422 
Du y 2'4°- Bt) =p 
“Dt & tani 
Ee é+u a D > {4 4 
Blt = en Ac a Cru (4.11) 
The incompressible approximation to Bit) is the following, 
him (¢> %) Bit) = W = (p-Pd/Pp ° 
In this approximation, Eq. 4.11 becomes 
Du aus k > - Pp = Oo 
ot + ay Ze (4.12) 
Eqe 4.12 was employed by Lambo/ in his theory of underwater explosions to 
describe the motion of the gas sphere. To integrate Eq. 4.12, he neglected 
Pe and assumed p to be the equilibrium pressure of the uniformly expanding 
gas sphere of volume Ya). If the term p, is retained in the last term 
of Eqe 4.12, one finds that instead of a monotonic increase in a(t) , oscik 
lations of the gas sphere occur a long time after the emission of the shock 
wave, Butterworth2/ and Herring®/ have given a satisfactory treatment of 
these oscillations and the secondary pressure pulses arising from them. 
Since they have no appreciable influence on the shock wave, we shall approx 
imate the asymptotic value of Bt) at long times by zero instead of P./o ; 
In order to integrate Eq. 4.11 for the initial stages of the 
motion of the gas sphere, we recall that the peak approximation is suitable 
for p and & as long as their magnitude is of any significance for our pur— 
poses, App.iying the peak approximation to the factors (4) and Dp/ Dtin the 
first two terms of B(t) and neglecting the relatively small term u*/(c -u), 
we obtain 
1g/ C. Hering, NDRC Division 6 Report Chesr 20 (1941). 
35 
