. 5 - 325 



acquired, since points representing cases in wnlch jravlty alone acts lie on the same curve as 

 those in which a free or rijia surface contriDutes appreclaoly to the total momentum. 



Fro- Fijure 1 it follows that the relation 



h^ - 3.57 (m^''^ - 0.008) (non-dimensional) (?) 



agrees very closc-ly with the results of numerical integration down to values of h of 0.015. 

 In equation (3) onV ■ the numerical value of h and m are to be taken; clearly the displacement 

 and moemtum will always have the same direct ion. (3). 



The Attraction of the Subtle to Various Surfaces^ 



It is seen from the foregoing tnat the general procedure in calculating the displacement 

 of a bubble at the end of Its first oscillation falls into two parts, voz: (i) the calculation of 

 the momentum acquiri?d by the bubble towards or away from the surface (to which must be added 

 vectorially tne momentum due to gravity), anfl (ii) the determiration of the dlsplacument h from 

 equatiorj (3) or from Figure 1. 



In the neighbourhood of an Infinite rigid plane the momentun may be calculated from 

 equations (l) and (2), while if the surface is a free plan* surface, equations (5) and (5a) of 

 Report 4 must be used. Similar "approximate' equations have been obtained for a number of surfaces. 

 In particular the sphere(»), infinite cyllnoer(u) ^na disc(3). In Report B It was shown that the 

 momentum acquired at the end of the first oscillation towards a rigid surface can be written 



m = (attraction coefficient of surface) (l-lla|„ z ) (») 



where a Is the non-dimensional itaximum bubble radius. This may be simplified still further by 

 using the approxifrat ion a z = 0.195, discussed above, giving 



m = (attraction coefficient of surface) L2i°l!iZ (»a) 



This attraction coefficient, which is a punjly geometrical factor, has been multiplied by 

 u X (distance) , and plotted in Figure 2, for the sphere, cylinder and disc. In the case of the 

 cylinder the attraction coefficient used is only an approximate expression for a certain Integral, 

 and Is not valid when the distance of the bubble from the cylinder axis approaches the value of the 

 radius of the cylinder. Over this region the curve Is drawn with a brokon line; It has been 

 put in by eye since it Is known that the curve must tend to the value unity when the distance of 

 the bubble from the cylinder's surface is very small. 



when the maximum bubble radius approaches the value of the distance between bubble and 

 surface these equations are no longer strictly valid. Since there are as yet no equations for 

 these surfaces in which still higher ordiir terms are considered, in such cases the approximte 

 equations will be the best cstirate. In this connection It is worth noticing that In the one 

 case Where such an Vixact" theory is available - viz. the Infinite rigid pUne as treated by 

 Schlffmann - the approximate theory Is not much in error even up to the point where the bubble 

 touches the plane. 



Formulae for Minimum Radius and Peak Pressure in Pressure 

 Pulse due to Bubble Collat^e when Surfaces are. Absent 



In Report A equations were given for a^, the minimum bubble radius, and P , the peak 

 pressure in the pulse emitted by the collapsing bubble. The equations were transcendental and 

 required graphical solution. Two alternat iv.; equations have been put forward by the U.S. Bureau 

 of Shlps(6) to represent the results of solving these transcendental equations. These alternative 

 formulae are thus more convenient for num'jrlcal work, and are given hero for completeness.. 



a^ = 0.UU6 z """ + C.198C (noo-dimensional) (5) 



