Thus 



35 223 



o>.'=^, [60] 



where Uo/Zn is the frequency of free undamped oscillation as deduced from 

 non-compressive theory. Hence the preceding equation can be written 



A particular solution of the last equation is easily verified to be 



d^R ^ u^ ^ dR , 2/D D ^ Pi . r<n 

 V ^0 ^ + '-o (« - ^o) = - Tp- eosa,t [6^ ] 



iJ - /?„ = = gz- (wo^ - oj^)cos(jt + — '^^ sincjt] [62] 



This solution represents steady forced oscillations of the bubble. 



The occurrence of sin wt in Equation [62], or of dR/dt in Equation 

 [6l], represents the effect of radiation damping or of scattering of the in- 

 cident wave by the bubble. The complementary solution obtained by solving 

 Equation [6l] with p^ = 0, represents a superposed damped free oscillation 

 that soon dies out. 



Values of the derivatives that occur in Equation [5^+] may now be 

 calculated and inserted from Equations [62] and [55], with i set equal to 0. 

 Furthermore, R may be replaced by R^ for a first-order theory. Since the 

 equation must hold at all times, the cosine terms must balance independently 

 of the sine terms on the two sides of the equation. Thus are obtained two 

 equations for the determination of c' and a 



2 c^ 2 2 _ 2 . 47rTtc RqU (a>() ~ tj ) 



c ct — D 



Rn'u 



2, .6 



.a 1 AnnRit u' 

 2u — 7 = 



iu,' - a>2)2 + 



A more interesting form is obtained, however, by introducing, first, the 

 fraction of the space that is occupied by gas, or 



f^^nnR^' [63] 



second, the ratio of the elasticity of water to the elasticity of the gas, 

 which will be denoted by N^, where 



t Zip (U)g 



