186 



EXPLOSIONS AS SOURCES OF SOUND 



is 1.32. It is not to be expected, however, that equa- 

 tion (24) will hold true indefinitely as the range is in- 

 creased, for its theoretical derivation assumes the rise 

 in pressure at the shock front to be instantaneous, 

 and neglects any dissipation of energy behind the 

 shock front. At large ranges neither of these assump- 

 tions is valid, and we should expect the decrease of 

 pressure with distance to obey a law similar to the 

 attenuation of sinusoidal sound (see Sections 2.5 and 

 9.2.1). 



The theory just mentioned also predicts that the 

 duration of the pressure in a shock wave should 

 slowly but continually increase as the range increases. 

 This effect is due to the fact that, according to Rie- 

 marm's theory, the more intense front part of the 

 wave should travel faster than the less intense tail. 

 Specifically, the theory predicts that if at large 

 values of the distance r from the charge the pressiu^e 

 in the wave is approximated by an exponential 

 V = Pmaxe^''*, then the duration parameter should 

 be given by 



-^ constant 



h'^ 



(25) 



where as before n is of the order of the radius ro of the 

 explosive charge. The experimental verification of 

 this relation is less conclusive than for the preceding 

 relation (24). Although a decided decrease in 6 has 

 been observed when r is decreased below a value 

 corresponding to pmax = 1,000 atmospheres,^" the ex- 

 periments of reference 2, which covered a range from 

 about 3 to 80 atmospheres, showed no measurable 

 increase of duration; yet an increase of the amount 

 given by the relation (25) should have been measur- 

 able. 



Under most conditions, given complete detona- 

 tion, the peak pressures and pressure-time curves ob- 

 tained at a given range from a charge of a given size 

 are quite accurately reproducible from shot to shot, 

 the deviations of individual peak pressures from the 

 mean being of the order of 2 per cent. However, for 

 asymmetrical charges, such as long cylinders with 

 detonation initiated at one end, both the peak pres- 

 sure and the duration of the shock wave, when 

 measured at a given distance, are different in dif- 

 ferent directions. Experiments on long cylindrical 

 charges have shown that the peak pressure is greatest 

 approximately at right angles to the axis of the 

 cylinder and is least on the axis off the cap end. Dif- 

 ferences as large as 40 per cent have been observed.'* 



The duration of the shock wave varies in the op- 



posite sense from the peak pressure, and in fact the 

 impulse Sv^i contained in the early part of the shock 

 wave is slightly greater off the cap end, where the 

 peak pressure is least, than in any other direction. As 

 one would expect, charges fired with an air cavity on 

 one side also show an anisotropy. 



When a charge is fired on the sea bottom, the peak 

 pressure received at a point above the charge is 

 somewhat greater than it would be in the absence of 

 the bottom. For sand and gravel bottoms this in- 

 crease in peak pressure has been found to be 10 to 

 15 per cent." In directions near the horizontal the 

 ampHtude tends to become smaller, as one would 

 expect from the shadowing effect of irregularities on 

 the bottom. The pressure-time curves from shots 

 fired on the bottom are not only likely to be rather 

 irregular, as mentioned previously, but tend to be 

 less consistent from shot to shot than is the case for 

 explosions in free water. 



8.6 SECONDARY PRESSURE WAVES 



In Section 8.2 it was mentioned that because of the 

 inertia of the water which has been pushed radially 

 outward by an explosion, the gas bubble undergoes 

 radial oscillations. Many features of this oscillatory 

 motion can be explained by a very simple theory 

 which treats the water as an incompressible fluid and 

 the radial flow as spherically symmetrical.^' Let u{r,t) 

 be the radial velocity of the water, measured positive 

 outward at distance r from the center and at time t. 

 The volume of water which passes outward in imit 

 time across the surface of a sphere of radius r is 

 Aitr^u. At any given instant the flux across any two 

 concentric spheres must be the same, since otherwise 

 the amount of water in the shell between these two 

 spheres would be increasing or decreasing. We must 

 therefore have 



45rr'^M = function of t, independent of r. 



If ri,{t) is the radius of the gas bubble at time t, we 

 must have 



u(n,t) = -7, = n 

 at 



and so 



, ,-, rln 



J.2 



(26) 



We are now ready to apply the principle of con- 

 servation of energy. The energy of the water and gas 

 bubble consists of three parts, kinetic energy, po- 

 tential energy due to compression of the gas in the 

 bubble, and potential energy representing work done 



