REPORT ON THE THEORETICAL SHAPE OF THE PRESSURE-TIME 

 CURVE AND ON THE GROWTH OF THE GAS-BUBBLE 



S. Butterworth 



August 1923 



Summary , 



Lamb's theory of undsr water explosions is shown to give a pressure-time curve in very 

 fair agreement with that obtainea by experiment, provifleO that the two gas-constants are 

 appropriately chosen. An attempt is made to estimate the effect of the compressibility of water 

 by assuming that outside a sphere of radius 3»7 times that of the original charge the water has 

 its normal compress Ibul ity and within this radius is incompressible, the incompressible envelope 

 being introduced to cover that region for which the pressures are too great for the usual acoustic 

 equations to apply* 



It is found that, for an explosion in which a given quantity of .nergy is released, the 

 initial gas pressure for this case must be of the order 360 tons per square inch to account for 

 the experimental pressure-time curve, whereas the incompressible theory gives only 60 tons per 

 square inch for this pressure. The theory thus indicates that the raximum pressure may diminish 

 with distance very rapidly in the immediate vicinity of the charge. line's theory is extended 

 to include the effect of a constant external hydrostatic pressure, the result being that the 

 bubble tends to oscillate in size the time of oscillation being of the order of one second for a 

 lOO lb. charge. In deep water this nay result in a succession pulses which would, however, 

 diminish rapidly in amplitude. 



The only theory which has so far been published in rega^a to the shape of the pressure-time 

 curve due to an underwater explosion is that due to Lamb*. Lamb assumes that the water may be 

 regarded as incompr.ssible and tnat the products of detonation during expansion follow the law 

 PV'^ = constant whore y remains invariable throughout the expansion. He talies the case of 

 spherical symmetry and works out the form of the pressure-time curve for the values y = 1 and 

 y = 4/3. For the latter case his result is given in the following formulae. Let R be the 

 initial radius of the sphere containing the products of detonation and R the radius after time t. 

 Let P be the initial pressure in the sphere R . Then if p be the density of the surrounding 

 incompressible fluid and ^ = R/R the rtflation between /5 and t is 



± = J. /ST 0?. 1)4 (3/52^ 4/3^ 8) (1) 



"o 1= ' ^'o 



Also if p is the pressure at a tine t at a point distant r from the centre of the bubble 



Formulae (l) and (2) can be used to calculate the pri'ssure-t ine curve when P , R , p arc known. 



0*^ 



It should be noted that (2) will hold only so long as r is greater than R so that there are no 

 negative values of p. 



This formula may be compared with experimental results if we can assign a value to P . 

 Two methods suggest themselves. first if w is the energy released by the detonation of one gram. 



• Phil. Mag. January, 1923. 



