490 . , . 



The calcul ation of a bubbl". firtssurc pulse - Swnmary of formulae . 



For convenience in jse the relevant -iquations ana directions fer the use of the graohs In 

 obtaining the pressure oulse to Be exocctca from a given charge at a given asotn are collected 

 here together witn a njmerical exr.mtle. 



(i) The two oarameters c '\na rr, must first Ce calculated. For T.N.T, and aooroximately 

 for equivalent weights of other exolosives on an energy Oisis, c = 0,075 M ivher" M is th? Charge 



weight in ID. The momentum constant .nay Oe calculated from formulae in references 2, 3 and 4 

 for the case ef a charje in ooen water or near various surfaces. In ooen water an aoproximate 

 excression for m is 



Where 2 is the initial deoth below a coint 33 feet above sea-level, in non-dimensional units. 

 To convert to these non-dimensional units divide all lengths Oy L = 10 M and all times by the 

 unit of time (L/g)*. 



(ii) Tne minimum radius a in non-dimensional units and its value A in feet may be 

 read off the curve in Figure 1 for the calculated value of the carameters c (or m) and m. 



(iii) The oeak oressure d in non-dimensional units is next calculated from 



U 7T .^ 



Where r is the non-dimensicnal distance from the coint to the centre of the bubble. If the 

 distance is requirefl in feet and the pressure in 'ID-Zsq. i n. the right hand side of this exaression 

 should be mult idled By «3.t M . 



(iv) The h.\l f oeriod Tj , aofincJ js the time for the oressure to droo from Its maximum 

 to ono half Its maximum value, is read -ff the curve in Figure 3 for the oarlicular value of the 

 oarametsrs c and m. 



(v) Finally either of the two curves in Figure 2 miay be taken as giving the oressure 

 time curve (or strictly as half the pressure time curve since the curve is theoretically symmetrical 

 about the time of the oeak cressurc). 3y ir,ul tiolying the numerical values of the ordinates of 

 the curve by the value of the oeak pressure calculated in (ill) above and the values of the 

 abscissae by the half period T calculated in (iv) ibove this curve becomes the theoretical 

 pressure time Curve in ^.bsoluto units, 



AS an ixamole consider a charge of 200 lb. T.N.T. it 200 feet depth. Tne length seals 

 factor L - 37,5 feet and c = O.ICW, The norvdimenslonal depth is thus 233/37.5 = 6,21 = z , 

 From (1) above the value of m for open water {i,e. In the absence of surfaces) us O.OOUSH, The 

 factor 100 c^ = 1,094 so that m/ 100 c = 0.00415, From Figure 1 the minimum radius 

 a = 0,067 (icc)'*'^ so that a = 0.D713 and the minimum radius in feet Is 2.67, Inserting this 

 value of a In the formula for the oeak pressure tne quantity ro is evaluated to be 20,35. 

 The value of 43.4 K? is 613 so that in feet and lD,/sq.in. units the quantity rc^j^ is 12490. 

 This means for example that at a distance of 20 feet the peak pressure In the culse is 

 acoroxirMtely 624 lt),/sq,in. Finally from Figure 3 the value of T./m'" is seen to be 0,68 

 for this value of m and c, which gives T( - 4.0 milliseconds. Tne shape of either half cf the 

 pressure time curve is given by Figure 2. 



References . 



(1) Sir Geoffrey Taylor. -The vertical motion of a spherical bubble and the pressure 

 Surrounding it". 



(2) A.R. Bryant. 'The behaviour of an underwater explosion bubble. Approximations 

 based on the theory of "Professor G.I. Taylor". 



(3) ^.». Bryint. •.' simplified theory of tne effect of surfaces ^n the mctlcn ;f the 

 cxol;si:n bubble". 



(4) A.B. Bryint. "The behaviour -.f >n underwater explesl-n bubble. Further 

 appr:xlmati :ns". 



