83 97 



to move under the same type of proportional constraint as the plate. Let 

 the plate be mounted in a fixed plane baffle. Then the reduction principle 

 stated on page 75 can be utilized by the following trick. 



While the liquid is in contact with the plate, [11 6] holds; this 

 equation can be written 







= 2F,+ <P -Mi, - pjz^(t - j-)n{s)ds [153] 



When free, the surface is equivalent to a plate containing neither mass nor 

 stress forces; its equation can be formed from [153] by putting Af = and 

 <P = <P, where 



*. 



= jf(p,-p,)f(x,y)dxdy [15*+] 



and represents the effect of the difference between the hydrostatic pressure 

 Pq and the pressure p^ on the surface. The equation for the surface when 

 free is thus 



= 2F. +*.- p|z,(< -f) 77 (s)ds [155] 



Finally, to avoid discontinuous change, the pressure on the surface may be 

 supposed to change rapidly but continuously from the pressure exerted on it 

 by the plate Just before cavitation to the value p^. During this transition 

 process the equation for motion of the surface of the liquid may be written 







= 2F, + <P'- pjz^(t -j)r,(s)ds [1561 



where *' changes rapidly from <P - Mi ^ to *,; here z\ stands for the accelera- 

 tion Just before the transition begins. 



During the transition, high accelerations may occur, with the re- 

 sult that the velocity z^, of the liquid surface changes by 4z^; where, in 

 analogy with [150] when [/ = <», 



^Ki ^Wz'c -*+*.) ^57] 



'^ pcr]\0) ' ' 



•me reduction principle on page 75. which was based on [11 6], can 

 now be applied by noting that [153], [15^], and [155] can be regarded as suc- 

 cessive forms of [ll6] in which the constant Mis first replaced by 0, and 

 2Fi + is then replaced by an appropriate expression. In [136], let tj be 

 taken as the instant at which the transition to cavitation begins. Then, in 

 the integral in [136], during the transition 2Fj + * is replaced by 2F- + *', 



