356 SECONDARY PRESSURE WAVES 



+ ~ t/2(cos20 - i sin2^) 



[2 r^\dt ) r^ dt 2 r^ ' 'J 



The pressure distribution in the surrounding water is determined by 

 Eq. (9.3) if the kinematic history of the bubble is known, i.e., its radius 

 and displacement as a function of time, and the solutions obtained in 

 Chapter 8 can therefore be used directly to find the pressure. It is to 

 be noted that r and d are moving coordinates and calculations of the 

 pressure at a point fixed in space and different times must strictly be 

 found using different values of these variables in order to allow for the 

 motion of the center of the sphere. The complicated dependence of 

 pressure on the angle from the vertical is, of course, the result of the 

 vertical displacement which leads to radial asymmetry of flow and corre- 

 sponding pressure variations near and on the surface of the sphere. 

 The indicated variation of pressure over the surface (r = a) is of course 

 not real, and results from the artificial constraint of the bubble to spher- 

 ical form (see section 8.5). These effects are most pronounced near the 

 surface of the sphere and become larger for more rapid migration. If 

 this upward motion is neglected by setting ^ = in Eq. (9.3), one ob- 

 tains 



Po rdt\ dtj 2r'\dtJ 



The first term on the right of Eq. (9.4) represents the hj^drostatic 

 pressure Po at the depth z, and the last term is important onty at points 

 near the bubble surface by virtue of the factor {a/ry. Hence except 

 for points close to the charge, the variations of pressure are determined 

 primarily by the term (1/r) d/dt {0} da/dt). This term is also the domi- 

 nant one at sufficiently great distances when the velocity U cannot be 

 neglected, as the terms involving [/ in Eq. (9.3) all vary as the second 

 or higher powers of 1/r. 



A. The pressure in early stages of expansion. In the initial stages of 

 expansion, the vertical momentum of the gas sphere is small, and its 

 radial velocity da/dt is large. The approximate relation Eq. (9.4) ob- 

 tained for f/ = may therefore be used, and the last term is unimpor- 

 tant except near the bubble surface. If the additional approximation 

 is made of neglecting the internal energy, the analysis of section 8.5 

 gives the result 



(9.5) ^' = — J , ^' = 2V^a'5/2 



dt' V27ra''''' 5 



