490 WAVES OF EXPANSION. [CHAP. X 



We shall here only follow out in detail the case of n=l t which corresponds 

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



iS^acos 6 .................................... (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 



-^ = ae tv 'cos0 .............................. (v), 



for r = a. This gives 



The resultant pressure on the sphere is 



Ap cos 6 . 2ira? sin 6d6 ..................... (vii), 



I 



J 



where A^? = c 2 p s=p c?0/e& = i'0-p < ........................ ( vn i) 



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



This may be written in the form 



2 + Pa 2 du 



-- * 



where u(=ae l<r< ) 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=0. 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 a, so that the 

 vibrations of a sphere whose circumference is moderately small compared with 



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

 treatment of the problem of the vibrating sphere, see Poisson, " Sur les mouvements 

 shnultan6s d'un pendule et de 1'air environnant," Mem. de VAcad. des Sciences, 

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



f Poisson, I. c. 



