172 
III. PRESSURE WAVE | 12 
3. QUALITATIVE THEORY 
The theoretical picture of the pressure as a function of the time thus ob- 
tained is sketched in Figure 7. It appears to agree at least roughly with the facts. 
PRESEN) PSI [0 aren. i A a 
Radius r 
Figure 7 
III. PRESSURE WAVE | 
4. LAW OF SIMILARITY 
4. THE LAW OF SIMILARITY 
The exact equations for the motion of non-viscous fluids lead to the same 
law of similarity that was established experimentally by Hilliar (Section 1, preced- 
ing). If the linear dimensions of the exploding charge are changed in the ratio z, 
without other change, the pressure curve previously obtained at a distance r from 
the center of the charge should now be obtained at a distance zr, except that all 
times will likewise be changed in the ratio z. The energy and the impulse,|pdt , 
carried by the wave will, therefore, also be z times as great. Since the wave covers 
a spherical surface z* times as great, the total amount of energy is thus proportional 
to z° or to the weight of the charge. 
III. PRESSURE WAVE | 
5. A CALCULATION 
5. A CALCULATION OF THE FIRST IMPULSE 
The only available quantitative calculation of the first pressure wave 
seems to be that made in England by Penney (7). He starts with a spherical mass of 
TNT having a radius of 1 foot and hence a weight of 390 pounds (specific gravity 
= 1.565). Instead of solving the detonation problem, however, Penney substitutes an 
idealized initial condition; he assumes that at a cértain instant the TNT is all ex- 
ploded within its original volume, the exploded gas being at rest but under a pres- 
sure of 1,300,000 pounds per square inch. The genesis and propagation of the pres- 
sure wave in the water, and the motion of the globe of exploded gas are then worked 
out by numerical methods, for times up to 0.7 millisecond from the start. 
In the beginning, the pressure at the interface between the exploded gas 
and the water is found to drop instantaneously to about 500,000 pounds per square 
inch, the water and gas at the interface acquiring simultaneously an outward velocity 
of about 3000 feet per second. A shock wave then proceeds outward into the water 
while an expansion wave travels back into the gas. After the lapse of 0.7 milli- 
second, the distribution of pressure p and of particle velocity u (taken positive 
