275 
-3- 
Now pf'(o) is the local velocity of sound ¥. Hence for a point moving with the velocity 
gr s pt't(o) + u = Veg, 
at 
dp = ~ 2 fe(p) dt - - 24 vat (6) 
r r 
Thus, if the values of p and u are known everywhere at time t, thus determining the 
values of P and Q everywhere at this time, the value of P everywhere at time t + 7 may be found 
from (6). All that we have to do is move the value of P at point r to point r+ (V + u) 7, 
and decrease the value by 2uV7/r. A similar procedure glves the now function Q, as the 
following analysis shows. 
Multiply (4) by f'(0) and subtract (2). Then 
29 29 = ~ prey oY oe _ 2 pr 
ae Lou Pile aa dodr sa 
rc) % “ ah = ; 
> $8 Lar { pt'(e) - u) at J 2E rrp) at 
Hence for a point moving with velocity 
ors pt'(o)-u = -v+u, do = - 22 trip) at = -Bvat (7) 
dt r r 
To find the new function Q, therefore, at time t + 7, we move the value of Q at polnt 
r to polnt r — (Yu) 7, and decrease Q by 2uW/r. 
From the new values of P and Q, the values of p and u are everywhere determined, Thus 
at any point 
u = #(P-Q), fle) = & (PF + Q), 
and from the curve giving f (9) as a function of p, the value of p may be read off, 
in tne problem which we are considering there are two fluids Involved, The analysis 
given above applies in both fluids, with thaeappropriate functions P, Q, V, etc. The boundary 
conditions at the interface are that the pressure and velocity should be continuous, The boundary 
condition at the origin is that tne velocity is zero, and the boundary condition at the outer limit 
of the disturbance In the water may be expressed as a relation between p and us. This last 
condition we shall discuss later, when we consider the details of the step-by-step calculation. 
SECTION II. NUMERICAL VALUES OF AUXILIARY 
FUNCTIONS. 
We now consider how to estimate the values of the functions f(o) and V in terms of the 
pressure as Independent variable, both for water and the exploded gases. Clearly what is 
required is the relation for v, the specific volume (ise. Wp), in terms of p In doth cases, 
The Exploded Gas. 
First let us obtain f(0) for the exploaed gases. We make use of some results obtained 
by Dr. H. Jones for the adlabatic of the gases, the initlal T.N,T. naving a density 1.565 ym./cc, 
Jones ylves numerical valpes of the pressure in terms of the volume, the Initial volume veing 
445 cc. 
A TEW eoees 
