96 82 



For a closing-edge, the same calculation applies except that here 

 Ai is fixed by conditions in the cavitated region ahead of the advancing edge, 

 and the impulsive change Ap in the pressure at the surface of the liquid is 

 to be found. As closing occurs, the velocity of the liquid surface suddenly 

 changes from some value Z; to the velocity Zp of the plate. The liquid sur- 

 face behaves like a plate of zero mass, hence it alone changes velocity in 

 the impact. Hence, from the first part of [1^8], 



2 - — 



Ap = pc(l--^) Uz,-i^) [151] 



U 



If Z; exceeds Zp ahead of the edge, the liquid surface will usually 

 meet the plate at a finite angle 0. Then in time dt the edge will advance a 

 distance U dt over the plate of such magnitude that U dt tan © = (z, - Zj,)dt. 

 Hence for a closing-edge of the type under consideration 



U = \' ~y [152] 



and an edge can advance as a closing-edge moving at speed U^ c only if 

 Z; - Zp ^ c tan 9. Exceptionally, it might happen momentarily that e = 

 and z , = z . 



If conditions are not such as to cause the edge of the cavitated 

 area to travel at a speed equal to or greater than c, it seems clear that the 

 edge will usually stand still, except as it may be carried along by flow of 

 the liquid parallel to the plate. For, propagation of pressure waves from or 

 to the free surface of the liquid should prevent the occurrence of large dif- 

 ferences of pressure in the liquid near the edge. Hence, if p^^ < p^, pres- 

 sures so low as Pj cannot occur at the edge, and further cavitation cannot 

 occur. Impulsive changes of velocity are likewise impossible; if such im- 

 pulsive action begins, but the edge moves at a speed less than c, the im- 

 pulsive pressure developed will produce such a redistribution of velocities 

 in the liquid as to equalize Zj and Zp on the cavitated side of the edge. As 

 an exceptional case, the liquid surface might perhaps roll onto the plate 

 like a rug being laid down on a floor. 



Otherwise, under the assumed conditions, the edge will move only as 

 it is carried along by the liquid in its particle motion. In a strict linear 

 theory, therefore, in which all particle velocities are assumed to be negli- 

 gibly small, the edge of the cavitated area must stand still except when it 

 can move at least at the speed of sound. 



CAVITATION WITH DOUBLE PROPORTIONAL CONSTRAINT 



Something more can be Inferred, including useful relations with 

 non-compressive theory, if the surface of the liquid is arbitrarily assumed 



