Pressure Pulses in Liquid-Bubble Mixtures 



27 



(■/- 1) ^w Co 



(7- i)/2r 



(19) 



(Note that the propagation velocity c increases much more rapidly with the 

 pressure ratio p/p^ than it does when the whole fluid is a gas, in which case 



c = (7p/^)i/2-p(>-i)/2r.) 



The preceding exact equations for very long waves may be helpfully simpli- 

 fied if attention is restricted to fairly weak pressure pulses. Writing 



— = l + P , 



Po 



we assume P to be a fairly small fraction and hence obtain from (17) and (19) 

 the approximations 



-'^'^ 



(20) 



and 



Pw Cf 



P . 



(21) 



Consider now a wave propagating in the x direction toward an undisturbed 

 region. After a time the frontal parts of the wave will be covered by backward- 

 going characteristics along which the invariant v - f has the value zero deter- 

 mined at the origin of these characteristics in the undisturbed region ahead of 

 the wave. Thus we have v+ f = 2f , and on substituting (20) and (21) we get from 

 (11) 



3t 



1 + 



y+ 1 



27 



Pw Cf 



T- = . 



(22) 



Writing, for the sake of neatness, 



fy+l Po \ 



and reintroducing the dimensionless variables T and X, we then have 



-^ + — ^ + Q —^ 

 BT 3X BX 



. 



(23) 



The third term on the left side of (23) is the required simple representation of 

 the nonlinear effects. 



Both Eqs. (8) and (23) have precise counterparts in water-wave theory, and 

 various schemes have been developed in that context for simultaneously obtain- 

 ing approximations to the effects of both frequency dispersion and nonlinearity. 



133 



