281 
-9- 
As a particular examole, let us take p = 60, Then, using the results described in 
Section 11 we find u = 480, Q= 1630. After time 107? seconds, the Q value 1630 has moved to 
r= 26.42, Hence at time 107° seconds, the value of p at r = 26.42 is 60, and the value of u 
is 480, the units of Q and u being metres/second, and of p, 10? kam/em*. 
Figure 1 shows the distribution of pressure and velocity obtalned by the above 
approximations, after time 107> seconds. Tne step-by-step process may now be applied to obtain 
the distribution at later times. There are three points that need further explanation. They 
are the fitting of the boundary conditions at the interface, the shock wave and the orlgin. 
Let us take the conditions at the interface first. 
Suppose that the interface is at radius r at time t, and that the pressure and velocity 
there at this time are p and u respectively. After a small further time 7, the interface, to 
first order of small quantities, has moved tor+uT. The step-by-step method, using the 
functions P and Q, enables us to get the pressure and velocity throughout the main bulk of both 
fluids, but not in the narrow strip on either side of the Interface, bounded by radius 
r+utT-V, 7 on the inside and radius r+ uT + Mb T on the outside, where V. and \ fr are the 
local velceltise of sound in yas and water respectively. The velocity and pressure distribution 
Curves, however, may be extended backwards Into the strip from both sides In such a way that 
they join up smoothly. 
Let us now take the conditions at the shock wave. If the shock wave radius at time t 
is R, and the velocity is U, then after a smal) further time 7, the shock wave tas moved to 
Re Ut. There is no difficulty about constructing the P curve at the new tlme, because the 
P curve moves outwards, and the rate of movement of any P value, according to the Ideas of 
Section |, is VY + u, where V is the local velocity of sound and is greater than U. The Q curve 
in the neighbourhood of the shock wave is not immediately obtalnable, because the Q wave is 
moving inwards, and the value of Q is not conserved in passing through the shock wave, However, 
In compensation for the lack of knowledge about Q, we have tke dynamical equations of the shock 
wave which jive us a relation between p and u. Thus, at the new radius af the shock wave 
R + U7 we have the value of P, i.e. one relation between p and u, while another relation Is 
provided by the shock wave equations. The two equations for p and u may be solved, and therefore 
the new value of Q at the shock wave found. By joining up this new Q polnt to the nefghbourlng 
Q curve obtained by the usual method, the complete Q curve Is found. For low values of p, It 
wll] be seen that the Taylor's expansion for u and ft, (P) begin with the same term inp. Henee 
Q at the shock wave begins with a term inp“, Even for p = 4, Q at the shock wave is less than 
1 m./second, 
The condition at the origin is that u = 0, and therefore that P = Q There is no 
especial difficulty about getting Q at the orlyin, but in some of the steps the arlthmetic was 
lengthy, because of the term 2uVr/r in dQ. This term, of course, Is always finite, but u 
Changes very rapidly as the rarefaction reaches the centre, and several Interpolated steps were 
made in the region of the origin, to improve the accuracy of the values obtained for Q ana P in 
this rejion. Even so, we do not claim much in the way of accuracy, particularly as time proceeds, 
In this region. The steps should be taken very much smaller than we have taken, Nevertheless, 
an error in this region will not affect the pressure-time curves at distance 50 feet from the 
Charje until about 0.6 x 10°? seconds after the shock wave has passed. 
In all we carried through 27 complete steps, and many intermediate steps over a small 
region of rf when this seemed necessary, AS a result we obtained the pressure as a function of 
radius at time 0,70 x 10°? seconds. At this tlme, the shock wave radius is 174 cm, and the 
interface radius 58 cm. The pressure difference at the two sides of the shock wave Is 1.92 in 
our units, or 12.6 tons per square inch. 
Figure 2 gives yraphically the results at the end of the 27th step (t = 0.7 x 107 second) 
Notlce the chanyes in the sign of the velocity u, and the fluctuations in the pressure curve. 
The pronounced minimum in the pressure curve near r = 100 cm has arisen from the great release 
in pressure in the water near the interface accompanying the expansion of tne Interface In the 
earlier stages of the motlon. (In the earlier Stages near the interface the pressure Is such 
that water is aporeclably less compressible than the exoloded yas). 
Section Vivsees 
