MOTION OF THE GAS SPHERE 807 



deviations from the acoustic result of Eq. (8.35) are not large, being 

 greatest at and near the stage of greatest contraction. This analysis, 

 which we shall omit, shows that the most important correction term to 

 be added to the right side of Eq. (8.35) is 



oadt\_ \dtj J 



Coa dt 



Substituting in Eq. (8.33), using Eq. (8.34) for X(a), and rearranging 

 gives 



d^a , 3 



\dt ) aco dt I \dt ) J pco dt \ c„ dt ) J 



d^ 



P 



The left side of this equation can be written as a derivative, and if den- 

 sity variations are neglected so that p can be replaced by po, the density 

 at the depth of explosion, we can integrate over a. Taking the lower 

 limit to be the maximum radius a^ for which da/dt = 0, we obtain 



(8.36) 



\dt / \ Zcodt / J |_ Po poCo dt \ Co dt /J 



In this expression, the terms in 1/co represent approximately the 

 effect of compressibility, and if these terms are neglected the energy 

 equation previously derived (Eq. (8.5)) is obtained. The factor 

 (1 — 4/3 1/co da/dt) changes sign when the bubble passes its maximum 

 radius, as da/dt becomes negative, and hence introduces a dissymmetry 

 in the radius-time curve. This is, however, not particularly important 

 as da/dt, the radial velocity, is very much less than the velocity of sound 

 Co over nearly the entire pulsation. The term involving dP{a)/dt repre- 

 sents the acoustic radiation of energy, and will be further discussed in 

 section 9.4. 



The equation (8.36) in principle permits a solution for the radius- 

 time curve and the corresponding pressure at the surface of the bubble 

 (which is determined for a given a), but no calculations of this kind 

 have been made. It seems quite certain that the deviations from non- 

 compressive theory will be insignificant over most of any one cycle, al- 

 though the radiation term does lead to a large energy loss near the mini- 

 mum. 



B. The effect of gravity. In the theory developed by Taylor, the 

 gas bubble is assumed to remain spherical despite its vertical migration 



