94 80 



of pressure occurs. Since In the neighborhood of such a point the pressure 

 differs only by a quantity of the second order, cavitation will then at once 

 occur at neighboring points as well. Thus the edge of the cavltated region, 

 advancing over the plate as a breaking-edge, will move at first at Infinite 

 speed. Eventually It will halt and return toward the cavltated area as a 

 closing-edge, leaving the liquid behind It In contact with the plate. 



Let U denote the speed of propagation of the edge In a direction 

 perpendicular to Itself, and let c denote the speed of sound In the liquid. 



If U^ c, the phenomena at the edge are essentially local In char- 

 acter and the analytical treatment Is easy. For effects can be propagated 

 through the liquid only at speed c; hence no effects propagated from points 

 behind the edge can overtake It, so that Its behavior Is determined entirely 

 by conditions ahead of It, and these conditions. In turn, are entirely unin- 

 fluenced by the approach of the edge. 



Consider, first, a breaking-edge. Let dn denote the perpendicular 

 distance from the edge to a point P ahead of It. 



Then the pressure, which is Pj at the edge, is 



p + -^ dn 

 * on 



at P, where dp/dn denotes the gradient of the pressure P in a direction per- 

 pendicular to the edge. The pressure at P will sink to p^, and the edge will, 

 therefore, move up .to P, in a time 



dp 



■^ — dn 



dt= ^" 



dp 



~ dt 



where dp/dt is the time derivative of the pressure in the liquid Just ahead 

 of the edge. Hence 



dn 



Thus U ^ c only if -dp/dt ^ c dp/dn. 



As the edge passes P, the pressure on the liquid surface, previous- 

 ly pj,, becomes p^ . If Pc> p^, the sudden Increase in the value of p in [103] 

 requires a compensating negative Increment of the Integral In that equation. 

 This Increment can arise only from high momentary accelerations of the liquid 

 surface. Hence, as the edge passes P, there occurs an Impulsive change in 

 the velocity of the liquid surface perpendicular to the plate. This change 

 is easily calculated. 



