232 DYNAMICAL THEOKY OF SOUND 



The most interesting case is where the radius a of the 

 sphere is small compared with X/2?r, where X is the wave-length. 

 In the immediate neighbourhood of the sphere kr will then be 

 small, and the formula (13) is, for this region, practically 

 identical with (2). It follows that 



<7=2iraM, ..................... (14) 



nearly, and further that the lines of motion near the sphere 

 will have sensibly the configuration shewn in Fig. 73. The 

 apparent addition to the inertia of the sphere has very 

 approximately the same value f Trpa? as before. On the other 

 hand, at distances r which are comparable with, or greater 

 than, X, the motion of the fluid is altogether modified by the 

 compressibility. At sufficiently great distances we have, by 

 (13) and (14), 



<t> = ^ika s A e ^cos0, ............... (15) 



or, in real form, 



S ff, ......... (16) 



corresponding to a velocity 



U = Acosnt ..................... (17) 



of the sphere. The amplitude now varies ultimately as 1/r, 

 instead of 1/r 2 , as in the case of (4). 



The investigation so far discloses nothing analogous to 

 a frictional resistance, whereas we know that owing to the 

 generation of waves travelling outwards a continual abstraction 

 of energy must take place. To calculate either the dissipative 

 resistance, or the work done, at the surface of the sphere, we 

 should have to use the complete formula (13); but the emission 

 of energy may be ascertained independently from the formula 

 (26) of 76. The strength of the equivalent double source 

 being given approximately by (14), we find 



W = %7rpk*a 6 cA 2 ................... (18) 



If p' denote the mean density of the sphere, its energy when 

 vibrating under the influence of (say) a spring will be 



