1184 
= 0h= 
wnence since Ba is large and R, > a we see that W is large of order 8 R, or greater, Hence it 
can be shown that 
| 2,2 2 | 
Lert #3 = inves) it Zid 1 “ 1 (46) 
| " : | F Ry e Bra 
and we thus have from (42) 
aye ip ce 2 - 8 47) 
I [ es “| ai ( 
where 
i 1 
| s,| = ola: azz | or less (48) 
From equation (16) we have 
(1+ 8) Coal = (1+ 8,) eae 2BR, Pleiies: (1 +8.) 7 
Po 
and thus from (47) we find after simplication 
' = I 
a BL eee, Je MOE Ne 0 (84) 
aioe €-1 €- 1 (BR, - na/c) 
Bre 8 iv eek roe a) } BV ot? - r= nt Pee 
+ = lt aR bee oer FW 5 
Bct-nvV t--re/ee (Bet —-nvV te rice 1 
(49) 
Under the assumptions Ga and € large and nt'/G a small the above equation (49) thus gives 
the solution correct to 0 (6,), i.e. the errors are of tne second order. : 
1f we now approximate further we obtain from (49) 
BoWewomm EY ct sr + Ba 4 | - ae} (50) 
€ 
2 Po ct n Ro 
which is correct to fractional errors in cach term of the first order 2 > Ba » provided [: - | 
° 
is not small. As shown in Appendix |, c/n R_v il) usually be of order z or less for underwater 
explosion waves of small amplitude and thus (1 — c/n R) Is positive of order unity. If 
equation (50) is truc up to the time of maximum velocity corresponding to p =o then this time is 
given by 
ie FR IBS ay age 
-BV c*t* Fr +Ba+ nt’ a, ct 
e = 
R 
° 
(51) 
where a, * 
Cc 
G- ae) 
Co) 
'n order to solve this implicit equation we assume ct’/ Ry is small ana retalning only the 
predominant terms, we obtain as solution 
log a, reel (Log a)? 
nt’ = = 
€ 2 Bae 
(52) 
SINCE) secs 
