598 



(v) Effect of taking into account the compressibility of water. 



There still ranains another likely source of error to account for the poor agreement 

 between the theoretical and experimental volumfs of dishing. So far we have assumed incompressiple 

 flow whereas it irould be more correct to allow for the compressibility of water. 



As shown in the Appendix the rquation of notion taking into account the compressibility 

 of water is given by 



1 d'^ 

 7 ^ 



F(-J) 



c^j(x) ■J"(T- x) dx 



(A. 35) 



K being a numerical constant and <j;, and i rather conplicated functions. 



Integrating this equation by parts three times gives 

 ^2, 



l-y^ 315 J dT^ 6 dT 



!2 e-^T 



Pi 



"ji" (x) -1 (T - x) dx 



= -ii e 

 Pi 



cf, (x) J (T - x) dx 



< T < I 



(«.33) 



This equation vas numerically integrated by Admiralty Computing Service for two extreme cases, 

 viz. 650 lbs. T.N.T, at 167 feet and 5 lbs. at 10 feet correspnding to charge distances of 88 

 and 27 charge radii respectively. 



The effect of this more refined treatment by allowing for compressibility is very small, 

 the calculated volumes of dishing being thereby reduced by only about 2% as shown in Table 4, 



General di scussion and conclusions. 



c original calculated values and the observed values of dishing are compared in Table 2 

 t .". large discrepancy ranging from 201- i»OJ overest imat ion by the theory. As possible 



Th 

 and exhib 

 sources of this discrepancy we have considered 



(a) Increased energy absorption under dynamic loading as compared with static 

 loading. 



(b) Use of the American formula (B). based on more modern data than Wood's formula 

 (6), for the parameters of the pressure pulse. 



(c) Allowance for the compressibility of the water. 



Of these the third possibility (c) introduces only a very small correction of order 3% and 

 Is only therefore a minor factor in the discrepancy. The second possibility (b) however produces 

 an appreciable Improvement in the agreement but still leaves a remaining discrepancy of order 

 10*- 30*. If this is prim^.rily due to the first possibility (a) we should require an overall 

 increase of order 25J - 35* of resistance to dishing under dynamic as opposed to static loading. 

 This seems unduly high for the annealed copper used In the diaphragms since, so far as Is known, 

 there is no evidence to indicate an increase of more than about 15* for copper due to dyramic 

 loading. 



There ranain, of course, other possible sources of error In the assumptions which we have 

 not examined, on account of their implicit complexity. Thus we have neglected diffraction effects 

 round the gauge and bodily motion but it seems unlikely that Such are important for the present 

 purpose in view of the observed fact that the two diajtiragms on opposite sides of the gauge exhibit 

 the same overall variation of dishing with charge weight etc. The assumption that the dish is of 

 parabolic shape throughout is, of course, only an approximation and will certainly not be valid 

 in the early stagos of dishing when the deformation will correspond more to a plastic wave 

 travelling inwards from the circumference. It seems unlikely, however, that the difference In 



