490 WAVES OF EXPANSION. [CHAP. X 



We shall here only follow out in detail the case of ?i = l, which corresponds 

 to an oscillation of the sphere, as rigid, to and fro in a straight line. Putting 



S l = a cos 8 .................................... (iii), 



where 6 is the angle which r makes with the line in which the centre 

 oscillates, the formula (ii) reduces to 



(iv). 

 The value of C is determined by the surface-condition 



-^Loe^cos* ........ , ..................... (v), 



for r=a. This gives 



_ _ 



The resultant pressure on the sphere is 



X= - I Ap cos 6 . 27ra 2 sin 6dB ..................... (vii), 



J o 

 where A^ = c 2 p s = p c^/c& = iVp &amp;lt;/&amp;gt; ........................ (viii). 



Substituting from (iv) and (vi), and performing the integration, we find 



2 + Fa 2 -iFa 3 . iat .. . 



X= --|7r Po a 3 . j-j-p-p ivae 1 ** .................. (ix). 



This may be written in the form 



du 



where u(=ae l&amp;lt;rt ) denotes the velocity of the sphere. 



The first term of this expression is the same as if the inertia of the 

 sphere were increased by the amount 



_ , ., 



whilst the second is the same as if the sphere were subject to a frictional force 

 varying as the velocity, the coefficient being 



In the case of an incompressible fluid, and, more generally, whenever the 

 wave-length 2?r/^ is large compared with the circumference of the sphere, we 

 may put ka = Q. The addition to the inertia is then half that of the fluid 

 displaced ; whilst the frictional coefficient vanishes f. Cf. Art. 91. 



The frictional coefficient is in any case of high order in ka^ so that the 

 vibrations of a sphere whose circumference is moderately small compared with 



* This formula is given by Lord Bayleigh, Theory of Sound, Art. 325. For another 

 treatment of the problem of the vibrating sphere, see Poisson, &quot; Sur les mouvements 

 simultan6s d un pendule et de 1 air environnant,&quot; Mem. de VAcad. des Sciences, 

 t. xi. (1832), and Kirchhoff, Mecluinik, c. xxiii. 



f Poisson, I.e. 



