136 



if the effect of gravity is neglected. 



or 



1.8?/ 5.9 X 10* 



= 0.03 



(25) 



1.44 X W 



accoralriij to the calculations illu.^trated in Figure 2. 



If the same charge haa Been exploded at a greater depth tht bubble would not have expanded 

 to so great a radius; it would therefore not have risen so much and F would have Been greater. 

 It would be interesting to repeat the calculation for the sarae charge for a range of depths, i.e. of 

 values of z' . 



Pressure di stri bution. 



The prassjre in an incompressible fluid due to the motion of an expanding or contracting 

 sphere when the centre moves with volocity U in a straight line is obtained from Bernouilli's 

 equation. The velocity potential <^ is 



<^ = s!l * i il|!cos0 

 r 2 r 



where is the angle between the radius vector r and the direction of motion, & written for da/dt, 

 (f: referred to axes which move with velocity U so that the pressure is given by 



£ - gz 

 P 



5J 

 3 t 



P - ^ (u^ * v2) 

 3 X 2 



(26) 



where u and v are the velocity components referred to fixed axes and B (^/3 t Is the time rate of 

 variation of <^ at a point which is fixed relative to the centre of the sphere and therefore moves 

 with velocity U. 3 cp/B x is the space rate of change of ^ In the direction of U. The velocity 



components u and v are racial and tangential, so th^t 



-i^ = 4^ * 2^ cosS 



3 r 



T 



i i^ = 1 2! U sin( 

 r Be 2 P 



(27) 

 (28) 



— i = u COS d - w sin I 

 o X 



+ lif (cos^i9-^ sln^ 



The complete expression for pressure is therefore 



1 a 



2 7 



1-JX + i i, (ail + 5au) cos e + ^ U^cos^ .9 - 4 sin^ |9) 

 r 2 r'^ r^ ' 



+ S^ SU cose ♦ i Sj uMcos^e* ^ sin^e) (29) 



This may be expressed in terms of the non-dimensional variables defined in (3) and («). 



-£- -z- 



30L 



tlj (a'lj' ■*■ sa-j-) cos e * ^\^' (co% e - \ s\r? d) 



S'U' cos e + - 



where u' = - H£ = -i- , v is 2S , ind U' is 2ii or 

 dt* /gL df df 



d'z' 

 df^ 



U'^ (cos^e + ^ sin^e) 

 (30) 



(30) is now expressed in a form suitable for computation. 



