222 3^ 



and the displacement and acceleration of the surface of the bubble are, re- 

 spectively. 



R-Ro=l 



dR , Po I c . . ,\ 



— a t = — *— =— cos w < + sin w t 



dt pcu \ uR^ 



d'R p^jc ,, ,,\ 



= -!-^[ — coswt — (jjsmut 



dt^ pc ^R^ I 



It Is easily verified from these equations that the value of p^ at the sur- 

 face of the bubble can be written In terms of R as follows 



V, = ^^^^2 [58] 



Here w has reference to the Incident waves, so that if X is their 

 wave length in homogeneous water, 



(jj _ 2n u)Rq _ „ i?o 

 c k ' c X 



But according to our assumptions, R^/k must be small. Hence the term u^R^/c^ 

 can be dropped and Equation [58] can be written 



D d R , p 2d 2 dR rcnl 



dt' c dt 



The total pressure is now the sum, p -^ Vt'' ^'^^ ^* ^^^ bubble this 

 must equal the pressure of the gas. The volume strain of the bubbles is 

 8V/V= 6{R^)/R^ = 38R/R= 3(i?- «o)/^o- Hence, if E' is the elasticity of 

 the gas, the pressure of the gas is 



Po-3^'^^ 



Hence 



-3E(R - R,) ^ 

 P + Pe= o + Po 



"■0 



and by Equations [57] and [59]. 



D '^ R -t- P 2e, 2 dR 3E , r, D \ , 



P^oJlY + -'^ Ro JT ^^^^-^0^^ -P.cosujt 



The corresponding equation for non-compresslve theory is obtained by letting 

 c become infinite. If p, is also replaced by zero, the usual equation for 

 free oscillation is obtained, namely, 



dt' ^ pR,'^^ ^o' '^ 



