-5- 273 



Pressure along the Axis of the Plate . 



If we tai<e the motion of the plate as given, we may calculate the pressure due to this 

 motion at points along the axis, and in particular at the centre of the plate, without approximation. 

 If u(t) represents the velocity of the plate, x the distance along the axis, arti r the distance 

 from this point to a typical point on the plate, then we have, by the usual formula, 



2 2 * 



Pp^eaf'"'"' r'7^M drd0=-p„C u{t - S ) - tft - (il-^i-T) ^ (9) 



n" C(j.l.r «...= -..< .1.-1) -.(.-'ip;>) 



which is rigorously correct along the axis. In our case U vanishes for negative times, so the 

 two terms in equation (9) vanish for negative values of the argument. In addition to this, we 

 have the pressure due to the incident pulse, and the pressure duo to the pulse reflected by the 

 rigid wall, expression (9) giving the modification to the latter due to part of the wall t)eing 

 movable. We thus have for the total pressure the complete expression 



■ ^ " * c' * p„- 5 '* ■ c' - P„ c U(t - J) + p„ c u 



--0 C --o 



(-'^') 



(10) 



By reans of this expression, we can discuss the complete history of pressure variations at points 

 along the axis. The condition for cavitation is that expression (lO) must be zero or negative. 

 By the approximation used in optical diffraction theory (that the plate may be taken as a portion 

 of a sphere with the point in question as centre), one can also calculate the variation of pressure 

 at points whose distance from the axis is small compared with x, but a complete expression for the 

 pressure history at an arbitrary point does not seem to be possible, except as a definite integral 

 or a series, even if we assume U to be given 



The Onset of Cavitation . 



If we use equation (3), or its approximate solution (u) , in conjunction with equation (10), 

 and put X = 0, we get the condition for the occurrence of zero pressure at the central point in 

 the form 



p-t (i-g^)- jal/^-a/^) p^l . il.LMh.LA p-'[t*p]- H (u) 



(ai + ,?) {2a * 0) 



\ ce v ) 



which reduces to Taylor's cavitation condition (2) for /? very large. 



This transcendental equation has been solved for various values of a and /3, and the results 

 are plotted in Figure 1. It will be seen that tor P> 10 the plate may be effectively regarded 

 as infinite, but that for smaller values the cavitation time begins to shorten appreciably. We 

 now have to consider what happens for values of the non-dimensional time greater than ■^so that 

 the final term in equation (lO) becomes non-zero. This term always contributes a positive pressure, 

 so tending to prevent cavitation. It can, in fact, be shown that if the pressure at the centre 

 just vanishes at the critical time t = j when the diffraction wave has just arrived at the centre, 

 the time derivative of the pressure will be positive. This confirms Kirkwood's (6) suggested 

 criterion for cavitation, that it occurs either before t = ^ , or not at all. Equation (lO) shows, 

 without difficulty, that cavitation is more likely to occur at the centre than at any other point 

 on the axis. It also follows that the centre is more favourable than other points on the plate, 

 because the diffraction wave arrives at any other point before it does at the centre. (it can 

 be verified rigorously that for points near the centre the term - p CU is unaffected, but that 

 the second term in equation (9) has to be modified to allow for the edge being nearer). 



2 B 

 Insertion of tho value -:=- for the non-dimensional time in equation (ll) gives us a 



critical curve relating to a and/?. The following values have been worked out, and compared 



with what one gets if one uses Taylor's expression (2) t = \^^l' '") '<"■ tlie cavitation 



tirre, i.e. neglects the effect of finite /3. 



Table l 



