22 HYDRODYNAMICAL RELATIONS 



spherical divergence. The pressure in the gas sphere, however, de- 

 creases as the vokime determined by the spherical boundary increases, 

 and the strength of this source must decrease. Outward accelerations 

 of the water near the boundary will thus decrease, but as long as there 

 is a pressure excess over hydrostatic, outward flow continues. 



The change in character of the motion can be made more explicit by 

 considering the relation between pressure at any point in the fluid, and 

 at the gas sphere of radius a. We have that 



P(r,0-P„ = V^-^) 

 and on the gas sphere 



P{a,t)-Po=^f(t--) 

 a \ Co/ 



which we can also write 



\ Co / a \ Co/ 



Comparing these equations 



(2.13) P{r, t) - Po = -^ [p(a, t - '^\ - pJ 



The pressure P{r, t) at any point in the fluid is thus determined by the 

 pressure P{a) on the boundary of radius a at a time (r — a)/co earlier. 

 This difference is just the time required for a pressure at a to be trans- 

 mitted to r and the time {t — (r — a)/c„) is commonly referred to as a 

 retarded time. 



In the later stages of the motion, for which the excess pressure 

 P(a, t) — Po is small and changes slowly with time, we can write Eq. 

 (2.13) as 



P{r,t) - Po^-lp(a,t) -P.I 



and the velocity Ur is, from Eq. (2.12), given by 



Uo = -^A [P{a, t) - Po] dt 



Ur 



a 



= ~, b^ait) - Ua{0)] 



This result, however, simply expresses the conservation of mass in non- 

 compressive radial flow, for which ^irrHr/di = iira-da/dt and hence 



