STRUCTURE AND DECAY OF SHOCK WAVES 



185 



in which the pressure dies off with the time t approxi- 

 mately as t~*'^. This law cannot of course hold in- 

 definitely, since the momentum integral Spdt must 

 be finite; the theory of bubble motion to be discussed 

 in Section 8.6 predicts that the excess pressure should 

 eventually go through zero and become weakly nega- 

 tive as the gas bubble expands. For many purposes 

 this tail is unimportant, but its contribution to the 

 momentimi integral may exceed that of the earlier 

 part of the shock wave. Detailed experimental infor- 

 mation on the tail is almost entirely lacking since 

 spurious signals due to the impact of the shock wave 

 on the cables leading to the pressure gauge usually 

 mask the latter part of the tail. 



'1 



20 



10 



• 



Figure 6. Peak pressure and speed of the .shock front 

 near a spherical charge of cast TNT. 



Under some conditions small secondary peaks or 

 fluctuations may appear in the measured pressure- 

 time curve. These may be due to any of a variety 

 of causes. Sometimes the effect is spurious, being due 

 to instrumental factors — for example, diffraction of 

 the pressure pulse around the hydrophone and its 

 supports or shock excitation of vibrations in the 

 hydrophone.^ Irregularities genuinely present in the 

 explosive wave itself have been observed, however. 

 Sometimes these occur under exceptional circum- 

 stances, such as for shots fired on the bottom,'* for 



cylindrical charges detonated at one end rather than 

 in the center,"''"' for charges surrounded by an air 

 pocket,'"" and for charges which fail to detonate com- 

 pletely because of inadequate boostering.'^ More- 

 over, even for spherical charges detonated at the 

 center, the tail of the shock wave shows a small but 

 reproducible hump or shoulder, who.se magnitude de- 

 pends upon the type of explosive. 



In accordance with the theory of Section 8.4.2, it is 

 to be expected that the speed of advance of a shock 

 wave at great distances from the explosion will be the 

 normal speed of sound, but that at close distances 

 the speed of advance will be considerably greater. 

 Moreover, we should not be surprised to find other 

 departures from the usual laws of acoustics. The 

 upper diagram of Figure 6 gives some experimental 

 values of the peak pressure in the shock wave as a 

 function of the distance r from an explosion and shows 

 fitted to these points a theoretical curve which, 

 though only approximate, should give a reasonably 

 reliable extrapolation of the peak pressure for smaller 

 values of r.'^ If the disturbance followed the ordinary 

 laws of acoustics, the curve would be a horizontal line. 

 The lower diagram of Figure 6 shows the velocity of 

 the shock front as a function of distance from the 

 charge, this curve being related to that of the upper 

 diagram by equation (16) of Section 8.4.2. It will be 

 seen that as r increases the pressure becomes more 

 and more nearly inversely proportional to r, as it 

 should be in the acoustic approximation. It has been 

 shown theoretically, however, that even in a prac- 

 tically ideal fluid the peak pressure in a shock wave 

 is not asymptotically proportional to 1/r at large 

 distances, but that instead 



Constant 

 Pmax ■ 



'-(:;)]■' 



(24) 



where ri is a quantity of the same order as the initial 

 radius r^ of the explosive charge.'^ This deviation 

 from acoustical laws is due to the dissipation of 

 energy in the shock front. The relation (24) has been 

 confirmed experimentally- at NRL for No. 6 detonat- 

 ing caps at ranges of 1 and 31.3 ft. The ratio 

 ('■Pmax)l ft /(r^max)31.3 ft was found in these experi- 

 ments to be 1.31 + 0.04, while the ratio 



W^ 



31.3 ft 



['<J' 



1ft 



