411 
R, only that part of the wave emitted by the gas sphere after the time t. (It) 
is realized, the crest of wave emitted in the interval 0 $ ts T being 
destroyed at the shock front as the wave progresses outward. 
‘ 
For small intervals of time ¢ after the wave front has passed 
a point R » we may expand Y% in a Taylor series, 
’ 3035 
t > t 2 i 5 ( ° ) 
He ( ot | 
Yy It /R 2 
where i is the time at which the front reaches the point R » and / 
is the partial time derivative of ? at time t, e An expansion of this type 
could be made not only at the shock front R but also at an arbitrary point fF 
behind the front, if desired, From Eq. 3.30, we obtain for Y ’ 
R 
uo (T) t da ' 
Ve Gps ale dr (3436) 
Cau c oz : 
al?) 
It is to be expected that @ is a decreasing function of 7 at constant Yr 
since it is a decreasing function of f) and G, (&) is in our applications 
a decreasing function of time. Thus Y will be greater than unity for suf- 
ficiently large r. This means that the t scale is broadened relative to 
the T- a scale, and the wave profile at Ris broadened relative to the 
time profile of the wave during emission from the gas sphere surface. 
At sufficiently large distances R from the center of the charge, 
SL(R,C) , though not of course LOGE ) , can be approximated in acoustical 
form by P/ Po 5 where A is the density of water at zero pressure, With 
this approximation and the approximation, Eq. 3.35, Eqe 3.33 become 
p = (a,/R) P(u+t7y), 
alt) 
Ptt) = fo a OE (te) E (3.37) 
° 
2h 
