250 MR. C. J. T. SEWELL : EXTINCTION OF SOUND IN A VISCOUS ATMOSPHERE 

 of which the mean value is 



2/vr 5 [(-)-A,t- + '] (5). 



n=0 



This last expression, then, represents the loss of energy to the primary waves in 

 consequence of the presence of the obstacle. 



From the value of A n obtained in (12) of 3 we see that the summation (5) consists 

 of a series of terms arranged in descending order of magnitude. Consequently, in 

 determining its value we shall limit our attention to the first two terms of the 

 summation. Hence the total loss of energy to the primary waves is given very 

 approximately by 



(6). 



Let us first consider the case when \a is small. In this case we have, from (17) 

 and (19) of 3, 



c~ 



Hence 



[tAo+AJ = -fj.Tf'IM- -rr TT (log iXa + y) [(logXa + y) 2 + -n i -7r 2 }~ 1 . 



Now for small values of the radius a the first term of this last expression is small 

 compared with the second, and consequently may be neglected. Hence the loss of 

 energy to the primary waves is given approximately by 



9. 



. 1 _21 -1 /7\ 



^18* 1 ('). 



in the case when \ft is a small fraction. 



The ratio of this last expression to p^a/c, which represents the rate at which 

 energy is incident in the primary waves upon the obstacle, is given by 



) 2 +- 1 V 2 }- 1 ..... (8). 



We may notice from this last result that, when \a is small, the proportion of the 

 incident energy, which is lost to the primary waves, -is very nearly proportional to 

 the reciprocal of the radius, since the logarithmic terms will change more slowly as 

 the radius changes. Hence, as in (7), the total energy lost to the primary waves is 

 almost independent of the dimensions of the obstacle, provided these are small enough 

 to satisfy the conditions under which the results (7) and (8) have been obtained, and 

 provided also that we limit our attention to obstacles whose dimensions are of about 

 the same order of magnitude. It might have been anticipated that the energy lost 

 by friction in the neighbourhood of the obstacle would, in the case of very small 

 obstacles, alter very slowly with the dimensions of the obstacle, and consequently 



