349 
== 
OP oP = 2 uc 
Py + (c+ u) =e = (1) 
Seay ee Se at (2) 
ot or r 
where u is the particle velocity, t the time co-ordinate and r the distance co-ordinate, and 
P=f+u, Q= f—-uand f is the Riemann function f = f pe ¢ is the velocity of sound, 
fe} 
fe) 
p the density and Po the density at atmospherical pressure. kirkwood's(3) method is based 
essentially on two assumptions:- 
(a) That the function G= r fs + 3 f? is propagated outwards unchanged with a 
Pp 
° ; E 
velocity c + f. (In the earlier reports f was replaced by u in these expressions). 
A similar function is propagated inwards into the gas. 
(b) That the important part of the pressure pulse can be regarded as exponential. 
In both methods due account has to be taken of the fact that at the shock front energy is 
being dissipated because the shock-front travels slightly slower than and thus "eats" “up the remainder 
of the pulse, the relation between shock front pressure and velocity being determined by the 
Rankine-Hugoniot equations. The dissipation of energy due to this “overtaking effect" represents 
the effects of viscosity and thermal conductivity, which cannot be neglected at the steeply—sloped 
shock-front, even if they can be neglected in the remainder of the pulse. 
The objection to the step-by-step method is that if the steps are made too smal) errors 
accumulate, while if they are taken toc large the second-order terms are not negligible. Penney and 
Dasgupta attempted to allow for these second order terms by their “backwards and forwards" process. 
This process was not used in the present calculations, because it was considered that the additiona) 
labour and possibility of error entailed by working over three very similar sets of figures was likely 
to counterbalance any gain in atcuracy. It was preferred to keep the steps small at stages where 
special difficulties occurred (at tne beginning, at the instant when the rarefaction wave in the gas 
first reaches the origin and at the instant when a discontinuity in the Q function is about to set in, 
as described later), and to obtain a rough check on the over-all accuracy by computing the kinetic 
and potential energies of gas and water at a number of instants, adding to them the energy dissipated 
at the shock-front and seeing how nearly the total remained constant. The results suggest that the 
accuracy remains fairly good, at least in the early stages of the work, and are given in Tables | and 
ile 
TABLE |. 
Energy balance for T.N.T., expressed in calories per gram of explosive. 
Position of Shock- 
Front. 
(Charge Radii). 
Total 
Table It sssee 
