SHOCK FRONTS 



175 



Most of the commonly used high exiilosives are re- 

 markcibly similar (o one another in the amount of 

 energy th(>y release per unit mass, antl in the relative 

 amounts of energy which go into the shock wave and 

 the oscillations of the bubble. Of the total work done 

 by the gas on the water in its initial expansion, iibout 

 40 to 50 per cent remains as kinetic and pot( ntial 

 energy in the oscillations of the gas bubble and sur- 

 rounding water, part of this energy being ultimately 

 converted into heat by dissipative actions in the 

 neighborhood of the bubble and part being radiated 

 as acoustic energy in the secondary pulses. The re- 

 maining 50 or 60 per cent of the original energy is at 

 any stage divided between energy present in the 

 shock wave and energy which has been converted 

 into heat by dissipative processes occurring at the 

 shock front. Dissipation of the latter kind is espe- 

 cially rapid in the early stages so that by the time 

 the shock wave has advanced a distance of the order 

 of ten or twenty times the original radius of the 

 charge, about a quarter of the original energy has 

 been dissipated into heat, and the other quarter 

 continues to be radiated outward in the shock wave. 

 From this time on, the dissipation is much slower, 

 although not negligible. 



In the preceding discussion, the phenomena have 

 been described without reference to the size of the 

 charge of explosive which is used. This is possible 

 because explosions of all sizes are similar. If the range 

 is not too great, the intensity and form of the shock 

 wave, and many of the features of the bubble oscilla- 

 tion, can be predicted exactly for one quantity of 

 explosive if they are known for another quantity of 

 the same explosive substance. To give a precise state- 

 ment of the rule by which this prediction can be 

 made: suppose two experiments are carried out with 

 the same explosive material, the shape of the charge 

 and the position of the detonation being the same in 

 both cases, but the linear dimensions of the second 

 charge being /3 times as great as those of the first. 

 Then the rule states that if the pressure is p and the 

 velocity of the water is u at a distance r from the first 

 charge, at time t after the detonation starts, the same 

 pressure p and velocity u will obtain at a distance /3r 

 in the corresponding direction from the second 

 charge, at a time fit after the detonation starts. This 

 rule can be applied to the shock wave provided that 

 the range from the explosion is sufficiently short so 

 that the dissipative or dispersive efifects responsible 

 for slowing the time of rise to maximum pressure (see 

 Section 9.2.1) have not had an appreciable effect on 



the pressure-time curve. The applicability of the rule 

 to the later oscillation of the bubble is more limited 

 and will be discussed in Section 8.(). The physical 

 basis of the rule will be taken up in Section 8.4.3. 



The phenomena which occur when a propellant 

 charge is set off under water are similar to those just 

 described for high explosives, with the important ex- 

 ception that because of the comparatively slow burn- 

 ing of the explosive, the pressure transmitted to the 

 water builds up gradually over a period of time, and 

 does not usually create a steep-fronted shock wave. 

 Thus instead of the sort of pressure-distance graph 

 shown in Figure 1, a propellant would give a graph 

 more like Figure 2. The division of the disturbance 



I 



BOUNDARY OF GAS BUBBLE 



I RESIDUAL FLOW 

 AR OUND BUB BLE 



OUTGOING PRESSURE WAVI 



DISTANCE FROM CENTER OF EXPLOSION 



Figure 2. Pressure distribution in the water a short 

 time after ignition of a propellant charge. 



into an outgoing pressure pulse and a residual bubble 

 oscillation can usually still be made, but the propor- 

 tion of the total energy which appears in the pressure 

 pulse from a propellant is much smaller than that 

 which appears in the shock wave from a high ex- 

 plosive, and the maximum of the pressure is very 

 much smaller.' The exact characteristics of the pres- 

 sure pulse depend upon the rate of burning of the 

 charge, which varies greatly depending on the type 

 of propellant and the grain size. 



The following sections discuss in greater detail the 

 previously mentioned features of the disturbance due 

 to a high explosive. 



8.3 



SHOCK FRONTS 



The steep-fronted shock waves mentioned in Sec- 

 tion 8.2 represent a form toward which all very in- 

 tense pressure disturbances tend to develop. In this 

 section and the following section we shall show why 

 this is true and shall show that many of the charac- 

 teristics of shock waves, such as the velocity of 

 propagation of the shock front and the rate of dissi- 

 pation of energy into heat, can be expressed as func- 

 tions of the pressure jump, that is, the amount by 

 which the pressure immediately behind the shock 

 front exceeds the pressure in the undisturbed water 

 in front of it. Characteristics of shock waves from 



