184 



EXPLOSIONS AS SOURCES OF SOUND 



This must be equated to the product of the expres- 

 sion (19) by the mass of water which unit area of the 

 shock front traverses in unit time, that is, since we 

 are assuming the shock wave to be weak, by pic: 



2juc-(pi — poY 



Picki ) 



\po Pl/ 



By solving this for 5 



2/iCpipo 



5po 



2/iCpo 



" w ^ w ^ (22) 



K(Pl — Po) K(Pl — Po) 



With the value given above for k and the value 

 fi = 0.01 cgs unit characteristic of pure water at 

 room temperature, equation (22) becomes 



2 X 10-' 



5 (in cm) 



(pi - Po)(ingm/cc) 



4.5 X 10^ 

 (pi — Po)(in d3nnes/cm2) 



(23) 



More refined calculations of this type have been 

 made'' and indicate that, in fact, nearly all the 

 calculated jump in pressure occurs within an interval 

 of the thickness given by equation (23).* 



If we are to regard the relations (22) and (23) as 

 giving a valid estimate of the order of magnitude of 

 the thickness of a shock front in sea water, we must 

 assume three things: 



1. That the Hugoniot equation (18) is sufficiently 

 accurate. 



2. That the curvature of the adiabatic for sea 

 water is roughly the same as for pure water. 



3. That the rate of dissipation of energy is of the 

 same order as that due to shear viscosity. 



The last of these assumptions is known to be true 

 for sinusoidal disturbances in pure water at fre- 

 quencies from 1 to 100 mc.'^ Since, according to Sec- 

 tion 5.2.2, the absorption in sea water seems to ap- 

 proach that in pure water at frequencies near 1 mc, 

 it is reasonable to expect this assimiption to hold in 

 the sea for values of 8 between say 0.1 and 0.001 cm, 

 and perhaps for much smaller values. The second 

 assumption would fail if the water contained many 

 bubbles, but calculations show that this would 

 happen only for an unreasonably large concentration 

 of bubbles. The Hugoniot equation depends for its 

 vaUdity on 5 being very small. Although no reliable 

 calculation of its range of validity has been made, it 

 is not hard to show that the equation (19) derived 

 from the Hugoniot equation should be correct as to 

 order of magnitude for explosive waves from ordinary 

 sources when the pressure amplitude (pi — po) is 



greater than about 100 atmospheres. Unfortunately, 

 at 100 atmospheres amplitude the value of 8 given 

 by equation (23) is 4.5 X 10"* cm, a value so small 

 that it is conceivable that assumption (3) might fail. 

 Thus, about all that can be concluded from the pre- 

 ceding analysis is that with a typical explosive source 

 the thickness of the shock front at a distance where 

 the pressure amplitude is 100 atmospheres is probably 

 not greater than about 0.001 cm and may be much 

 smaller. 



At greater distances from the explosion, a probable 

 upper bound to the thickness of the shock front can 

 be set by neglecting the Riemann overtaking effect, 

 which tends to make the shock front steeper, and 

 imagining the pressure pulse to be propagated out- 

 ward according to the laws of acoustics, subject to 

 the same attenuating mechanisms as sinusoidal 

 sound. By the method of Fourier analysis (see Sec- 

 tion 9.2.4 and Figure 13 in Chapter 9) it can be esti- 

 mated that to avoid inconsistency with the values 

 given in Section 5.2.2 for the attenuation at high 

 frequencies the thickness of the shock front should 

 not exceed a fraction of a centimeter after it has 

 traveled 50 yd through homogeneous sea water. A 

 greater thickness could be produced only by in- 

 homogeneity of the medium or by a highly nonlinear 

 absorption, that is, by some mechanism which would 

 be much more effective in absorbing energy from a 

 disturbance of large amplitude than from a weak 

 disturbance. 



8.5 STRUCTURE AND DECAY OF 

 SHOCK WAVES 



When a shock wave from an explosion passes a 

 given point in the water, the initial behavior of the 

 pressure as a fimction of time consists ordinarily in 

 a roughly exponential dropping off, as shown 

 schematically in Figure 1. The time required for the 

 pressure to fall to 1/e times its value just behind the 

 shock front is of the order of 15 /tsec for a Number 6 

 detonating cap and 600 jusec for a 300-lb depth charge. 

 This decay time depends somewhat upon the type of 

 explosive being used, and it may depend slightly 

 upon the range; for any given explosive it varies as 

 the cube root of the charge weight, in accordance 

 with the similarity rule given in Sections 8.2 and 

 8.4.3. After the pressure has fallen to ^{5 or }/[o its 

 peak value, however, the decay of pressure is much 

 more gradual than would correspond to an exponen- 

 tial law. Theories of the shock wave" predict a "tail" 



