TARGET STRENGTH OF SIMPLE TARGETS 



359 



•'n 



e~'"*'<^rf<^ = 



-wixip* 



- 2Trix 



TtX 



Thus equation (42) becomes 



r = 201og— , 



(45) 



(46) 



and if equation (43) is substituted for x, the target 

 strength reduces to 



This expression is valid only when the distance from 

 the source to the sphere is at least several times 

 greater than the sphere diameter. When R is very- 

 much greater than A, equation (47) simply becomes 



A 



T = 20 log 



2' 



(48) 



which is identical to equation (10) in Chapter 19 

 derived on the basis of ray acoustics. At shorter 

 ranges, equation (36) is still applicable, but must be 

 evaluated more accurately. It may be noted that the 

 value of T in equation (47) is based on the assump- 

 tion that the transmission loss to the nearest point 

 of the sphere is used in equation (35). If the trans- 

 mission loss to the center of the sphere is used in- 

 stead, T must be increased by 40 log (1 + A/R) and 

 increases as the range becomes shorter. 



As already pointed out, equation (47) may be ap- 

 pUed whenever the first Fresnel zone of a reflecting 

 surface is spherical in shape, and has a radius of 

 curvature A much larger than the wavelength X. The 

 result is independent of the wavelength. Equation 

 (36) could be evaluated more accurately to find a 

 dependence of T on wavelength. This dependence 

 would be appreciable only when the wavelength was 

 no longer much smaller than the sphere radius A, in 

 which case the total number of Fresnel zones would 

 no longer be large. Since the accuracy of the Fresnel 

 method is doubtful under these conditions, the wave- 

 length dependence found in this way would not be 

 very reliable unless confirmed by a much more 

 elaborate investigation. 



20.4.2 General Convex Surface 



More generally, the curvature of a surface cannot 

 be described by a simple single radius of curvature. 

 In such a case, the boimdary of the first Fresnel zone 

 will not be a circle, as was the case for a spherical 

 surface. In a more general case, this boundary will be 



elliptical in shape, and the surface intersected will 

 have two principal radii of curvature Ai and A^, 

 which will usually differ from point to point. 



These radii may be defined as follows. Let be a 

 particular point on the surface and let OC be a line 

 perpendicular to the surface at the point 0. Any 

 plane containing OC will intersect the surface in some 



Figure 3. Reflection from any convex surface. 



line QOQ', as in Figure 3. In the neighborhood of the 

 point this curve is approximately a circle of radius 

 A. However, as the plane intersecting the target is 

 rotated about the line OC, the radius A of the curve 

 QOQ' will vary. It will have a maximum value Ai and 

 a minimum value Ai, in general, as the plane rotates 

 through 180 degrees. Furthermore, according to dif- 

 ferential geometry, these two radii will be 90 degrees 

 apart. These two quantities Ai and A^ are called the 

 principal radii of curvature of the surface at the point 

 0. If they do not change rapidly with position on the 

 target surface — more particularly, if they are ap- 

 proximately constant at all points in the first Fresnel 

 zone — the target strength of the surface may be 

 computed. 



The derivation is more complicated than that in 

 Section 20.3.1 and will not be given here. The result 

 of the analysis is in the following equation 



