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

 Now, from (17) and (18), 7, we have 



a- 2 -3X- 4 cr 4 )} . . . (5). 



Let us first consider the case when \a is small. Since /i 2 /X 2 or or/c 2 is always small, 

 it follows from (5) that in this case we may write approximately 



6 -f 3 v/20-VvA 

 />/-/. v ' / 



When Xa is great, it is necessary to include one other term of (5), and we may in 

 general write in this case 



- .... (6). 

 ca/ 



Comparing this last result with that obtained in the case when \a is small, we see 

 that we may take it as a sufficient approximation in almost all cases. For small 

 values of the radius the first term in (6) will be negligible. 



Substituting from (6) in (4) we obtain for the total loss of energy to the primary 

 waves the expression 



1/DoO-VcTra 2 . (|o-V/c 4 + 3 v/2o-' - V 2 /c + 6 Y 



ca 



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

 <r 2 /CTra 2 , and hence the ratio of the lost energy to that incident upon the obstacle is 



3 x /2cr 1/ V' 2 /c+6 + |o-V/c 4 (7). 



CO, 



The first two terms of this last expression represent the proportion of the incident 

 energy lost by friction. The last term of (7) gives the proportion lost by scattering 

 to a distance, and is the same as is obtained in the theory of a frictionless air. 



When X is small, the most important term of (7) is the second 6 . Hence we 



CO/ 



see that in the case of small obstacles the ratio of the lost energy to the incident 

 energy varies inversely as the radius of the obstacle, and consequently tends to become 

 very great as this radius is diminished. On the other hand, the actual amount of 

 energy lost varies directly as the radius of the obstacle; and diminishes with the 

 radius. It is to be noticed that in the case of sufficiently small obstacles the energy 

 lost to the primary waves is independent of the wave-length of the incident sound. 



When Xa is great, the most impoi'tant term of (7) is the first 3 v /2o- I/ V /2 /c. Hence 

 we see that in this case the ratio of the lost energy to that incident upon the obstacle 

 is very nearly independent of the radius of the obstacle, provided the order of 

 magnitude of this ratio is altered by the viscosity. Consequently for sufficiently large 



