1054 
~2= 
We have 
Do = Op + ou do (6) 
ot ot or 
also Oo 2 ,2 | (7) 
ot ot 
where ¢* is a function of P or 0 and the carticular oarticle under consideration. 
Then from (3) ana (6) 
a0, Do Op do Ch) a 22 Be 
se ut Oe = (Db, mee tc | tt 
at Ot or lor 2 or ot or 
(b, - u) Oo, ee [-3 zo (Fp + u2e | using (2), 
and finally 
ap Op Ou 2p cu 
ahe (hh. on) es 6 Spas where r = R 8 
ot Wh or Pp or r (3) 
Again from (1) and (5) we have 
du, Ju Ou Gu 1 ip Ou 
— = —— + DOD U——-—-- —— + D — 
t t Lor i jd Olt 1 or 
(vb, - u) OU. 5 AC reset (9) 
Or por 
Eliminating a from (8) ana (9) we have 
r 
do 00 du,. 1 A0 2pce 
Pet) 2 Sane ke alee a ee ie a) 
t or at por r a 
dp au 2 2 = 
(b, - u) -—+ + cp — = Op UU emeeee Ne P cu (d, u) 
at ot or 1 r 
where r = R (10) 
Since all terms in (10) refer to tne coint immeadiately behina the shock wave we may 
rewrite (10) as 
Qo du Jo 2 p,6.2u. (bd, =u.) 
(b, - “l= + SPL = 2 a { (by - uy)? - c,? )- Rb 
(11) 
where cy is the value of c aporopriate to yas just behina the shock wave. 
The Rankine Hugoniot theory(1) of the shock waves gives 
a a, (0. - 0) 
7s OE ayieey +a) ic, 2) 
a fy 1) o+ly+ 1) Dy 
(> (13) 
oO 
° 
u 
” | 
o oO 
fe} 
— 
= 
> 
