- 5 - 



259 



For a jiven value of x, this vanishes if; 



e_a 



' I -a |_ a 



ez 



loj sinh \-~\ ♦ a cosh , 



(7) 



Kote that positive x and positive y are in opposite directions. 



For X = equation (7) gives us t = , J^ - loj (=) for the tine at which cavitation starts. 

 (t is called the cavitation tirrw^ Dy Kirkwood and others, and the compliance time t)y Kennard. We 

 denote it Cy 6Z. The velocity of the water at cavitation is the sum of the velociti^5 due to 

 the incident and reflected pulses, th^t is simply twice the velocity due to the incident pulse 

 (sines the pressures are equal ^nd opposite at this instant), so that we hive:- 



2 p^ -B-m 



2 P, 



Po' 



a cosh - 

 \6 



+ sinh 



a 



x(?-a) 



(8) 



{Bz] ' [ez 



this velocity oeinj towards the plate. 



By differentiating equation (7) we obtain tne velocity of the propagation of cavitation:' 



M 



1 + a coth 

 1 + a. 



(5) 



The cavitation front is thus always supersonic, a particular case of a result found by KenrBrd(l). 



(3 ) Energy considerations . 



Equations (7) and (8) are all that we require for settins) up a theory of the second phase 

 of the motion, after cavitation occurs, but before we do this we snail compute an upper limit to 

 the extra energy that may be ^iven to the plate Dy this mechanism. If we put i - Oz in equation 

 (il) we obtain:- 



lis 



Po' 



a 



Po' 



1 - (1 * a) a 



Td 



The kinetic energy of the plate, per unit area, is given Oy; 



2 n 1+a 



— 7 — 2 — °- 



p7^ 



2 P„ 



P Z 



r<t 



(10) 



(11 ) 



This energy will certainly appear as damage, whether or not any more is absorbed from the cavitated 

 water. Taylor leaves the theory at this stage. 



In Figure 1 rough pictures of the state of affairs at the time the plate leaves the water, 

 and at a slightly later time are given. It will be seen that, when the cavitation front reaches 

 Infinity, it will also have "eaten up* the part A C 3 of the reflected pulse, and the whole of the 

 incident pulse, but that the part C E of the reflected pulse always remains ahead of the 

 cavitation front, anJ is thus lost. The total available energy thus consists of three parts:- 



(a) 

 (b) 

 (c) 



The Taylor energy already given in equation (ll). 



The energy in the incident pulse from t = 5c to infinity. 



The energy in the negative portion A C B of the reflected pulse, at t = dZ, 



(b) is easily computed. It is simply /O ^-^ 



°.k 



^ 



ez 



c d t, the factor ■- being 



omrtted because we want potential plus kinetic energy. Thus we have:- 



' Pc "^ 



& 



(12) 

 (a) .. 



