SIMPLE. HARMONIC WAVES. DIFFRACTION 243 



In the case of the sphere we found Q / = JQ = | 7 ra 3 , and the 

 expression (6) therefore reduces to 



J(*a).iro' ......................... (7) 



In other words, the sphere scatters only the fraction | (&a) 4 of 

 the energy which falls upon it. For example, if the wave- 

 length be a metre (which corresponds to a frequency of about 

 332), and the diameter of the sphere 1 mm., the fraction is 

 roughly 7 '6 x 10~ n . In the case of the circular disk, where 

 Q' = f Tra 8 , Q = 0, the ratio of the scattered to the incident 

 energy is ^(Ara) 4 . 



The mathematical theory of the scattering by cylindrical 

 obstacles is more difficult. We will merely quote the result, 

 based on Lord Rayleigh's calculations, that when plane waves 

 are incident on a circular cylinder of radius a the fraction of the 

 incident energy which is scattered is f Tr^&a) 3 , approximately, 

 it being assumed as usual that ka is small. For a wire of 

 diameter 1 mm., and a wave-length of a metre, this 



= 1-15 x 10- 7 . 



It is to be observed however that in the case of very minute 

 obstacles the order of magnitude of the results may be con- 

 siderably modified by viscosity. The determining element here 

 is the ratio of the diameter of the obstacle to the quantity 

 h which was introduced in 66 as a measure of the thickness of 

 the air-stratum, at the surface of the obstacle, whose motion 

 is appreciably affected by the friction. When the ratio in 

 question is moderately large the influence of viscosity on the 

 results will be very slight. 



The distribution of velocity in the immediate neighbourhood 

 of the obstacle will be sensibly the same as in the case of a 

 uniform current of incompressible fluid flowing past the body. 

 In the case of the sphere it can be determined completely, but 

 the following approximation will be sufficient. We assume 



(8) 



where the first term represents the incident waves, and the 



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