NONLINEAR PKESSl RE WAVES AND SHOCK FRONTS 



177 



From what has been said previously it would ap- 

 pear that any niafhoniatifal theory of the propaga- 

 tion of a shock front would have to he based on 

 hydrodynamical equations of sufficiently complicated 

 form to include the effects of viscosity and heat con- 

 duction. Fortunately, however, a very simple analysis 

 based on the laws of conservation of mass, momen- 

 tum, and energy suffices to determine the relation 

 between pressure, particle velocity, temperature, and 

 similar factors, just behind the shock front, and the 

 velocity of propagation of the front. A detailed ap- 

 plication of the laws of viscosity and heat conduction 

 turns out to be necessary only if we are interested in 

 phenomena in the shock front itself, that is, phenom- 

 ena taking place in the very thin layer of water within 

 which the abrupt rise of pressure takes place. 



The simple analysis just mentioned, due to Rankine 

 and Hugoniot, will be described at length in Section 

 8.4.2. It will be shown there, among other things, 

 that in ordinary fluids a negative shock is impossible, 

 in other words, that a discontinuity in pressure and 

 density can only be propagated toward the region of 

 lower pressure, and that the velocity V with which a 

 shock front advances, relative to the undisturbed 

 fluid in front of it, is greater than the velocity of 

 sound [the value of (dp/rfp)'] in the undisturbed fluid 

 ahead of the shock front. 



The thickness of the region within which the pres- 

 sure rises from po to pi is of course determined by the 

 dissipative phenomena, namely, viscosity, heat con- 

 duction, and any other sources of dissipation, such as 

 bubbles, w-hich may be present in sea water. A precise 

 mathematical treatment of these factors would be 

 difficult, but order-of-magnitude considerations to be 

 given in Section 8.4.4 indicate that close to the ex- 

 plosion this thickness should be very small; at a 

 distance where the pressure jump is 100 atmospheres, 

 for example, as is the case at a range of about 1 ft 

 from a Number 8 detonating cap, the shock front 

 should be less than 0.001 cm thick, perhaps much 

 less. In this region the thickness of the shock front 

 is a function only of the magnitude of the pressure 

 jump and decreases as the latter increases. It might 

 be supposed that the time of rise would be connected 

 with the time required for the detonation wave to 

 travel across the explosive charge, but this is not the 

 case unless the charge is extremely elongated, since 

 the Riemann "overtaking effect" will succeed in 

 making the shock front vertical before the shock 

 wave has advanced an appreciable distance from the 

 charge. 



With increasing distance from the charge the pres- 

 sure amplitude becomes .small, and eventually the 

 overtaking etTect will become negligible in comparison 

 with the dissipative processes which tend to .smooth 

 out the abrupt rise in pressure. In homogeneous sea 

 water, however, the thickness of a shock front should 

 remain quite small until it has traveled a considerable 

 distance. Thus, for example, it can be shown that the 

 thickness of the shock front at 50 yd from the ex- 

 plosion of a detonating cap should probably be only 

 a fraction of a centimeter, corresponding to a few 

 microseconds or less for the time of rise of the pres- 

 sure at a given point. 



Experimental information on the thickness of 

 shock fronts, or equivalently the time of rise of the 

 pressure at a given point, must be treated with cau- 

 tion. The measured value of time of rise can easily 

 be completely falsified by inadequate frequency 

 response characteristics of the hydrophone and re- 

 cording equipment; in particular, the finite size of 

 the hydrophone seems to have rather more effect on 

 the time of rise than one would at first suppose. A 

 very careful series of experiments has been conducted 

 at NRL2 on shock waves from detonating caps con- 

 taining about half a gram of high explosive, at 

 ranges from 1 to 30 ft. In these experiments the time 

 taken for the pressure in the shock wave to rise to 

 its maximum value was measured under the best 

 conditions as about 0.3 microsecond (Msec) at all 

 ranges, and since this was about the same as the 

 estimated resolving time of the apparatus used, these 

 experiments support the theoretical expectation of 

 the preceding paragraph that the time of rise should 

 be exceedingly minute. Other experiments supporting 

 this expectation have been made at the Underwater 

 Explosives Research Laboratory at Woods Hole,' 

 using 3^-lb charges and ranges of the order of 10 ft; 

 however, the resolving time of the apparatus for these 

 experiments was only about 4 fisec. Measurements 

 made at ranges of the order of hundreds of feet, 

 however, seem to show quite an appreciable time of 

 rise;^-'' this effect, which is probably instrumental 

 but may possibly be related in some way to oceano- 

 graphic conditions, will be discussed at length in 

 Section 9.2.1. 



8.4 THEORY OF NONLINEAR PRESSURE 

 WAVES AND SHOCK FRONTS 



The four following sections will be devoted to a 

 mathematical discussion of some of the topics which 



