306 MOTION OF THE GAS SPHERE 



where the variable X has been defined as the divergence of the velocity 

 u,X = (l/r^) d/dr (r^u), and the following relations have been employed: 



ri*=-ji<""Kt) 



= I ^ ':rr I + I r — dr (integrating by parts) , 



an J dt 



[du du~\ _ d^a 



'dt Trjr^a ~ de 



If we assume that the density is a known function of pressure P, 

 neglecting irreversible processes induced by the preceding shock wave, 

 the right side of Eq. (8.33) is a known function of P(a), the pressure in 

 the gas sphere. The departures from incompressible theory are repre- 

 sented by the terms in X. The equation of continuity supplies a means 

 for evaluating X(a) in terms of P{a), however, for we have from section 

 2.3 that 



(8.34) m = [div«],=<. = \- i^l = [- i-^' 



[_ p atjr=a L ^"P dt Jr=a 



where c^ = {dP/dp)s is the velocity of sound. With this relation, 

 Eq. (8.33) becomes an ordinary differential equation for a in terms of 



^ a 



Pip), except for the integral I rd\/dtdr. This term is, moreover, 



the only one which can account for loss of energy by radiation of a pres- 

 sure pulse, as the others are unaffected by a change in the sign of 

 u{a) = da/dt. If the motion of the water is sufficiently small for acous- 

 tic theory to be applied, this integral would be given exactly by a simple 

 analysis. For in the limit of small amplitudes, the velocity u, and 

 hence its divergence X, must satisfy the wave equation A-X — il/co~) 

 d^X/dt"^ = 0. Hence for spherical symmetry X is of the form (1/r) 

 f{t — r/co). We therefore have r d\/dt = —Cod/dr (rX), and the inte- 

 gral in question becomes 



(8.35) f r — dr=-Co[ ~ {r\)dr = Coa\(a) 



J dt J dr 



In the present problem, the amplitude of motion is sufficiently large 

 that the acoustic theory is not valid, but Herring has shown that the 



