389 
Lanbe/ has presented a solution of the problem in the incom- 
pressible approximation, remarking that a more elaborate analysis appears 
hopeless. The moderately good agreement of Lamb's theory with available 
experimental data for TNT is completely removed when an appropriate equa- 
tion of state such as that of Wilson and Kistiakowsky/ is employed for 
the gas sphere instead of the ideal gas equation of state. Lamb's theory 
has been improved somewhat by Butterworth, 5/ who attempts to fit an 
acoustical solution of the hydrodynamical equations for large distances 
to the incompressible solution for small distances at a distance equal to 
several times the radius of the gas sphere. A thorough amlysis of the 
problem has been given by Penney.0/ Penyjey's calculations are based upon 
the numerical integration of the Riemann equations, valid if dissipation 
terms in the equation of motion of the fluid can be neglected. This direct 
method suffers from the disadvantage that a laborious process of numerical 
integration must be repeated for each explosive. 
Kirkwood and Bethel/ have developed an explicit theory of the 
shock wave produced by an underwater explosion. The theory can be employed 
to calculate the pressure-time curve of the shock wave at any distance from 
a spherical charge of explosive, given the thermodynamic properties of the 
H, Lamb, Phil. Mag., 45, 259 (1923). 
4/ G. B. Kistiakowsky and E. B. Wilson, Jr., OSRD Xeport No. 114 (1941). 
/ S. Butterworth, British Report, S.R.E., Summary No. M.S. 584/36 (1936). 
6/ W. G. Penney, Btitish Report RC 142 (1941). W. G. Penney and H. k. 
Dasgupta, British Report RC 333 (1942). 
7/ J. G. Kirkwood and H. A. Bethe, OSRD Report No. 588 (1942). 
