166 



At a distance d non-dimensional units Delow a free surface or above a rigid surface the 

 rise h is altered to h' where 



(non-dimensional) (I2a) 



Agreement of approxtmattons with exact solutions . 



Recently the full numerical integration of Taylor's equations has been carried out for a few 

 depths and charge weights, covering the range of non-dimensional depths from 1 to u, and charge weights 

 from 2 to 100 lbs. Comparison of the above approximations with these figures, and with some 

 unpublished figures for a 1-oz. charge at 6 feet depth shows that the agreement over the whole range 

 is satisfactory. The period of the motion and the neximum and minimum radii agree within 1 to 5 per 

 cent, the maximum vertical velocity and the rise of the bubble within 5 to 10 per cent, and the peak 

 pressure within 7 to 20 pet cent. 



PtRT 1 1 . 

 Derivation of the Approximate Solutions . 

 Taylor's equations of motion of the bubble, when expressed in their non-dimensional form, 



are (l) 





(non-dimensional) (l3) 



W is the total energy of the motion, Ga the potential energy of the gas in the bubble 

 at radius a. For T.N.T. Taylor expresses the potential energy term (2) 





0.075 M^ I 



During the numerical integration of equations (13) by a step by step process it is noticed 

 that at different stages of the motion some of the terms in the equations either remain sensibly 

 constant or become negligible in comparison with the remaining terms. The following approximations 

 arise from these observations. 



The maximum radius of the bubble - a_ 

 _^ m 



During the first half period of the bubble the vertical trotion remains small, so that in 



equation (13) the term -^ z remains substantially equal to i z (the initial depth), while the 



1 ( 5t\ *^™ '^ negligible. The maximum radius of the bubble is obtained by setting 5S = in 

 equation (l3) simplified by these assumptions 



3 . 



ip^-pd - ca^~ ) (non-dimensional) (u) 



The 



(1) "Vertical motion of a spherical bubble and the pressure surrounding it". Equations 5 

 and 6. Taylor distinguishes all non-di nensional quantities by dashes, which are omitted 

 for convenience. 



(2) Above report. Equation 19. The exponent i is strictly true only for T.N.T. 



