- 9 - 235 



The stabi I ity of the surface of the bubble near the position of mimmum 

 cont raction . 



when the bubble contains gas the radius contracts at increasing rate until the volume decreases 

 sufficiently to raise the internal pressure above the undisturbed hydrostatic pressure at the level of the 

 centre. The radial velocity then decreases and reverses sign at the minimum radius. The curvature of 

 the (a, t) Curve is concave towards the axis of t during the greater part of the first oscillation so that 

 the acceleration of the surface is directed from the liquid inwards towards the gas. During this part of 

 the pulsation small disturbances of the surface should be stable. when the pressure of the gas rises the 

 radial velocity decreases and the (a, t) curve becomes convex to the t axis, the surface being accelerated 

 outwards from the gas towards the liquid. under this condition the surface may be expected to be unstable 

 in the same way that a horizontal liquid surface is unstable when the liquid Is above and the air below, so 

 that gravity acts from the liquid toward the air. 



These conclusions are borne out by the photographs of a bubble tal<en near its minimum radius. 

 For example, in the photographs shown in Figure 8b, it will be seen that up to photograph No. 8 the surface 

 of the bubble is very smooth. Indicating stability of small disturbances, and Figure 9 shows that when 

 photograph no. 8 was taken, the surface had net begun tc accelerate outwards. 1 he part of the (a, t) curve 

 of Figure 9 which is convex dbwnwards'^ccurs just befrrc- phjt'jgraph Nc. 9 was taken, so that instability might 

 be expected to show itself in the photograph. it will be seen that in ph.jtcgraph No . 9 the surface has in 

 fact become ragged L«ing to the Instability. 



appea 

 at 



It will be noticed in photographs Nos. 9, lO. U and 12 of Figure 8b that the instability which has 

 leared in no. 9 seems to disappear very rapidly on the lower part of the bubble but continues and increases 

 the top. 



The appearance of the free surface . 



In the photographs of Figures sa, b and c, the frc-o surface appt-ars rather out of focus above the 

 bubble; in photographs nos. 1 to 12, the free surface has moved very littlt, but in nos. 17 to 22, it has 

 risen to a considerable height. 



A ppendix. The effect of a f ree or rifjid su rface on th e mo tion of a spherica l 

 bubb l e, 



in order to compare the results of ttiese experiments with calculation of thf observed rate of rise 

 and pulsations of the bubble, it became necessary to extend the calculations of Report a so as to include 

 terms representing the effect of the free surface. Th'^ condition which enables the rise of the bubble 

 under gravity in one pulsation to be comparaOln with its maximum diameter n<^cessarily involves a free 

 surface effect which is comparable with the gravity effect. 



Herring gave =1 formula for the effect of a distant free or rigid surface on the position of the 

 centre of a bubble. if u is the velocity towards the surface his formula is 



a' dt \^ dt 1 u dt V ot y 



(A.l) 



a = the radius of the bubble at time t, 



h = the depth of the explosion, 



1 df?^ CtJ 



U = I and R 1^ defined in Herring's paper, 



dt 



and the upper sign refers to 3 rigid surface while the lower refers to a free surface. since only terms 

 of order i/h are retained It setms that equation (A.l) may be written 



•^ , 3 , - 3a^ d / da \ 



