SECONDARY PRESSURE WAVES 375 



in heat as the wave passes to infinite distance, and turbulent flow 

 around the bubble. 



A. Energy loss hy radiation. Herring (46) has shown that, if verti- 

 cal motion and internal energ}^ of the gas sphere are neglected, an inte- 

 gral of the equation of motion can be obtained which takes account of 

 compressibility of the water to a first approximation. This expression, 

 the derivation of which is outlined in section 8.7, is 



(9.26) 2irpoa^ (y\ - 47r r [Pa - Po\aHt = — f ^ aHa 



dm dm 



where a-m is the maximum radius of the bubble. Pa is the pressure on the 

 gas sphere of radius a, and Co is the velocity of sound. In deriving this 

 result, a number of approximations have been made. In particular 

 terms of order (1/co) (da/dt) have been dropped, which requires strictly 

 that the flow velocities everywhere be much less than the velocity of 

 sound. 



The terms on the left side of Eq. (9.26) represent the kinetic energy 

 of the gas as the radius contracts from a^ to a. Except for a constant, 

 this side is thus the conservative energy of noncompressive motion, and 

 changes in its value determined by the right hand side, which is always 

 negative, must represent energy radiated by compression of the water. 

 An exact solution of Eq. (9.26) would require a second equation deter- 

 mining P{a) in terms of radius a. If, however, it is assumed that the 

 difference from noncompressive motion is not great, the solution already 

 obtained for a{t) in this approximation may be used to evaluate the 

 right hand side and thus obtain a rough estimate of the energy loss. It 

 is evident from the preceding section that this term can only be signifi- 

 cant while the bubble is near its minimum size, and the major contri- 

 bution to the integral can come only at these times. This conclusion 

 is readily verified exphcitly by substituting expressions for dPa/dt 

 = dP a/ da -da/dt, and a{t), but this analysis is omitted here and only the 

 approximation to a{t) suitable for small a will be considered. 



The gas pressure P{a) for an adiabatic equation of the form 

 Pa-a^y = constant is conveniently written as Paa^^ = Pd-a^'^, where Pa 

 is the maximum pressure corresponding to the minimum radius a, and 

 hence dP/dt = { — 3y/a) Pa {a/a)-^y~^ {da/dt). Near the minimum, the 

 buoyancy of the gas sphere is negligible and the energy equation is 

 2Tr poa^ (da/dty -\- E{a) = Y, and hence 



dt \2xp, \ Y 



