263 
NOTE ON THE LATERAL EXPANSION BEHIND 
A DETONATION #AVE 
G. I. Taylor and M. Jones 
May 1942 
* * * * * * * 
When a detonation wave travels along the length of a cylindrical explosive charge, the 
front seems to be a surface which is nearly perpendicular to its direction of motion, i.e. to the 
surface of the charge. Immediately behind the detonation front the products of combustion can 
expand laterally. If the explosive is contained in a tube the inertia of this tube is in most 
cases sufficient to ensure that the pressure across any sectioo of the expanding tube is nearly 
uniform. When the wall of the tube is very light or when the charge is uncased the conditions 
are more complicated and the lateral as well as the longitudinal component of velocity in the 
expanding gas must be considered. Though the motion of the expanding gases from a cylindrical 
explosive cannot be described completely without great complexity of analysis, the motion in the 
region close to the point where the detonation wave meets the surface of the cylinder can be 
analysed. 
The problem is simplified in the following way. we consider a semi-infinite block of 
explosive bounded by, and lying below, the xy-plane. Co-ordinates are taken moving with the 
uniform detonation velocity Up in the positive x direction. The lower half of the yz—plane then 
represents the detonation wave front and the motion of the expanding gases is steady relative to 
these moving axes. On account of the Chapman condition, we know that, referred to this system, 
the products of the detonation wave front are moving in the direction of the negative x - axis 
with the local velocity of sound C. The flow of the hot gases and the positlon of the shock 
wave which is produced in the surrounding medium can readily be calculated by the methods 
described by Taylor and Maccoll (Aerodynamic Theory, W.F. Durand Vol.!I1). To obtain numerical 
results the law for the adiabatic expansion of the explosion products must be known, and this has 
been given by Jones and Miller, and Jones for T.N.T. at densities 1.5 gm/en? and 1.0 gm/en? 
respectively. 
Referring to Figure (1), OA denotes the detonation wave front, which is at rest in the 
Co-ordinates considered, with gas passing through from right to left with velocity c where 
2a, (it 
2G) : 
p is the pressure and p the density, and the suffix o denotes values taken over the plane OA. 
The equations of motion for two dimensions are 
eat Hho Sy a) 1p (2) 
or r 300 r pwore 
u iad + u ov + Oy == me i op (3) 
or r 06 r p dé 
where r and @ are polar co-ordinates with respect to 0 and the reference line &, and u and v 
are the components of the velocity parallel and perpendicular to r respectively. 
We assume the existence of a solution (verifiable a posteriori) in which p, u and v are 
functions of @ oniy. Equation (2) then becomes 
du 
v =(= (4) 
Ge 
and (3), together with the equation of continuity, 
d 
yo + — (vp) = 0, 
dd if 
Gives seoee 
