487 



THE THEORETICAL SHAPE OF THE PRESSURE PULSE PRODUCED 

 BY AN UNDERWATER EXPLOSION BUBBLE. 



A. K. Bryant and LI. G. Chambers 



Naval Construction Research Establishment 



1950 



Summary , 



This reoort oresents a methoa ^heret)y tne actual shaoe of the orsssure Dulse producea ay 

 an uncerwater explosion OuODls may De Oelineatea with little labour. It is shown that a single 

 universal curve may, with aoprooriate i^.Ojustments of the pressure ano time scales, De made to 

 fit reasonably closely the thsorstic-il curves ootainsa by lengthy numerical integrations tor a 

 wide range of charge weights ana oeoths, A formula for the peak pressure in the pulse and 

 Curves for th^ minimum buDDls radius and the half-period of the pulse are ^iven. 



Introduction . 



The theory developed by Taylor (reference l) describes the oscillation and the rise of 

 the DuDOh' pr:duced by an un9erwat9r explosion in terms of two simultaneous nor>-linear differential 

 equations whicn so far have only been integrated for a very limited number of cases. In 

 reference 2, a nufnOor cf :;QProximats solutions were developed for certain maximum and minimum 

 values of the vsrixolcs. It was founa possible for instance, to Jerive an analytic exoressicn 

 for the peak pressure in the pu1s-j produced when the bubble passes through its minimum size. 

 The present oaorr oxtonds this work by presenting an approximate method for calculating with 

 little effort the actual shape of the eressure culse-. For convenience in actual use, the 

 important formulae -^no the use cf the graphs is summariseo witn an example, in the last section, 



Theo ry . 



If a is the non-pi mens tonal radius of the bubble at time t, and z is its depth relative 

 to an origin 3^ feet above se^i-level, tne differential equations aerived by Taylor are 



« ^ =3 , , ,^ ,3 12 , 7T ,3 ',2 ^ , _ G , . 



(2) 



In these equations norv-aimens icnal lengths ar? used, derived from real lengths in feet 

 Dy dividing by the length scale factor L = 10 M* where M is the charge weight in ID, T.N.T, 

 {or equivalent weiyht of T.N.T. en an energy basis tf some othsr explosive is considered). 

 Ncrv-oimensional tirrcs art converted to real times Dy mul t iolying by the unit of time/{L/'g), 



In equation (i) the quantity G/w is the ratio of the energy left in the gas to the total 

 energy of the rnotion ana for T.N.T. it can oe written. 



3/» = ca-* (3) 



1 

 wnere c = ;. 075 n^ (ii) 



Two assumptions are now made which reouce equation (l) tc an equation with only one 

 dependent variable. It is assumed first th>.t for times fairly close to the instant when the 

 bubble reaches its minimuiT' size tne integral in (2) may oe regarded as sensibly constant at a 

 value m wnicn has been called the "momentum constant" since, aoart from 3 numerical factor, it 

 is equal tz the i^;rrentum -,f the water in a vertical olrectipn associated witn the mjving bubble. 



The 



