HYDRODYNAMICAL RELATIONS 21 



outgoing wave about the center. The form of an infinitesimal spherical 

 wave thus does not change as it spreads out with the speed Co, but its 

 amplitude diminishes as r increases owing to the factor (1/r), which 

 results from the greater area over which the disturbance is spread as it 

 advances outward. 



C. The after flow. The relation of Ur to P{r) can be obtained by 

 integration of the first of Eqs. (2.10). Inserting the value of P, we 

 have 



(2.11 f^^lf(t-L\+±.jU-'-\ 



dt poCo r \ Co/ PoT^ \ Co/ 



Integrating from a time to to variable time t at constant distance r gives 

 (2.12) Urit) - Urito) = ^^^^ + — f [P{r, t') - Po] dt' 



poCo Par J . 



to 



If the time ^o = is chosen to precede any disturbance, it is seen 

 that the velocity in the fluid at a later time is a function, not only of the 

 pressure at that time but of all the previous changes in pressure after a 

 disturbance first reaches the point. These changes are such that, in a 

 radial disturbance, the water will be left with an outward velocity, or 

 afterflow, following passage of a positive pressure wave. This after- 

 flow remains, even though the pressure has returned to its equilibrium 

 value, and will be brought to zero only after the pressure falls below the 

 equilibrium value. 



A complete evaluation of the pressures and flow velocities behind the 

 front of an advancing pressure wave clearly can be made only by con- 

 sidering the properties of the spherical source, or agency by which the 

 pressure wave is generated. The conditions at the front, no matter 

 how far it progresses, are determined in the acoustic approximation by 

 the initial motion of the source. The conditions behind the front, how- 

 ever, depend on the later behavior of the source, and any physically 

 realizable source must in turn be affected by the motion of the fluid 

 surrounding it. 



To 'clarify the nature of the afterflow term, it is appropriate for 

 underwater explosions to consider the source as a spherical boundary 

 in the fluid containing gas initially at high pressure. The initial pres- 

 sure in the pressure wave is determined by the initial gas pressure. This 

 initial compression leaves behind it outward flowing water in an in- 

 creasingly large sphere. If the compression is to be maintained in this 

 volume, increasingly large displacements of water near the source are 

 necessary, despite the weakening of the initial pressure at the front by 



