513 



Harmonic Analysis of Bubble Shapes . 



The silhouette photographs of the bubple give profiles which in general have the shape of a 

 cross-section through the axis of symmetry. it was desired to determine the co-efficients in the 

 series. 



DP (cos 5) + b P (cos 0) + etc. 



(1) 



in which R is the radius vector to the surface of the bubble from a given origin on the axis, 6 the 

 angle included between the radius vector and the axis. 



AS a first attempt it was decided to determine that curve in which only the first seven 

 co-efficients were non-zero, and which fitted the observed section at seven equally spaced values of 



the angled, viz. 0°, 30°, 60°, 90°. 120°, 150° and 180°. The radii vectors R R, at these 



angles were measured giving seven simultaneous equations, of which a typical one is:- 



3 + b^p^ (cosfi^) + b^p^ (COS0J + + b^P^ (cosf*^) 



(2) 



These equations were solved once and for all for the seven co-ef f icients in terms of the radi 

 R to R . The solution is given here for reference. 



.OJU r^ + .2U3 r^ + .U67 r^ + .256 r 



.086 r + .660 r + .687 r 



.152 r^ ♦ .776 r - .267 r - .661 r 



.237 r^ + .513 r^ - 1.36U r^ 



.«42 r^ - .27U r - .9U5 r + .777 r 



.677 r^- 1.173 r^. .677 r^ 



.•372 r --.7115 r + .745 r - .372 r 



(3) 



R, + R, 



R. + R, 



A fairly extreme example is illustrated in Figure 6. The full line is the observed cross- 

 section of a bubble, the broken line is the curve given by the seven Co-eff icients as calculated by the 

 above method. The agreement between the two curves is well within tne limits imposed by the actual 

 photographs. The above method which is the simplest that could be devised, was accordingly used 

 in analysing all the photographs obtained. 



Analytically, the ideal method of calculating any given c:-efficient in the series (1) say 

 b is to use the known relation 



6 = io-li [-^-180 R p (cos 0). d (cosi 



(1) 



