SECONDARY PRESSURE WAVES 



187 



•^r 



—r- 



-1 r- 



¥ - 1- 



ri uj uj 



7 UJ UJ 



^ U. li. 



(J I I I I 1 I I I I \ I I 



o I 1 1 1 \ 1 1 1 1 1 1 1 1 1 1 1- 



o ^ o 



o -o <> 



500 



TIME IN MILLISEC 



Figure 7. Ideal radius-time curve for the gas bubble from an underwater explosion. Full curve computed assuming pres- 

 sure of gas in bubble to be given by 



p = 0.064 ( -^^ 1 atmospheres. 



Dashed curve computed assuming pressure of gas in bubble to be zero. Scales: (a) No. 8 cap at 50-foot depth; (b) 1 lb 

 TNT at 50-foot depth; (c) 300 lb TNT at 50-foot depth. 



against the surrounding hydrostatic pressure, which 

 we shall denote hy pa>, since it represents the pressure 

 in the water at a great distance from the bubble at the 

 same level. Consider the kinetic energy first; since the 

 mass of the gas in the bubble is negligible, practically 

 all the kinetic energy resides in the water, and the 

 amount per unit volume is 3^pM^. The total kinetic 

 energy is thus 



must be constant in time in the approximation we 

 are using 



4 



2Trprlrl + -irp^rl + G{n) = W. 



(28) 



U Tb 



.3 "2 



^pu'-^rrMr = 2t: prlrl 



(27) 



by equation (26). The potential energy of the gas is a 

 function of its volume, and can be represented by a 

 fimction G{ri^ ; it is a negligible fraction of the total 

 energy except when the radius of the bubble is small. 

 The work which has been done against the external 

 pressure is represented by pco times the volume of the 

 bubble, or (4/3)irpco»'6. The sum of these three terms 



The behavior of r;, as a function of the time is thus 

 determined by solving equation (28) for h and 

 integrating. The result can be expressed in a form 

 which is independent of the amount of explosive in- 

 volved, manifesting a similarity rule of the same 

 form as that given in Section 8.2 for shock wave 

 pressures. For if, as before, we let n be the radius (or 

 equivalent linear dimension) of the original charge of 

 explosive, G will be r% times a function of the ratio 

 Tb/ra, and W, which represents that part of the origi- 

 nal energy which is not dissipated or carried away by 

 the shock wave, will be proportional to r%. Using 

 these facts,. it can be seen from equation (28) that 



