res 
353 
In addition to the formulae already derived, it is neceessry, as will be explained later, to have 
an expression for the distance eg moved by the shock front in the interval t, - ty. 
For this purpose, U, the shock wave velocity, has to be introduced in the expression 
t 
1 
following relation, which is satisfactory for smal) pressures:— 
1 
= + = 
U e. (1+ ax) where a co 
As given above:- 
= A_de = M_dx 
ake 0 - ia Ge eo ae 
cMT (¢ eal ere (1 + x)Pt y2 
Ue : 2 i 
== = M dx 
| uate gor - | Co SSD oa ee 
ty mt ele (a> x) x2 
ry : 
2 
=a ‘ —Sar- to | ong 
ent i x (1+ x) ot 2 xX) (1 + x)et x 
= M 
iia orn [. baie ata) (9) 
A wi 
0. 
2 2 , 
= ax x 
where J (x,) = | Sar | at 
X2 (1+ xt x? Xo (1+ x)Mt x 
t 
ra 
i Udt. The decrease of U with time is not known, but it is related to the pressure by the 
The upper limit of 2 in the integrals has no special significance, but is introduced in order to 
simplify tabulation. 
Method of Calculation. 
Knowing the pressure (p,) distribution in water when the shock-front has moved out toa 
distance Ry in time ty we then assume tne shock-front to have moved to a further distance R, and 
use equations (5), (6) ana (7) to find the pressure distribution in the new position at time t+ 
The various steps of the calculations are as follows:- 
(a) The integrals !(x) and J(x) are tabulated for a series of values of x < 2. 
(b) Knowing p, hence c, at R, for the shock-wave, we can find M by using equation (7) 
with r= Ry c= Cy 
(c) At this stage it cannot be assumed that the leading point of the shock-front at time 
(d) 
t, will always remain the leading point (the pressure thus decreasing according to 
equation (5)) and on this assumption use R, and the value of M found by (b) to find 
the value of Co in equation (7). In actual fact the velocity U of the shock-wave is 
rather less than that of the leading point, so that the method just described would 
eventually give Po at a point beyond the actual position of the shock—-wave. Instead, 
we must make use of equation (7) which was evolved from an expression for U. Thus, 
knowing 8, - Ry, M and x,, we find J(x5), hence x, C, and py at R,. 
The next step is to find t, - t, by equation (8) using the values of M, Xj» X> NOW 
known. The time to- ty is a fixed time basis for all further calculations. 
SO secee 
