264 
gives 
ve ace (5) 
Also (3) and (4) with the condition that when p= p., u= v and v = c give 
p 
iP 2 Cow so ff 1 ap (6) 
Pp 9 
by less ; dp v4 ted f 
From tabulated adiabatic p, v relations, the values of and J = dp have been calcula or 
p 
fo} 
various pressures. Thus the values of u and v for a given pressure are obtained from equations 
(5) and (6). The angle specifying the radius vector along which these values of u, v and p 
occur is given by equation (4) which may be rewritten 
a-f ta (7) 
° 
This integral has to be evaluated numerically. 
The direction of the stream lines is denoted by the angle ¢ relative to OA as reference 
line, which is given therefore by 
p= OF tani (8) 
u 
Tables (1) and (2) give the angles @ and ¢ and also u and v for various pressures below the 
initial value p,, for the loading densities 1.5 and 1.0 respectively. Curves in Figures (2) 
and (3) show the angles @ and ¢ plotted against log,9 p. 
The Shock Wave in the Surrounding Hedium. 
When the expansion takes place into a medium such as air or water the solution can be 
obtained as follows. The solution already given holds over the region AOB. Along 0B, the 
pressure is constant and the stream lines are all parallel to OD, i.e. the angle AOD is the 
value of @ corresponding to the value of 6 AoB. In the region BOD the pressure is constant and 
the stream lines are straight and parallel to OD. O& represents the shock wave front in air; 
the angle E0A we denote by W. It is known from the theory of oblique shock waves (cf Taylor 
and Maccoll loc. cit.) that behind OE the stream lines are straight and the pressure is constant. 
The solution is determined therefore by the following two conditions. First, the constant 
pressure in the region EOD must be equal to that in BOD, and secondly, the direction of the 
stream lines behind OE must be given by @, i.c. they must be parallel to 0D. There is a 
tangential discontinuity in the velocity along OD. 
Let U(p) be the velocity with which a shock wave is propagated into the outside medium 
in the direction of the normal to its plane when the pressure benind the shock wave front is p 
and the medium in front of the shock wave is at rest. Let u(p) de the corresponding particte 
velocity behind the shock wave front. The first condition may then be expressed by the equation 
sin w-F) = at (9) 
i) 
Since the velocity of the medium relative to the shock wave is U(p) - u(p) the second condition 
is expressed by U(p) - u(p) = Up cos ane tan (WY -@) or sin eS - cos a2) tan 
2 2 
(b-q) = Ute may. be weitvenvini the Torn 
i a y 
tan (=) = Tae (10) 
De . 
(u 
The functions U(p) and u(p) depend upon the medium into which the expansion takes place. for 
a perfect gas at pressure Pa in which the velocity of sound is cy we have 
U{p)ieesivete 
