13 



Giltnore^^ extended these computations to include higher order compressibility terms 

 as well as the effects of viscosity and surface tension. Instead of using the acoustic approxi- 

 mation (i.e., all disturbances propagated with the velocity of sound c), Gilmore assumes the 

 Kirkwood-Bethe hypothesis,^'* which assumes that disturbances are propagated with the 

 velocity c + u, u being the local fluid velocity. Gilmore then derives the equation of motion 

 of the bubble wall in the form:* 



dR\ C) 2 \ W) \ C] C dR\ cj 

 where R is the radius of the bubble, 



C is the sonic velocity in the liquid at the bubble wall, 



H is the enthalpy difference |= r2 ] between the liquid at pressure P at the bubble 



wall and at pressure p^ at an infinite distance from the bubble, and 

 U is the velocity of the bubble wall. 



The values of H and C are derived from the experimentally developed formula for isentropic 

 compression of liquids. 



P^ + B [pj 



where B and n are constants for each liquid (for water, B = 3000 atm and n = 7). Thus, he 

 finds that 



and 



'\P^ + Bj 



^_ n(p^ + B) r / P + B ^^ -^ 



The effects of surface tension and viscosity are included in the boundary conditions by 

 writing the pressure at the bubble wall as 



"--^-'(^l 



where P ■ is the pressure of the internal gas, 

 a is the surface tension, and 

 11 is the dynamic viscosity. 



*Only a very few results of Gilmore's paper are abstracted here. For full details see Reference 21. 



