p = p 



Pc O 



„ a 



p = p + p .p 



234 



the boundary remains at 

 rest, except, of course, 

 as It may move slightly 

 with the particle veloc- 

 ity of the water. The 

 incident waves are re- 

 flected as if at a free 

 surface at which the 

 pressure is always p^; 

 see Figure 4. This case 

 will occur, for example, 



whenever the incident waves are waves of tension but are not of sufficient 

 strength to cause fresh cavitation. 

 If, on the other hand, 



Vc 



u 



Figure U - A Plane 

 Stationary Boundary 



Figure 5 - A Plane Forced 



Closing-Front advancing 



toward the Right 



p' > -p-(p, + pcv^) 



the boundary advances toward the cavitated region as a forced closing-front; 

 see Figure 5. For the pressure p and the particle velocity v of the unbroken 

 water Just behind the front, the latter taken positive toward the side of 

 cavitation, and for V^, the speed of advance of the front relative to the 

 cavitated water ahead of it, the following formulas are obtained (2) 



_ _ (1 — r;) pcw^ 



_ W + 7] (c — W) 



^ ~^c- ^ 2w +r,(c -21/;) 



y. = c 



w 



m; + 7] (c — w) 



where 



'^ = oc"(2p' - p,) - V, 



[5] 

 [6] 

 [7] 



[8] 



p is the density of water and c the speed of sound in it, and r\ is the frac- 

 tion of space that is occupied by bubbles. 



According to Equation [7], V^ = c if r\ = 0. The boundaries at 

 which 77 = constitute, however, a singular case which will usually be of 

 momentary duration. 



The most interesting example of such a boundary is a breaking-front 

 which has just ceased advancing. Usually the advance ceases because V^ has 

 sunk to c and would go below this value if the front advanced farther; then, 

 by Equation [4], 77 = at the front. Furthermore, by Equation [3], in which 



