SECONDARY PRESSURE WAVES 361 



wave. The component of velocity normal to the plate in the wave, in- 

 cluding the afterflow velocity, is exactly equal and opposite to that in 

 the incident wave, and as for plane waves the excess pressure at the 

 plate is doubled by reflection. Similar considerations must hold for 

 more complicated structures and finite amplitude waves, the effect of a 

 spherical or any other wave being determined by the resulting pressure. 



Fig. 9.2 Variation of the radius a and pressure function o? — during the 



at 



bubble pulsation. 



This does not imply that the later portions of a pressure wave can be 

 disregarded; their effect is determined by the magnitude of the pressure 

 and the dynamical properties of the structure. 



C. The pressure in later phases of the motion. As the gas sphere ex- 

 pands, the pressure falls off more and more gradually, reaching a mini- 

 mum value at times near the instant of maximum expansion. At this 

 time, the pressure at points in the vicinity of the bubble is less than 

 hydrostatic, corresponding physically to the limit of radial oscillation 

 in which the gas pressure has fallen below the equilibrium hj^drostatic 

 value. Mathematically, the minimum pressure, except in the im- 

 mediate vicinity of the bubble surface, is determined by the minimum 

 value of d/dt {aHa/dt). The nature of variation of this function is 

 shown qualitatively by Fig. 9.2. The radial velocity da/dt decreases 

 continuously from its initial positive value to a large negative value at 



