eg SUC a EG, 
> 2 
Du 3u° (27)""3, 6 “t/8, 
Dt , 
a 
a) Qa, wW, Ae,+P, oe 
ie aa AL On a 7*,* : 
Bi feed Ev tN, (4013) 
ate 
B WW, (o*s A en ) 
1 Cc, = u, é 2, 
t 
where the slowly varying factor (a,/a ) a is retained in B¢ t) for con- 
venience in integration. Writing Da/pt for WU and integrating once, we 
btai = 
; ge = ay [i+ eapeiie € a 
Zs a (4.14) 
Hy = Blu, , fy = 28 7, 
Again writing Da /Di tor U and inte Eqe 4el4, we get 
(=) = (I- 2 Hott Mes + 8, Sas eT cole 
nal 
pie byte 28 5 Hop 72 (4.15) 
2 7 u, I+ Pops 
The asymptotic forms of Eqs. 4.14 and 4.15 at large times are 
G)2 
ar ers a ig, (1+ Jto|4) , 
(4.16) 
(a/a, ween ee S/[u/2?X it £/4). 
The asymptotic constancy of ah y and variations of a/fa, as is as 
hold rigorously when Po is zero and P tends to zero more rapidly than Vt" 
The constancy of Ss LT; implies conservation of the kinetic energy integral 
of the fluid. At all times of significance to us p, may be treated as zero, 
since we are not concerned with the secondary pulses which have already heen 
adequately treated by others. 
The second term Q(t Mutey/2 in 6, (t) becomes with the use 
of Eqs. 4.14 and 4e15, 
36 
423 
