132 - 2 - 



and a Decomee strall, / a'dt becomes much larger than a^ so that - dz/ot becomes large compared 

 with 2gt. Thus the centre of the bubDle will De displaced upwards at a rate which rapidly 

 increases as a diminishes, in fact tht analysis by wnich Conyers Herring showed that the hollow 

 is displaced upwards rapidly ceases to apply. Herring describes this rapid increase in 

 displacement as an "illustration of the gentral instability of the contracting stage'. 



Though instability would probably prevent the hollow from remaining approximately spherical 

 during the contracting stage, yet photographs of the bubble from the explosion of a detonator by 

 Edgertonrf have given measurable radii over rather ;iore than 2 complete pulsations, so that the 

 hollow appears to remain sufficiently coherent for this time to pulsate as though it were spherical. 



This experimental result encourages one to pursue the subject further and to calculate 

 what the motion would be if the bubble were constrained to be spherical by some iiind of constraint 

 which absorbs no energy and no inertia. (l) is still applicable. The other equation which must 

 be satisfied is that which ensures constancy of energy. The kinetic energy of the water 

 surrounding a sphere of radius a is 



2.,a^[^| ^ JT.pa^[§f 



{no terms containing the product ^ . g^ appear) 

 so that the equation of energy is 



:^npa^{gz) * 2vp^^\^y * \-np^^ [%X = * - 2(a) (2) 



where %pz is the pressure at the level z so that z is measured from a point 33 feet above the sea 

 surface, W is the totai energy associated with the motion and G(a) is the work which the gas would 

 do on the walls o' the bubble if expanded ad iabat ically from radius a to infinity. 



Rec'uction to non-dimensional form . 



For convenience of calculation (2) may be reduced to non-dimensional form by means of the 

 length L which is defined by 



L = {^)* (3) 



Setting a = a'L, z = z'L t = f /I (4) 



a', z* and t' are non-dimensional and (2) and (l) can be written 



aa-1^ . _! \,_2M\ .1 l^^-y _ £z- (5) 



df 277 a*-^ 1 w r 6 I df 



dz' 



A' 



:t5 



(6) 



It will be seen that if G(a) could be neglected compared with w, (5) would be non-dimensional 

 so that the complete range of solutions of (5) and (6) would only involve the one variable parameter 

 which is introduced during integration, namely z' , the value of z' at the level of the explosion. 

 For this reason it is necessary to estimate the volue of G(a)/W. 



Value cf G(a)/V . 



The ratio G(a)/w can be calculated from the adiabatic relationship for the gaseous products 

 of the explosion. Using the adiabatic relationship calculated by H. Jones and Miller for T.N.T, 

 it is found that below a pressure of about 300 atmospheres the gases expand according to the law 



pv^-2^ = constant = (1.725 x lo') (S39.3)''^^ = 7.830 xio' (7) 



where ..... 



