236 



The subsequent closing of a gap, provided it does not contain an 

 appreciable amount of air or other foreign gas, will result in the usual 

 water-hammer effect. If the gap closes against a rigid boundary moving at 

 fixed velocity i'", and v' is the particle velocity of the advancing water, 

 the pressure rises instantaneously from p^. to p^ + pc{v' - v"). When a gap 

 closes in the midst of the v;ater, however, with a difference v' - v" in the 

 particle velocities on the two sides of the gap, the impact pressure is only 

 -ppc(f' - I'"); here the pressure at the gap rises instantaneously from p to 

 p^ + 2pc(v' - v"). The action is, in fact, the same as if the two masses of 

 v/ater had Impinged simultaneously and from opposite sides upon a thin solid 

 sheet moving with the mean velocity of the water or a velocity - {v' + v"). 



CAVITATION AND DYNAMICAL SIMILARITY 



Cavitation in the midst of a liquid differs in its effect upon re- 

 lations of similarity from cavitation at the surface of a solid. 



A glance at the differential equations of sound, or at some of the 

 equations written in this report, shows that, in constructing a possible mo- 

 tion similar to a given one, but on a different scale, it is necessary to 

 preserve unchanged at corresponding points the values of the two dimension- 

 less quantities 



P _V_ 

 pv' pc'- 



where p is the pressure referred to any chosen datum or zero of pressure, 

 V is the particle velocity, 

 p Is the density, and 

 c is the speed of sound in the liquid in question, here water. 



In a given liquid, with fixed p and c, it follows that both p and the par- 

 ticle velocity must be preserved at corresponding points. The only transfor- 

 mation that is possible is thus the simple one, familiar in the discussion of 

 underwater explosions, in which all linear dimensions and all times are 

 changed in the same uniform ratio. The occurrence of cavitation at fixed 

 values of p^ and p^ alters nothing in this conclusion so long as cavities of 

 appreciable size do not form. 



If large gaps occur, however, gravity may play a role in their 

 neighborhood. Then, from such equations as s = ^gt^ and p = pgh, where s is 

 the displacement in time t or /; is the static head, it is evident that, for 

 similarity to hold, an additional quantity must be preserved. This may be 

 written in various forms, such as gs t /s or 



gl 



„2 



