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



Hence by use of the expressions (13) and (15) 6 for \\i n () and /(), we have 

 approximately, when ha is small, for all values 



I 



" 



2 2 



We shall find it necessary to obtain a closer approximation to the value of A!. 

 Writing n = 1 in (14) we have 



(ka)} . (16). 

 Now from (15) 6 we have approximately, since ha is small, 



/ (ha) = h~ l a~ l (l-iha-\h 2 a 2 }, 2h 2 a% (ha) = Gh^a' 3 (1 +i/iV+- 2 ViV). 



Substituting these expressions in (16) and making use of (13) 6, we obtain 



Writing Xae~'"" r for ka we obtain finally 



a- 3 )~] . . . (17). 

 By a similar process we obtain from (8), 



A, = -^V(l-f^ 2 a 2 +A l /i 3 a 3 ) ....... (18). 



8. Having obtained approximate values for the various constants involved in the 

 expressions for the secondary waves, we may now proceed to estimate the additional 

 loss of energy consequent upon the presence of the obstacle. The method adopted is 

 exactly similar to that of which we made use in the case of cylindrical obstacles. 

 As above, it is easily seen that the total additional dissipation of energy due to the 

 presence of the obstacle is given by 



^dS+^p^dS ......... (1) 



where the suffixed letters have the same meaning as in 4, and the integration is to 

 be taken over the surface of a sphere of radius II concentric with the obstacle. As 

 before, we shall suppose that R is great compared with the wave-length of the 

 incident sound, and yet such that o-VR/c 3 is a small fraction. By this assumption we 

 are enabled to neglect the imaginary part of hR and also to regard the motion as 

 sensibly irrotational at the boundary r = R. 



