MOTION OF THE GAS SPHERE S4I 



Kennard (55) has made extensive calculations, based on formulas 

 equivalent to the initial equation (8.66) of the development given here, 

 but including the internal energy in computing the integrals. These 

 calculations indicate that near either surface the effect varies nearly as 

 l/l when close to the surface, the variation as 1/P being approached at 

 greater values of I. It is of course to be remembered that all such cal- 

 culations become increasingly questionable too close to the surface, a 

 criterion suggested by Kennard being that the distance I should be at 

 least twice the maximum radius am for reasonable accuracy. Kennard 

 has also considered the case of a free surface and rigid bottom both 

 present, which is only approximately a superposition of their separate 

 effects, and the case of an infinite rigid surface at an arbitrary orienta- 

 tion. For a vertical wall, a net horizontal attraction of the bubble 

 occurs as one would expect, in addition to its rise under gravit3^ For 

 further details Kennard's report (55), w^hich gives a comprehensive dis- 

 cussion of the features of bubble migration under a variety of condi- 

 tions, should be consulted. 



It should be mentioned that a number of boundary conditions other 

 than of infinite plane surfaces have been considered by other writers. 

 Savic (98) has obtained approximate forms of the equations of motion 

 under the influence of a rigid sphere and an infinite rigid cylinder. A 

 report from the Road Research Laboratory (94) treats the case of a 

 rigid disk, and it is found that its effect is, as would be expected, rather 

 less than that of an infinite rigid surface. These problems all involve 

 applications of potential theory, using the method of images or other- 

 wise, which are similar to the cases already described. Although these 

 calculations are of interest as idealizations of practical situations in ex- 

 plosion damage, their detailed developments are rather lengthy and 

 have not been as yet of much practical application, and so will not be 

 given here. Temperley has discussed the much more complicated case 

 of a deformable target (112), which can appear to act as either a free or 

 rigid surface depending on its displacement. Situations of this kind 

 are of considerable importance in assessing the effect of bubble motion 

 on explosion damage, a topic which is considered briefly in Chapter 10. 



8.11. Measurements of Periods and Migration near Surfaces 



A. Period measurements. A number of studies have been made of 

 the effect of the free surface and rigid bottom on the period of oscillation 

 for small charges (3^ pound of explosive). If the boundaries are far 

 from the bubble, theory predicts a variation of period with hydrostatic 

 pressure according to the formula T = KW^'^ {d + 33)~^^^, where W is 

 the charge weight in pounds, d is the distance below the surface in feet, 

 and the pressure at the surface is 33 feet of water. The predicted de- 

 crease in period near the sea surface and increase near the bottom are 



