-'- 511 



(d) Shape Change of the BubDle :- In Figure u the second harmonic shape co-efficient b 

 has been plotted as a fraction of the co-efficient of zero order a up to just before the mininnm. The 

 following table includes figures for the higher order co-efficients also. It will be seen that b 

 is always small; the small shift of origin which would have to be nade to make it vanish exactly 

 should not affect the other co-efficients appreciably. 



TABLE 3 . 



Harmonic shape Co-efficients during Contracting phase. 



For comparison with the observed measurements of the second harmonic shape co-efficient b , 

 the value calculated by the Nautical A)manac office using Temperley's equation has been plotted in 

 Figure U. Temperley's equation contains various time derivatives of the first co-efficient a, and 

 the linear velocity u of the bubble. These values were taken from the earlier calculation in which 

 the bubble was assumed to remain spherical; I.e. it was assumed that a and u are not perturbed by 

 the growth of the higher shape co-efficients b etc. 



It will be seen that though the general form of the b , versus time curve is roughly the same 

 experimentally as is given by the n.a.O. calculation, b . attains appreciable magnitude much earlier 

 than in the theory. It seemed possible that this was due to actual linear velocity u being greater 

 than the theoretical one in the early stages of collapse of the bubble. Accordingly Temperley's 

 equation was reintegrated, using the observed value of the velocity u together with the theoretical 

 values of the mean radius a and its derivatives. This result has also been plotted in Figure ». 

 It will be seen that this b . curve is now somewhat nearer the observed curve, but there is still a 

 considerable' difference. It is thought that most of this discrepancy could be due to error in the 

 velocity-time curve used. A final check of the correctness of Temperley's equation must therefore 

 await a more accurate determination of the velocity-time curve for the bubble. 



Conclusions . 



(1) During most of the first oscillation the bubble remains very nearly spherical and 

 agrees reasonably well with Taylor's theory when allowance is made for the presence of the free surface. 



(2) During the last five milliseconos before the first minimum the bubble flattens, its 

 vertical diameter shortening, and the upper surface becomes flatter while the lower surface remains 

 approximately hemispherical. Close to the minimum the upper surface becomes concave and the bubble 

 resembles an inverted mushroom. 



(3) 



