- 5 - 167 



The period of the osci I lattc^r P 



In the step Cy step integration of (l3) it is observed tnat the time T taken by the bubble 

 to reach its minimum radius at the end of the first oscillation is within a few per cent of twice 

 the time taken to reach the maximum radTus. It is also found that omission of the term •— in 

 integrating (l3) causes very little difference in the time taken to reach the maximum radius, 

 although the actual value of the maximum radius is quite considerably altered. The value of the 

 non-dimensional half-period i* is accordingly obtained from 



= J dt = f / 1 2 , (non-dimensional) 



/ 2 7T 



Where the integral on the right is taken from a = c to the maximum value of a. 

 The value of this integral has been given by Lamb (l) and Conyers Herring (2). The result, expressed 

 in the non-dimensional variables used in this note is: 



This value of the period (3), converted into real units, is 

 1 



0) 



The vertical momentum consiorit - m 



In the step by step integration of equation (l3) it is found that the non-dimensional 

 quantity J a dt becomes substantially constant when the bubble has contracted to about a half of its 

 maximum radius, and remains constant up to and beyond the time when the bubble radius is a minimum. 

 This constant value may be put' equal to m, and is proportional to the vrrtical momentum of the water 

 surrounding the contracted oubole. It can be shown that the vertical -^omentum in lbs, feet/second 

 units is given by 



7 

 Vertical momentum » It^l L m lbs. feet/second (6) 



A knowledge of the value of m enables several quantities associated with the motion of the 

 bubble to be calculated, so that an approximation to m is desirable. It is clear that the value of 

 m depends mainly on the radius tim^-curve whrn the bubble radius is large. An approximate 

 evaluation of m may be made if it be assumed that the effect of altering cither the depth z or the 

 Charge weight (i.e. the parameter c) is mainly to alter the maximum radius and the period of the 

 motion, without appreciably altering the shape of the radius time curve, at least in that portion 

 when the radius is large and the vertical momentum is mainly acquired. This is equivalent to 

 as.suming that the radius time curves could be superposed in this region if the length and time 

 scales were adjusted to make the maximum radius and period agree. .Mathematically, this assumption 

 is that in the equation: 



(*)' 



f ij^) (non-dimensional) (l7) 



for < t < U* 



the function f '-^) is independent of depth and charge weight. 



T 



(1) Hydrodynamics. H. Lamb, p.lin. 



(2) Theory of the Pulsations of the Gas Bubble Produced by an Underwater Explosion. 

 Conyers Herring. 



(3) 



The period may be expressed in a form valid for any explosive. If A. be the energy of 



the motion in calories per gram of explosive, M the weight of the charge in lbs. (?) becomes 



T (seconds) = °-^^' "^ ^^ 



i '^''' 



This expression enables k to be calculated frorr. experimental measurements of tnr period T. 



