412 
These relations, together with Eqs. 3.34 and 3.36 ) provide an an asymptotic 
relation for the pressure as a function of time t at a fixed point R in 
terms of the radius of the gas sphere a (t) and the enthalpy of the water 
at this surface at time T +t / ae 
The foregoing analysis reduces the problem of propagation of the 
shock wave from the gasesphere surface a(t) to an investigation of the 
slopes C of the characteristics of G An exact determination of € or. 
even a rigorous proof that c has the properties hypothetically ascribed 
to it appears to be prohibitively difficult. However, it is possible 
to get information about the asymptotic behavior of , to assign a 
reasonable approximation to C » and to estimate the error entailed in this 
approximation. From the acoustical approximation, Eq. 3.23, we remark that 
c approaches ©, , the sound velocity at zero pressure, as }~ increases, 
The asymptotic value Cis not adequate for our purposes. As a possible 
approximation to é » the local sound velocity C + U in the moving medium 
suggests itself as reasonable. In order to investigate this approximation, 
we write, 
@ = (dG/dtI/(IG/r), 
gC pomieag a (3.38) 
gp ny 
where g is the oie ate ef. the contours of slope U#u. 
Expressing IG/dor in terms of 0G/0t and 4 » writing IG/dt as 
Gt) (dt /ot), and using Eqs. 3.13 and 3.19 for the calculation of 9» 
we finally obtain 
25 
