BY SMALL OBSTACLES OF CYLINDRICAL AND SPHERICAL FORM. 251 



that the effect upon the primary waves of very small obstacles would be almost 

 independent of the dimensions of the obstacles, provided they were limited to be of 

 the same order of magnitude. 



Let us now turn to the case when \a is great. In this case we have approximately, 

 from (17) and (19) of 3, 



A, = Jur/r'a 2 [1 + v/2 (\a)-' -U 2 a 2 (log 

 A = _fr/ t V{l+i/<V(logi/ 

 Hence we find approximately 



v 



/2 (X)-' +* (Xa)-'}. 



Substituting this value for [tA + A,] in (6), we obtain, when A/f is great, for the 

 total loss of energy to the primary waves the formula 



hW { v/2 (Xft)- 1 + 1 (X)-'}. 



The ratio of this to p,,o 2 a/c, which represents the rate at which energy is incident 

 upon the obstacle, is given by 



which gives the proportion of the incident energy which is lost to the primary waves. 



The first term in (9) is independent of the viscosity of the medium, and is obtained 

 in the ordinary theory of a non-viscous air. The second and third terms of ( ( J) 

 represent the additional loss of energy to the primary waves consequent upon the 

 viscosity of the medium. Further, since the ratio of the second to the third term of 

 (9) is of order Xa, it follows that the latter may be disregarded. Hence we see that, 

 since a does not enter into the second term of (9), the additional proportional loss of 

 energy consequent upon the viscosity is almost independent of the magnitude of the 

 obstacles. In other words, the actual loss of energy in the primary waves due to 

 friction is proportional to the radius of the obstructing cylinder if this be sufficiently 

 large. This last result is clearly what might have been expected on physical 

 grounds. 



It remains to consider the case when X is neither very small nor very great. In 

 this case it is impossible to approximate to the values of the Bessel functions involved. 



From (13) 3 we have 



A l|TT-7) 2 ,- 2 ~_2| / 



AI -^LTTfia ' 



D u (ka) 



Hence it follows that 



= *j w ;1 - P'^ 



(\ae^) Do (Xae-' 

 2 K 2 



