56 42 



with this assumption, the tendency for the motion to approximate 

 ultimately to the non-compresslve type can be formulated mathematically. At 

 the Instant of cavitation, an Impulsive decrease may occur In the velocity of 

 the liquid surface, but this Is not of much significance. For It may be In- 

 ferred from the reduction principle as developed In the Appendix that, within 

 a time of the order of the diffraction time T^ after the onset of cavitation 

 at a certain time tj, the velocity of the center of the liquid surface will 

 approximate to the value 



4 = 2;+!^/ (2F. + *.)dt [62] 



Here F, and M, are given by Equations [31a] and [36], respectively; 



<P. = jj (P^- Pc) f i^^y) <^^dy [63] 



where p^ is the total hydrostatic pressure in the liquid at the level of the 

 point I, y on the cavitated surface, p,. is the pressure in the cavity, and 

 the integral extends over the entire surface of the liquid under the plate; 

 and, finally, z^' stands for the velocity of the combined plate-liquid surface 

 at a time that precedes the onset of cavitation by an interval of the order 

 of the diffraction time; see Equation [158] in the Appendix. 



The value of z^ given by Equation [62] differs from the value given 

 by non-compresslve theory only in that the initial velocity z/ is not taken 

 at the instant of cavitation. If cavitation occurs very soon after the ar- 

 rival of the pressure wave, z/ is practically the same as the value of i.^ 

 Just before the arrival of the wave. 



Similarly, after closure of the cavitation at a time t^, the veloc- 

 ity of the combined liquid-plate surface soon becomes 



where M, * and F^ are as in Equation [29], z^p is the velocity of the plate 

 Just before Impact, and zj, is the velocity of the liquid surface at a time 

 that precedes t2 by an interval of the order of the diffraction time. See 

 the Appendix, Equations [l6l] and [l62], where an explicit expression for zj, 

 is given. 



This is again nearly the non-compresslve result. The last term In 

 Equation [64] represents the change in velocity of the liquid-loaded plate 

 that is caused by the applied forces. If z^', were replaced by the velocity 

 of the liquid surface at the moment of impact, the first two terms would 



