262 - « - 



one of them is a and the other specifies the strength of the spring. The remaining physical 

 quantities can oe combined into multipliers in various ways by a suitable choice of non-dimensional 

 units. One possible choice is given belcw. 



(a) The case g = 0. 



2 p - ^ 2 p - ^ 



In this Cdse we have v. = g— ^ e "^ and equation (18) becomes:- y = g— f * ^ (t - i) 

 '' ^0 "o 



so that numerical integration an be carried out quite easily. It will be noticed that if 

 (J = the equations are both satisfied by taking x to be perrranently zero, meaning that, if the 

 plate is unrestrained, it moves forward uniformly and never collects any water on it at all. Any 

 restraint on the plate, hcwever small, will eventually result in its coll<?cting all the cavitated 

 water, so that there seems to be a definite singularity in the solution of this equation at o) = 0. 

 This peculiarity persists for all values of a, and may account for the difficulties experienced in 

 Obtaining a solution valid foroj small, the case we arc interested in. For example, one can use 

 successive approximations based on the solution y = fJs-l fora)= 0, and this leads to a solution 



in ascending powers of t' , but it is useless for calculating the maximum value of y because the 

 convergence becomes very poor If oit ~ l. A number of step-by-step calculations were carried out for 

 different vf.lues of Ci). These all suggested that x was nearly proportional to t, and an apprixinvite 

 solution based on this was also tried but failed for the same sort of reason. It is like tryinj to 

 calculate^ by equating the series for cos fi to zero. 



(b) The case g - 1. 



Equations (18) and (19) simplify somewhat in the apparently singular caseg.= 1; many of 

 the terms take an undetermined form, but can be evaluated without difficulty. The equations take 

 the form:- 



Af(i.x) il]. ^- exp[i-=-CiLl . sS = (21) N 



exp 



t - X + i.:i-i (22) N 



2 J 



dt [ dtj p 



r ^^ _... f 1 - .- ^A r . _ 1 - e- 2" 1 



where we have introduced non-dimensicndl units As follows:- 



Unit of t = 



Unit of X = C5 



Unit of y = 2 P^0lp^ c 



Unit of pressure = That pressure which acting against the spring under static 

 conditions would produce a deflection y equal to one non- 

 dimensional unit. 



S^ = a (J- 9^ 



Equations involving such units will be denoted by N. 



(6) Solut ions of these equations. 



A complete understanding of the problem would involve a set of solutions of these equations 

 over a range of values of the two parameters g and S , but only the cases a = and g = 1 seem workable 

 from a computational point of view (except to experts), For the present, however, it is doubtful 

 if expert assistance is needed, as we are Interested practically in small values of a (0.01 to 0.1, say) 

 and the solutions for g = should give a fair idea of the behaviour of thin plates. Five cases were 

 computed, for values of S ranging from 1 to 10"^. S = 1 represents the condition where the timti- 

 constant of the plate under plastic stresses is comparable with the time-constant of the pressure -pulse. 

 This Is an extreme cese, and would only hold, e.g. for a simll diaphragm gauge. The the?ry cannjt, 

 however, be applied directly to such a gauge, Dn account of diffraction effect jf the edges, which 

 would be very Important. An idea jf the behaviour of thick plates ci^n be ;btaine3 by s::lving the 

 equation for a = 1 in a representative case (actually g= 1, S = 10" ), The results obtained are set 

 out in Table 2, and the solut ions for S - 10~ , g = and g = 1 are plotted in Figures 2 and 3 



respect ively 



