98 84 



as in [156], but the resulting contribution to the integral is negligible be- 

 cause of the extreme shortness of the time Interval. Hence, the integral may- 

 be written simply as 



|(2F, + *,)d« 



from [155]> where tj„ is the time at which cavitation occurs. 



Hence, putting M = In [136], it may be concluded that, after the 

 onset of cavitation, within any time interval of length D/c the velocity of 

 the surface of the liquid will take on at least once the value 



= K(t'J + ^j(2F, + <P,)dt [158] 



Here t^ is the time at the end of the chosen interval and z<;(*ct> ) is the com- 

 mon velocity of liquid surface and plate at some instant that precedes the 

 onset of cavitation by an Interval less than D/c. A specific expression for 

 z^itl^} can be obtained by using [155] Instead of [136]. From this expres- 

 sion it is easily seen that, if cavitation follows the Incidence of a pres- 

 sure wave within an interval much less than D/c, then z^(t^„ ) is approximately 

 equal to the velocity of the plate Just prior to the incidence of the wave. 

 It will be noted that the value of z^, given by [158] represents 

 the value of z^, at time tj- as calculated from non-compressive theory, ex- 

 cept for the substitution of i^^^'cv^ ^°^ K^Kv^ ^^ ^^^ initial velocity. 

 For the non-compressive value can be obtained by Integrating the analog of 

 [126] for a free surface or 



M, ■z„ = 2F. +*, [159] 



In [158] the initial impulsive change of velocity has disappeared. 



During the reverse, process that occurs when the cavitation closes, 

 the velocity of the liquid surface changes impulsively from some value z^, to 

 the velocity z^ which the plate happens to have at that instant. Thereafter 

 [153] holds again; but in this equation some of those values of 'z\ ( t - s/c) 

 that have reference to times before the closure of the cavitation are now 

 values of the acceleration of the free liquid surface. 



During a time after the closure that is short relatively to the 

 diffraction time, [153] can be written approximately as 



D 



Mz^ = 2F, + <P + pcAd^i - ij - pjz,[t - j)r](s)ds [16O] 



