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APPENDIX B 
Theory of gas bubble motion 
Bi. The gas bubble is assumed to remain spherical throughout the motion 
but oscillates in size and rises under the influence of gravity. The 
motion is considered to take place in an unlimited mass of water. The 
free surface is taken to be sufficiently far away as to have no direct 
effect on the motion. (The surface will, however, enter indirectly, in 
so far as its position determines the hydrostatic pressure in the 
neighbourhood of the bubble). 
B2. The motion of the gas bubble involves two well-known types of 
incompressible flow. The first type is the bodily motion of a sphere 
through water in a fixed direction, namely upwards for the gas bubble in 
the present problem. The second type of incompressible flow is the pure 
radial motion of the gas bubble corresponding to an expanding or 
contracting spherical cavity. 
B3.- The following notation will be used for the analysis of the motion. 
t = time interval after the initiation of the explosive charge 
f = mass density of water 
a = radius of bubble 
g = acceleration due to gravity 
zZ = depth of bubble centre below a point 33 ft. above the 
water surface 
p = pressure inside the bubble 
Vv = volume of bubble 
G(a}) = potential energy of gas in bubble. This is equal to the 
work which would be done by the gas on the walls of the 
bubble if expanded adiabatically from radius a to infinite 
radius 
Qa, = fotal energy associated with the bubble motion 
BL. It is a standard result that a sphere moving linearly in water can be 
regarded as having, in addition to its own mass, a virtual mass equal to 
half the mass of the displaced water. Therefore, neglecting the weight of 
gas in the bubble the total effective mass of the bubble is the virtual 
mass, 2/3iipae The upwards velocity ot the gas bubble is -dz/dt so that 
the upward momentum is - Zeke dz/dt. This upward momentum is 
produced by the net hydrostatic force on the bubble. Neglecting the 
weight of the gas in the bubble, this force is simply the weight of the 
displaced water, L/ 31k pg. Therefore, by Newton's second law of motion 
3 3 
$Tpa c = & { - pe a | arele. | teats svete eters sate (B1) 
Integrating equation B1 gives 
ee a) oe a at PY |: A Sa, qgcren a ees (B2) 
B5. Another relation between a and z can be derived by forming the energy 
equation of the motion. First, the kinetic energy in the water round the 
bubble due to the combined radial and linear motion is given by 
otipa? (22) + Tape (ay 
