347 
CALCULATION FROM THERMODYNAMICAL AND HYDRODYNAMICAL 
CONSIDERATIONS OF UNDERWATER SHOCK-WAVES 
FOR SPHERICAL CHARGES 
H. N. V. Temperley and J. Craig 
April 1945 
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Summary. 
The results of calculations of the pressure pulses produced in water by spherical charges 
of T.N.T. and T.N.T./Aluminium 85/15 are presented. The equations of state for the explosion products 
were those calculated by Booth, while the equation of state for water was that used by Penney and 
Dasgupta which is practically identical with that used by Kirkwood and others(3). The gases are 
initlally assumed to be at rest and at uniform pressure, as Is done by Kirkwood and others(3) and in 
Penney's original calculations. 
The results of the various theories are compared with experiment and the calculations now 
available enable the causes of some of the discrepancies between the varlous theorles to be traced. 
With one or two exceptions, the experiments seem in good agreement with theory. The most serious 
discrepancy between theory and experiment seems to be in the single result obtained from the pressure— 
bar(1), which suggests a time constant much larger than the theory indicates. The most serious 
discrepancy between the theories seems to be the differing effects of the addition of aluminium to 
T.N.T. according to Kirkwood's theory and according to the present calculations. We have confirmed 
Kirkwood's assumption that the main portion of the pressure pulse is exponential, but it seems probable 
that Kirkwood's method of obtaining the constants in the exponential expression is too crude. 
Introduction 
It is desirable in explosion research to have some idea of the form of an underwater explosion 
pulse very close to the charge, say up to distances of 10 charge radii for a spherical charge. With 
the possible exception of Taylor and Davies’ pressure bar(1), (which so far has only been used for 
one shot) no instrument has been devised which will stand up to the very high pressures in this region, 
although there are indirect 7ethods of inferring the peak pressure which we shall discuss later on. 
It is therefore necessary to resort to theory in order to gain some idea of the pressures to be expectea 
from contact or near—contact explosions, to assist both in the design of instruments for measuring these 
large pressures and in the design of structures to resist sugh explosions. 
Methods available. 
Two methods suggest themselves. One might start with experimental results at great distances 
from the charge and attempt to extrapolate inwards or one might start from the equations of state of 
the explosion products and of the water and, assuming reasonable initial conditions, follow hydro— 
dynamically the variations with time of the pressure in the water and in the jas sphere, In this 
report we shall not use the first method, which has recently been used by Kirkwood and others(2) but 
we shall concentrate on the second. The success of this method naturally depends on the accuracy of 
our data. The equation of state of water over the pressure range required may now be regarded as 
fairly well known, covered as it is by recent experiments of Bridgman and others. The equation of 
state, of the explosion products cannot be observed directly, but must be calculated thermodynamically. 
This has been done by varlous workers. Eyen if one takes proper account of chemical equilibrium 
during the early stages of the expansion of tne explosion products, it soon appears that the temperature 
falls rapidly during the expansion, and at sone stage the explosion products will no longer be in 
equilibrium. Thus, at least one arbitrary assumption is involved in this calculation. A second 
difficulty Is that explosives such as T.N.T. form solid carbon, and it is difficult to decide whether 
or not it should be considered to be in thermal equilibrium with the gases. A similar question arises 
with the aluminium oxide that is formed when aluminised explosives are detonated. Penney has also 
Suggested that the aluminium might actually burn in the water near the charge. Fortunately, however, it 
Appears seeee 
