) . S 



-sm' 



R 2 



The quantity mi from equation (37) is then 

 8A(A + R) . .,0 

 "^= XR "^^2- 



(40) 



(41) 



For short wavelengths, sin <j>/2 will be very small in 

 the first zone and may be set equal to <j)/2; similarly 

 cos d in equation (36) may be replaced by one. There- 

 fore, if equations (38) and (41) are substituted into 

 equation (36), the target strength of a sphere becomes 



20 log 



2\Jo 



''27rAVrf0, 



where 



and 



2 A (A + R) 

 \R 



(42) 



(43) 



xct,l = 1. (44) 



The integration may be carried out and yields 



