39 53 



There remain then, for discussion, the process by which the region of cavita- 

 tion spreads over the plate, the subsequent motion of the free liquid surface, 

 and the final process by which the cavitation is destroyed. 



After cavitation has begun, the edge of the cavitated region will 

 advance over the plate for a time as a breaking-edge, enlarging the area of 

 cavitation; then it will halt and eventually return as a closing-edge; see 

 Figure 22. It must begin its advance from the initial point at infinite 

 speed; and it may happen that the cavitation spreads instantaneously over a 

 finite area. Similarly the cavitation may disappear simultaneously over a 

 certain area, in which case the closing-edge may be supposed to move at an 

 infinite speed. In other cases the edge will move at a finite speed. 



The process at the edge turns out to be distinctly different ac- 

 cording as U, the velocity of its propagation in a direction perpendicular 

 to itself, is less or greater than c, the speed of sound in the liquid. 



If U < c, it appears that no discontinuities of pressure or par- 

 ticle velocity can occur at the edge of the cavitated region, and U is mere- 

 ly the velocity with which the liquid next to the edge is streaming over the 

 plate. This velocity, in turn, is determined jointly by the incident wave 

 and by all of the diffracted waves emitted by various parts of the plate, 

 and no simple statement in regard to its value can be made. 



If U = c, on the other hand, the propagation of the edge is essen- 

 tially a local phenomenon, and mathematical treatment is easy. For effects 

 can be propagated through the liquid only at the speed c, and no such ef- 

 fects coming from points behind the moving edge can overtake it; thus its 

 behavior must be determined solely by conditions Just ahead of it, and these 

 in turn cannot be affected by the approach of the edge. For the same reason, 

 the analytical results are not limited now to small displacements of the 

 plate. Impulsive effects also become possible. 



For a breaking-edge moving in this manner, 



£e 



dn 



where dp/dt is the rate of change of the pressure in the liquid ahead of the 

 edge, as determined by the incident pressure wave and the motion of the plate, 

 and dp/dn is the gradient of this pressure over the plate in a direction nor- 

 mal to the edge; see Equation [l'^7] in the Appendix. Here, necessarily, 

 dp/dt < 0, Thus the edge of the cavitated area will advance toward the un- 

 broken side at the speed [/ ^ c provided - dp/dt ^ c dp/dn. 



