145 



ON THE CHANGING FORM OF A NEARLY SPHERICAL 

 SUBMARINE BUBBLE 



W. G. Penney and A. T. Price 



October 1942 



Introduction, 



When an explosive charge Is detonated under water the resulting bubble, as It grows In size, 

 rapidly approaches a spherical form. The degree of stability of this spherical form is a matter 

 of some Interest, especially In view of the fact that the bubble may contract again, as both 

 photographic and acoustic observations show. In the case of small charges several pulsations may 

 occur, and there is evidence of at least one complete oscillation being possible with a large charge. 

 On the other hand, for a very deep explosion, where no dome or plume appears, what finally reaches 

 the surface is not a gas bubble but an emulsion of gas and water; this suggests that the bubble 

 has departed so greatly from the spherical form that it has entirely brol<en up and disintegrated. 



The mathematical calculation of the changing form of the bubble would present considerable 

 difficulty even if there were no uncertain factors in the problem. A start can, however, be made 

 by considering the case of small deviations from the spherical form, and investigating the tendency 

 of these to increase or decrease as time proceeds. In the calculations which will now be described, 

 a first order perturbation theory is developed for a nearly spherical bubble expanding or pulsating 

 In an infinite Incompressible fluid. The radius vector from the centre of the bubble to its 

 surface Is expressed as a constant plus the sum of spherical harmonic components of different orders 

 n, the coefficient of each component varying with the time. Thus the departure from accurate 

 sphericity at any moment Is measured by the magnitudes of these harmonic components. 



Each harmonic component of order n is found to contain a quasi-periodic time— factor of the 

 form Cj e " 'n'*' + Cj e " f (-t) , where '_(') is a periodic function having the same period T 

 as that of the pulsation of the oubble, X. is a certain (complex) constant, and C, and C are 

 arbitrary constants determined by the initial conditions. The order of magnitude of the function 

 f (t) at any moment is roughly proportional to the reciprocal of the radius of the bubble; 

 consequently, in the case of larje pulsations, i.e. Intense explosions, the non-sphericity is 

 greatest when the bubble is small. Also, in virtue of the exponential factor e^n {K contains 

 In general both real and imaginary parts), the non-sphericity at any stage, say at the minimum 

 size, increases with each pulsation, indicating that the spherical form is ultimately unstable. 



The harmonics of high order are found to increase exponentially at first, and then oscillate 

 In magnitude. The higher the order n, the greater is the Initial magnification. Hence any needle 

 shape imperfections on the charge will become highly magnified as the charge explodes. This may 

 be the explanation of the prickly appearance sometimes observed In photographs of the early stages 

 of submarine explosions. 



The conclusions which can be drawn from these calculations are, however, restricted by the 

 general condition that the perturbation must be small and it ts clearly desirable to extend them so 

 as to avoid this restriction. 



Basic assumptions. 



Taking the mean centre of the bubble at time t » as origin, and assuming that the 

 departure from sphericity is small for all times to be considered, we write the radius vector R 

 to the surface of the bubble in the form 



i(t) 



CD 



* ^ On''' ^nl^' *'• ^ I On I (a < e) (1) 



n=l " " 



where S^ is a surface spherical harmonic of degree n, and e is a small quantity of first order.. 

 Sjj may be supposed analysed into zonal and tesseral harmonics in the form 



