354 



THEORY 



/ is the frequency and X the wavelength of the sound. 

 If the reflecting target itself is considered to be made 

 up of many point sources distributed over its surface, 

 p becomes p2, the reflected pressure; and the pressure 

 dp2 produced by all the point sources located in a 

 surface element dS is 



^102 



G 



-e' 



r 



2n(ft-r/\) 



dS, 



(8) 



or the pressure p2 produced by the entire target be- 

 comes 



■G 



s 



2iri(/(-r/X) 



dS 



(9) 



where G is essentially the average value of B in equa- 

 tion (7) for each individual source multiplied by the 

 number of sources per unit area; the integral is eval- 

 uated over the entire area S. 



Figure 1. Transformation to polar coordinates. 



This quantity G is a measure of the number and 

 strength of the point sources over the area; in general 

 G will vary over the target surface. The function G 

 must be chosen so that the resulting sound pressure 

 Pi satisfies the boundary condition (6) on the surface 

 of the target. 



The value of G at a particular point of the target 

 surface will be assiuned to be completely determined 

 by the incident sound pressure at that point. This 



assumption is not rigorously correct, but it leads to a 

 good approximation if the target has a surface whose 

 radius of curvature is every^vhere large compared 

 with the wavelength. 



First, a relationship between the value of G at any 

 point and the resulting gradient of pz at that point 

 will be derived. Then, the gradient of p2 may be re- 

 placed by minus the gradient of pi, because of the 

 boundary condition (6). In this manner, a direct 

 relationship will be obtained between the incident 

 sound field pi on the target surface, and the value of 

 G required to compensate the gradient of pi. 



Because of the assumption made that the gradient 

 of p2 at the point on the target surface is determined 

 primarily by the value of G at that point, a particu- 

 larly simple model may be considered and the result 

 generalized. The pressure gradient at the center of a 

 disk illustrated in Figure 1 will be derived, on the as- 

 sumption that G is constant over the surface ; in other 

 words, the density of point sources on the surface of 

 the disk is assumed to be uniform. If polar coordi- 

 nates p and 6 are introduced, the integral (9) for the 

 pressure on the z axis can be transformed as follows: 



= 27rG f e- 



Jp=0 



pdp 



n(.ft-T/\) 



or, since r^ = p~ -\- z^, 



Pi = 2wG 



J r= 



VK2+z: 



2niUl-r/X^ 



dr. 



(10) 



Equation (10) may be integrated directly and gives 

 for the soimd pressure on the z axis 



po = iXG\ 



2wH/l-z/\) 



(U) 



and by differentiating p2 with respect to z, the gradi- 

 ent at p2 perpendicular to the surface becomes 



5p2 r ? , 



(12) 



For the point on the surface where z = 0, the gradient 

 reduces to 



V dz A=o 



27rGe' 



2wift 



(13) 



which is independent of the radius R of the circular 

 surface. This result confirms the assumption that the 

 gradient of p2 at any point on the surface is deter- 

 mined only by the value of G in the immediate 

 vicinity of that point ; thus G is independent of possi- 

 ble variations in G at other points. Actually it is 



