17. In addition to the pressure, other quantities of interest in the 
study of underwater explosions are the particle velocity, the impulse per 
unit area and the energy associated with the pressure pulse. Formulae 
for these quantities are derived in Appendix a. The relative significance 
of these quantities will depend on the mechanism of damage. 
Particle velocity 
18. The particle velocity is the outwards radial velocity u communicated 
to the water by the pulse. It is essential to distinguish clearly 
between the wave-velocity c and the particle velocity ue. As a very crude 
analogy, the pressure pulse from an explosion can be regarded as "news" of 
the explosion transmitted through the water; the wave velocity c is then 
simply the speed of transmission of this news whereas the particle velocity 
u and pressure p represent contents of the news. It is shown in 
Appendix A that the pressure p is related to the particle velocity u by the 
approximate equation 
PES POU cnsr lines | ais eee \ see pies) “ieway MmetOS (9) 
Equation 9, which is exact for a plane wave, becomes increasingly accurate 
for a spherical pulse as it travels outwards and is one of the basic 
relations which will be assumed in much of the succeeding analysis for the 
effects of the pressure pulse from an underwater explosion. 
Impulse per unit area 
19. The impulse per unit area I transmitted by the pulse across the 
spherical surface at radius r is equal to the area of the pressure/time 
curve and varies simply as the inverse of the distance. 
Energy associated with the pressure pulse 
20. The energy associated with the pulse is defined as the energy per 
unit area transmitted by the pulse across the spherical surface at 
radius r. This energy varies inversely as the square of the distance 
if the relation in equation 9 is assumed to be accurate. Theoretically, 
the total energy E transmitted across the spherical surface is constant 
and is independent of distance. In practice, however, some of the energy 
is left behind as kinetic energy of the water after the pulse has passed. 
Nevertheless, at distances for which the pressure pulse from an underwater 
explosion can be considered as a sound pulse, these afterflow effects are 
small and will be neglected in the succeeding analysis. 
21.° All the preceding theory of sound pulses depends essentially on the 
assumption that the amplitude of the waves is small. The necessary 
criterion for the validity of this assumption is that the ratio u/c should 
be small. From equation 9 this indicates that the pressure p must be 
small compared withpc which is about 150 tons per sq. in. for water. 
Hence, the pressure pulse can involve pressures of the order of several 
tons per sqe in. and still be regarded as of small amplitude for 
theoretical analysis. In contrast, it is interesting to note that for 
air p co is only of order 20 lb. per sqe in. Thus, pressures which can 
be regarded as of small amplitude for blast in water are about 10,000 times 
greater than pressures of waves which may be considered to be of small 
amplitude for blast in air. This factor is largely responsible for the 
present emphasis on small amplitude waves for the pressure pulse in water 
as opposed to the emphasis on waves of finite amplitude for blast in air 
(Part 1, Chapter 4 of this Textbook). 
