13 
33. In theoretical analysis, this time 1/n tends to play much the same 
role for an exponential pulse as does the duration for a pulse of finite 
duration. Similarly, a corresponding measure of the characteristic length 
of the pulse in space is provided by the quantity c/n which has the 
dimensions of a length. A measure of the characteristic duration of the 
pulse from T.N.T. charges can thus be obtained by using one of the 
alternative formulae 16, 18, and 20. The same formulae can be used to 
provide a measure of the characteristic length of the pulse by calculating 
c/n. For example, using equation 16 and taking c = 5,000 ft. per sec. 
then 
g = 0.60 # Eee dee wee. Jee) Jece, 9664 ¥ seein (21) 
An estimate of the maximum particle velocity u occurring at the pulse 
front is given by using equation 11 and the approximate equation 9, ,For 
T.N.T. charges. 
We) 250 4 HO Jes BEGG, 606 G00 Goo 606) one!) coc (22) 
34e The use of the principle of dynamical similarity has not, in general, 
been employed in the empirical analysis of underwater explosion data to the 
same extent as for the problem of blast in air. Although the particular 
formulae 11 to 14 satisfy this principle, it should be noted that the 
experimental data from which they were derived can be rather better 
represented by formulae not satisfying dynamical similarity. It is not 
yet certain whether this departure from similarity is a true effect due to 
the fact that different size charges are never perfect scaled replicas, or 
whether it is mainly a spurious result arising from defects in the methods 
of measurement. 
Effects of finite amplitude of the pulse 
55. The simple theory of small-amplitude pulses is sufficient to account 
for the vropagation of the pressure pulse at distances where the maximum 
pressure is of order 2 tons per sq. in. or less. At much smaller 
distances from the charge the simple theory becomes completely inadequate 
and a more elaborate theory is necessary. 
36. Reasonably successful attempts have been made to calculate what 
happens in the neighbourhood of the charges In particular, the theory 
predicts a form and order of magnitude of the pressure pulse in good 
agreement with experimental results and the theory serves to indicate the 
magnitude of the pressures near the explosion where it is difficult to take 
measurements. Such more complete theory’ for underwater explosions is 
essentially similar to that for blast in air and it will suffice here only 
to emphasise some effects connected in particular with the sharp-fronted 
nature of the pulse. 
37- The passage of a finite amplitude pulse involves, in general, an 
irreversible heating of the water and a consequent dissipation of energy 
by conduction of heat through the water and by internal friction or 
viscosity. Both these effects are most pronounced in the steep front of 
the pulse where the most rapid changes occur and both tend to decrease the 
pressure in this front. On the other hand, the fact that larger pressures 
travel faster than smaller pressures (as in air) implies a building up of 
the pressure at the front. These conflicting effects tend to strike a 
balance and theory indicates thet the thickness of the shock front, that is, 
the distance in which the pressure in the water rises from its undisturbed 
value in the front of the pulse to its maximum value in the pulse, is of 
the order of a few millionths of a centimetre; this corresponds to a time 
*® Recent work has, however, shown that for shock waves, the principle of 
dynamic similarity does hold 
