360 



THEORY 



10 log 



AiAi 



2(1 + AJR){\ +A2/R)' 



(49) 



which reduces immediately to equation (47) when 

 Ai is equal to A2. While equation (47) was valid only 

 if Ai/R was moderately small, equation (49) is ap- 

 plicable even if A\/R is very large as long as A2/R 

 is still small. Equation (49) cannot be used, however, 

 when either Ai or A^ approaches the wavelength of 

 sound. 



20.4.3 Cylinder 



For an infinitely long cylinder, equation (49) may 

 be applied directly by letting one radius of curvature 

 Ai be infinite. The target strength found for this case 

 reduces to 



where A2 is the radius of curvature of the cylinder. 

 This equation is valid only when the wavelength of 

 the sound is much less than the radius of curvature 

 of the cylinder, and when this radius in turn is much 

 less than the range. 



For an actual cylinder equation (50) may be used 

 only if the cylinder is perpendicular to the sound 

 beam at some point, and if the cylinder is long enough 

 to include at least the first few Fresnel zones. The 

 expression may therefore be used only at moderate 

 ranges, since with increasing range the length of the 

 first Fresnel zone increases infinitely. 



To compute the range beyond which equation (50) 

 cannot be used, let the length of the cylinder be L, 

 and let the sound source he in a plane which is per- 

 pendicular to the axis of the cylinder and bisects the 

 cylinder. Then the path length r to the end of the 

 cylinder is 



The length of the cylinder will include many Fresnel 

 zones if r given by equation (51) exceeds R by many 

 wavelengths. Therefore equation (50) may be used 

 only as 



R<<^- (52) 



For example, for a wavelength of 4 in., corresponding 

 to a frequency of about 15 kc and a cylinder 10 ft 

 long, the range must be much less than 100 yd if 

 equation (41) is to be used. 



At long ranges, R is much greater than L'^/\, and 

 the computed length of the first Fresnel zone exceeds 



the length of the cylinder. In this case, instead of 

 using the approximation (30) an exact integration of 

 equation (22) over all the zones is possible, provided 

 the variation of cos 6 is neglected, and the target is 

 far away from the source. Thus in equation (36), in- 

 stead of one-half the integral over the first zone, we 

 may take the same integral over all the zones. With 

 the same approximation for u made in the previous 

 section, the target strength becomes 



At these longer ranges, the target strength is again 

 independent of the range, in agreement with the 

 comments made in Chapter 19. However, equation 

 (53) presents one case in which the target strength 

 varies appreciably with changing wavelength, even 

 when the wavelength is much smaller than the target. 

 For intermediate values of the length of the cylin- 

 der, both the first and last zones must be considered. 

 A more exact evaluation of equation (22) can be car- 

 ried through in this special case by use of particular 

 functions called Fresnel integrals, which have been 

 tabulated. 



20.4.4 Reflection at Close Ranges 



The formulas developed so far in this chapter are 

 applicable to many simple shapes provided the sound 

 source and receiver are not too close to the target. 

 The target strength at close ranges may also be found 

 directly from equation (36). Detailed results have 

 been worked out for cases of this nature, but will not 

 be reproduced here. In general, when R becomes 

 much less than the principal radii Ai and Ai, the 

 reflection can best be described as reflection from a 

 plane surface. In the hmiting case where R/A2 is 

 negligible, 



V2 



2i?' 



(54) 



as long as the sound field this close to the target obeys 

 the inverse square law; for a large directional trans- 

 ducer, this condition is not likely to be satisfied at 

 very close ranges. If equations (35) and (54) are 

 combined, the target strength becomes 



T = 201og^- (55) 



Formulas for the target strength of various types of 

 objects, such as two cones placed base to base, and 

 a circular disk placed at an angle to the sound beam, 

 are given in reference 6. 



