85 99 



where Sj is such a value of s that ( -Sj/c represents the time at which clos- 

 ure occurred and A is given by [1*^1]. Here z^, is a constant, and the part 

 of the Integral for < s < s, has been transformed into the term containing 

 A in the same way in which the similar term in [1^0] was obtained. 



The reduction principle can again be invoked in order to obtain an 

 expression for the final value of z^, the common velocity of liquid and plate. 

 If, in [136], (j is taken at the beginning of the transition process, the 

 transition itself again contributes nothing appreciable to the integral in 

 [156], which becomes here, from [153] used as a form of [n6], 



'/ V 



J(2F, + * - Mz^)dt = M[iStn) ' 2.(«/)] + / (2i=; + *)d« 

 'u 'n 



where (jj is the instant Just after the completion of the transition. In the 

 last integral tjj becomes replaced, as the time of transition is shortened to 

 zero, by the time (j or i ^^ at which the cavitation disappears; but ifj((,j ) 

 becomes 2^p(ic, ) or the velocity of the plate, not that of the liquid or 



Hence it follows from [136], with the M in that equation replaced 

 by 0, that, after the closure of cavitation at time t^, , at some instant with- 

 in any interval of length D/c the common velocity of liquid surface and plate 

 takes on momentarily the value 



z, = 



Af + M, 



M,z,,(t:,) + Afz,p((J+J(2F.+ *)dt 



[I6l] 



where z^^C «^, ) is the velocity of the plate at the instant t^, , whereas 

 ^ ci ( ^ ei ' ^^ ^'^® velocity of the liquid surface at an instant (^, that pre- 

 cedes tj, by less than D/c and usually by less than the diffraction time, T ^. 

 Here in [136] tj has been replaced by t^, and z^ftj) by z<.j{(jj). Actually, 

 the value of z^ that is obtained from [136] in the manner described is some- 

 what different; if it is denoted by zj, its relation to z^, as defined by 

 [l6l], can be written in the form 



. _ . , M 

 z, z. + 



M +Af 



[i,(t,) - z;] 



hence, since ilf/(ilf+ M, ) < 1, z^ lies between z[ and 'z^{tg), and, since the 

 velocity eventually traverses the entire range from z[ to Zj{(^), the value 

 Zj occurs also. The explicit expression for z^,((j,), obtained by using [135] 

 instead of [136I, is 



D 



KM'J = ^/(«o/) + m;J"[^c/(«c/ - f) - KMc] r,(s)ds [162] 



