67 
Secondly, changes in gravitational potential energy of the water can be 
simply allowed for by taking the bubble as having a negative weight equal 
to the weight of the displaced water. The potential energy due to 
gravity can thus be written as 4/318 epz. Finally, the potential energy 
of compression in the gas has been defined as G(a) whilst the total energy 
of the motion is Q, Hence the energy equation can be written 
' mn Sd Ae Bydm~ | bas 9 
Gp = 2iipa (3) + Zipe ($2) + Zieapez+ G((alyineistet Mite ate (B3) 
B6. The values of G(a) and Q, will depend on the nature of the explosive. 
For T.N.T. it was deduced from early measurements*® of the period of 
oscillation of the gas bubble that Q, = 0.5Q where Q is the total chemical 
energy liberated by the explosive which can be taken as about 880 calories 
per grem for T.N.T. In C.G.S. units the energy Q, of bubble motion for 
a T.N.T. charge of mass M grams is given by 
10 
Qa = 1285 x) 10 M ergs eco eee eee ere eve eee (BA) 
B7. An expression for G(a) can be obtained by using the relevant 
adiabatic relationship between pressure and volume for gaseous products 
of the explosion. Using the relationship calculated by Jones and Miller 
for T.N.T. it is found that 
G(a) = 2.189 x 10'x SOLeMoco oom, F001. ooo | S00 (B5) 
a 
where a is measured in cme and M in grams. 
B8. Equations B2, B3, By, and BD can be used to determine a and z, for 
aT.N.T. charge of given size. However, for calculation purposes it is 
convenient to put these equations in non-dimensional form. This can be 
done by introducing a length L defined by 
L= (ey S66. pool, Paco Woodes WROGUMEEGDO: 600: od (B6) 
Hence writing 
a=a'L 
z= zZ'L eee eee eee coe eos eee eee eee (BZ) 
L 
= ¢'/= 
t t Zz 
the equations for a‘ and z' are 
2 2 
da' =. ee - G(a) - 4 dz* = a , eco eee BS 
(2) 2na'? I Q, 6 (22) 3 = ( ) 
i 3 = 5 At at!- isc “hee ase, voce meee (B9) 
a 
*Later results suggest that a better estimate is: QL = 4 Q 
