354 
—e— 
So far we have found the pressure at the new position of the shock-front. We now have to 
find the pressure distribution in the reyion behind the shock-front. For this purpose, we select., 
a range of values of cy at distances ry when the shock-front is at Rye fach pair of -values 
(ry, cy) is then extended to the new values (es Ey) by the following methods:— 
(e) As before, knowing ry C, hence x,, We find the corresponding M from equation (7). 
(f) Using the fixed time basis t,— t, already found, equation (6) gives 1 (x5) hence 
Xp Cor Po 
(g) Returning to equation (7) in the form r, crane (c, = co) = M, we can therefore 
find r. 
Thus, the conditions at r, have been translated into the corresponding conditions at Toe 
This method is repeated for each chosen r, until the pressure-distance relationship for 
the shock-wave in its new position Ro has been completed. 
As before, most of the complications are due to the fact that we still have to take due 
account of the overtaking effect even at comparatively low pressures. The extrapolation process 
2 
is based on the constancy of the quantity ret (c = ¢,) for corresponding points on the pressure 
curve at different times. It is interesting to compare this with Kirkwood's method. The 
function that Kirkwood assumes to be constant is given by:— 
” 
Grae f op + : f? where f is the Riemann function, defined as before. 
Pp, 
Again using the equation of state for water, we find that this function is given by:— 
G pebetbs =e (c - c)? (10) 
= or cee = =¢ 
n= 1 (n - 1) C 
For low pressures, c is nearly equal to Cor and the main variation of the quantities G and M will 
be due to the variation of c — Co which occurs to the first power in both quantities, the second 
term in equation (10) being negligible compared with the first. It thus follows that in this 
limiting case the extrapolation process we have used is equivalent to Kirkwood’s. For pressures 
of the order of 2 kilobars ¢ — Sy is about 30% of Co» SO that the peak pressures would have to be 
greater than this for the two methods tc give significantly different results. Simple acoustic 
theory would give nearly as good an approximation. 
It was found that the pressures dropped to low values at points more than a few charge 
radii behind the shock-front, so that c became nearly equal to cy and the extrapolation process 
became inaccurate. Thus, apart altogether from the fact that Q was not negligible, it was not 
possible to apply the extrapolation process to points in the region of the water near the gas bubble. 
On the other hand, the overtaking effect continually destroyed the front of the pressure pulse. 
It was, therefore, not practicable to carry the extrapolation process beyond 30 charge radii, which, 
however, is enough to give us a satisfactory comparison with experiment and with the results of 
other workers. 
Difficulties encountered in the calculations. 
We have already mentioned the difficulties connected with the extrapolation. In the 
calculations proper, some further difficulties were encountered. 
(a) Once the rarefaction wave passing into the gas had reached the origin, it became 
difficult to maintain accuracy on account of the term - Zue which remains finite 
at the origin but cannot be determined grapnically with any accuracy. For points 
near the origin . was replaced by — but this quantity was also difficult to evaluate 
on account of the fact that the curve of u against r ends at the origin and ordinary 
methods ...06 
