324 - 4 - 



The Relatiov between the Momentum Constant m and the Pi si> Lacement at the 

 End of the First Oscil Lation . 



In orJer to calculate the aisplacwnent of a BuCDle it is desiraole to have some relation 

 between displacement at the end of the first period ^nd the monientum constant m. The method 

 adopted Dy Conyers Herri nj, and in Reports A and 9, was to assume that in any given case the 

 displacement with the surfact? prjsent Pore the same ratio to the free displacement under gravity 

 of a bubble at the same depth (with surfaces absent) as the momentum bore to that acquired freely 

 under gravity with surfaces absent. This in fact presupposes a linear relation between displacement 

 and momentum at constant depth z • 



The values of m and displacement h at the end of the first period, as determined from all 

 available full integrations of the equations of motion of the bubble have been collected together 

 In the following table. 



Relation between Momentum Constant m and Bubble 

 Pi spl acement h (non-dimensional units throughout) 



NOTE : In the last four entries, taken from O.S.R.D. 3841, d is the non-dlmenslonal distance 



above a rigid surface. In all other entries d is the distance below a free surface. Negative 

 values of m and h nnean that the bubble is moving downwards. 



It was shown- that when the bubble is near its minimum radius the dlff^^rent lal equation 

 of radial motion contains only the paraireters c and m. Since also tne linear velocity of the 

 bubble is a function only of m and the radius a. It scans 1 1 tely that the displacement will be a 

 function of m and c o ily. On plotting h' against m*'^, as in Figure 1, It is seen that all the 

 points lie on the same curve. Thus h Is practically Independent of c (which varies as the -^th 

 power of the charge weight). Moreover h' Is Independent of the manner In which the momentum Is 



acquired 



