209 



Now if <^ is the velocity potential ana V is the velocity of the sphere a*ay from the point 

 source then, assuming movement of the sphere small with respect to its radius, we have as boundary 

 condition at the surface of the sphere, 



2^ = - V Cos e, r = 

 3 r 



(8) 



whence it follows that 



2e 

 3r 



p -^^^t- = p £ 21- Cos 0, r =■ a 

 3r 3t a dT 



(9) 



Also if the total ntt force on the sphere in the direction away from the point source be denoted by 

 P, then 



2 77 a 



p Cjs Sin ^ d ( 



(10) 



Hence substituting from (9) and (lO) In equation (7) we have after performing th- 

 integrat Ijns, 



djp , 2 dP , jp ^ ""Pcs^ \'il * ^l 



OT' 



dT 



aT' dT 



U77 p^ a^ { r {^) *\ f (T) } 



(11) 



If the mass of the sphere be M then the equation of motion of the sphere is 



(12) 



dV . Pa 

 aT Mc 



(13) 



From ;,qjations (ll) ana (l3) the final equation determining P is thus, 



2-5 t 2k £f * 2K P = U77 p aM f (T) + T f (T) > 

 jT^ dT » 



2K= 2 * iIL^l£_ 

 3 M 



(1") 

 (15) 



For a given form of Incident wave, equation (it) is a simple linear differential equation 

 to determine p, whence the velocity V can be obtained from equation (l3). The displacement s Is 

 then given t>y 



V dt = 



V a T 



(16) 



The inltUl conditions for P are given by 

 P = 



^ = u 77 p a^ f (o) 



aT 



T - + 



(17). 



The first of these is in obvious physical requirement while the second corresponds Xc the 

 physical Condition of initl-.l complete reflection at tnt element of the sphere's surface nearest 

 the source. 



