265 
atts 
up) = 4 se (%) ‘ ra (a1) 
a 
eis 
ulp) = £9 Pa : eee RNY cere = (12) 
y 2° Pa zy 
where y is the ratio of the specific heats. 
For water, tables of the functions U(p) and u(p) have been kindly supplied to us by 
Dr. Penney. These functions are shown in Figure (4) in which U(p) and u(p) are plotted against 
10946 p where p is expressed in atmospheres. 
To determine the shock wave we take a given value for p and from the known value of U(p) 
we find W from equation (9) and also @ from equation (10). The points @, log p so determined 
are plotted in Figures (2) and (3). Curves (a) and (b) relate to air and water respectively. 
The intersections of the @. log p curves obtained in this way with those obtained from the 
values of tables (1) and (2) determine the angle and the pressure of the shock wave maintained 
in the given medium by the adiabatic expansion of the explosion products. 
A solution of this type in which the shock wave passes through the point 0, Figure 1, 
is no always possible. In a gas, for high values of the pressure such as we Consider here, 
(1 ~ u(p) U(p)) tends to a constant value (y- 1)/(y+ 1). Thus, if we denote for brevity 
tan @-9) by 7) and tan y - 9) by &, equations (9) and (10) give 
E-n = (Cay E (1+E7) (13) 
yrt 
The greatest value of 7 for which this equation has a real root in& is given by 
ES TT Lh Gia 
tan Gy - | = (y+ 1) - 1), (14) 
and the co responding va'ue of W is given by 
"max 
tan Wray = 5) = (LE8 (15) 
y-i 
For air, assuming y = 1.4 we find Prax = 135 degrees 35 minutes. Thus if the curve (a) Figures 
(2) and (3) has not cut the @, 10a p curve when @ has reached this limiting value no solution 
of the type sought is possible. The effect is similar to that of the formation of a shock 
wave by a moving wedge. When the angle of the wedge exceeds a certain critical value, the 
shock wave no longer passes through the vertex of the wedge but moves ahead. Figures (2) and 
(3) show that for T.N.T. in air the angles @ are less than the critical values so that solutions 
of the form shown in Figure (1) are possible. 
If the surrounding medium were helium yas (y = 1.67) the limiting value of q@ would be 
126,49 degrees so that in this case the shock wave would be ahead of the detonation wave for 
both densities. 
It is interesting to find the angle X at which the particles of the surrounding medium 
are thrown forward in the shock wave, i.e. with respect to the explosive at rest. X is the 
angle between the direction of motion of the particles and the direction of the detonation wave 
normal. The horizontal and vertical components of the particle velocity in the shock wave 
(i.e. perpendicular and parallel to OA Figure (1)) are respectively 
Up sin W -9) sin ( - 4) 
cos (y - d) Gs) 
and 
U (b - 3) sin @~-%) 
p cos (b- 5) sin @ 3 in) 
cos WW - $) 
Thus tan X = cot  -3) or X=m7-w 
The sesce 
