355 
=-9- 
methods of numerical differentiation are not applicable. An attempt to maintain 
the accuracy, by keeping the first few steps after the arrival of the rarefaction 
wave at the origin small, was made but it is difficult to assess the error duc to 
this cause. However, events at the origin are unlikely to affect the main portion 
of the pressure pulse for a reason that we shall discuss immediately. 
(b) The step-by-step method broke down in the gas shortly after the reflection of the 
rarefaction wave at the origin, a discontinuity in the Q function appearing about 
half-way between the origin and the interface. A similar effect secms to have 
occurred with Penney and Dasgupta calculations, but it is understood that it did 
occur in Penney'’s original calculations. The effect might be interpreted as the 
incipient formation of a shock-wave travelling inwards, but it is difficult to 
see why a rarefaction wave should suddenly reverse its direction of propagation 
and thus become a shock-wave. Such effects are usually associated with a 
boundary reflection, but no boundary is anywhere near. 'n another report we 
have described the phenomenon in detail, and have suggested that it may be an 
instance of an apparent "negative shock-front", such as has been observed by 
Libessart though the persistence of such a phenomenon for any finite time also 
leads to serious difficulties. 
For the purpose of these calculations, it is important not so much to elucidate the exact 
nature of the effect but to decide what effect it is likely to have on conditions in the water. 
For this purpose the calculations were continued in the following way (after we had first satisfied 
ourselves, by repeating part of the calculations with a reduced size of step, that tne appearance 
of the discontinuity was a real phenomenon). A continuati'on of the calculation, accepting the 
Riemann theory literally, would have given us a Q curve witn two branches overlapping one another 
and connected by an S-shaped portion, so that, for certain values of r there would have been three 
possible values of Q. The lower part of the curve, starting from the origin, was retained up to 
the point at which the S-shaped portion began to bend backwards, and all of the upper branch of the 
curve overlapping this portion was erased. It is clear that this procedure will lead to the total 
energy being under-estimated if the real phenomenon is (say) an ingoing shock-wave, and we find 
indeed from Tables | and || that there is a marked decrease in the total energy in the latter steps, 
the decrease in fact practically coinciding with the appearance of the discontinuity in Q 
Our reason for adopting this method of calculation was to try to find out whether the 
rarefaction wave could reach the interface, and thus affect conditions in the water, possibly 
producing a pronounced minimum in the pressure-time curve of the kind found by Penney. We obtained 
the reassuring result that, even with our extreme assumption, the discontinuity in Q was only propagated 
with a speec comparable with the advance of the gas-water interface, so that it would be a long time 
before it comla aff=ct conditions in the water. In fact, the discontinuity in Q was still far benind 
the interface when tne main portian of the pressure pulse had formed and the step-by-step calculations 
were broken off. Thus it would appear that conditions at the orijin are not likely to have any 
important effect on the pulse fn the water. 
Summary of results. 
In Tables IV and V the relation between pressure and time, which is what would be measured 
by a gauge at a fixed position in space, is given for distances of approximately 2, 3, 4 and 5 charge 
radii for both explosives. tn Figuresi and 2 the same data are plotted logarithmically, and it will 
be see that, with the exception of the case = = 3.08 for T.N.T., the pressure-time curve is a 
reasonable exponential even quite close to the charge, thus proving some justification for Kirkwood's 
second assumption. The time-constants if the exponentials given by the slopes of these curves are 
also given in the Tables. Owing to the "eatiny-up" of the pressure pulse by the shock-front, it is 
not possible to obtain satisfactory figures for the time constants beyond 5 charge radii. A rough 
comparison with Kirkwood's figures is made in Table VI. 
In Figure 3 peak pressure multiplied by number of charge radii is plotted against digtance 
for T.N.T. and T.N.T./Aluminium according to the present calculations, and according to Kirkwood's(2) 
latest figures (0.S.R.D.2022). Also included for comparison is Penney’s earlier work on T.N.T. 
Al) these curves are based on the assumption that the gas is at rest initially. Penney and Oasgupta's 
calculations include the effect of a detonation wave, and cannote therefore be compared directly with 
the others. The following differences between the various calculations must adso be noted:- 
(a) eeteeme 
