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



4. Now that we have approximated to the values of the various constants involved 

 in the expressions for the secondary waves, we can proceed to estimate the additional 

 rate at which energy is being dissipated in the space surrounding the obstacle. This 

 additional rate of dissipation will be equivalent to the rate at which energy is being 

 lost to the primary waves in consequence of the presence of the obstacle. Now, if we 

 consider a region bounded internally by the obstacle and externally by a cylindrical 

 surface coaxial with the obstacle and of radius R, it is clear that the difference of the 

 rates at which energy is being carried across the internal and external boundaries of 

 this region will be equivalent to the rate of dissipation of energy within it. 



Hence, if p , q t) denote the pressure and the radial velocity at any point due to the 

 primary waves alone, and if pi, qi denote the pressure and radial velocity due to the 

 secondary waves alone, it is easily seen that the dissipation of energy within a 

 distance R of the obstacle is given by 



where it has been assumed that q , q t are both measured inwards, and the integration 

 is to be taken round the boundary of the surface r = R. 



Now the dissipation of energy due to the primary waves alone is given by lp u q a ds. 



Hence it follows from (1) that the additional dissipation of energy due to the 

 presence of the obstacle is given by 



&ds ........ (2). 



Now the rate at which energy is being carried across the surface r = R is \p\qids, 

 and hence from (2) it follows that the total additional dissipation of energy due to 

 the presence of the obstacle is expressed by 



{ 



J 



(3), 



where the integration is to be taken round the boundary of the surface r = R.* 



Since crv/c 2 is in all cases a small fraction, it is clear from (6), 3, that R may be 

 great compared with the wave-length of the incident sound and yet such that crVR/c 3 

 is a small fraction. In this case we may neglect- the imaginary part of AR in 

 expressing the value of < and fa at the surface r = R. Also, if o-R/c is great, it 

 follows that | &R | or XR is very great, since their ratio is the small quantity (o-v/c 2 ) 11 *. 

 Since, when | &R | is great, D n (&R) approximates to the value 



1/2 



* This method of finding an expression for the loss of energy was kindly suggested to me by 

 Prof. LAMB. I had previously obtained the same results by means of the dissipation function ; but the 

 work involved was very cumbrous. 



