245 
Sec. 17 =) Eq. 107-108 
in air and in water, then the shock velocity is considerably higher, the 
pressure difference very much higher and the temperature rise lower in the 
water case. 
In order to gét these results quantitatively, we first put the 
Rankine-Hugoniot equation (Eq. (12)) ina rather general form, From purely 
thermodynamical considerations, one obtains the result that 
Bp = By pe (2), ney oe ¥ a2) ‘ 
ae) = = - ome 
oe aP/To 
1g = 
-f,? ft) a , (107) 
The second intesral can be evaluated and the last one can be integrated by 
parts. If this is done and the result inserted in the Rankine-Hugoniot 
equation Ey - Ey = 2 (Py, + Po)(71-Vo), the result is 
1 
NIH 
(25 =} (va Vo) vias Cc. aT-T ee aP 
- fm 4 + Sans OMe 
P 
if * v(t) aP = 0. (108) 
1 
This equation should be quite general for any fluid to which the fundamental 
equations are applicable. In applying it to water, empirical data for V as 
a function. of P and T can be used. This data has been obtained by Bridgeman. 
Professor J. G Kirkwood? has devised a very effective method of 
solving this equation numerically for water using successive approximations, 
He inserts the value of Pj - Py, and an assumed value of Tg into a modified 
form of the equation and obtains a better value of TZ. From P, and T,, Vo 
comes from Bridgeman's results. Table 16-2 shows the values of Pp, Vo, Ip 
and D computed by Iirkwood, 
1j. Rarefaction Wave Following Shock Wave. 
Suppose that the piston is accelerated instantaneously to a constant 
velocity w, so that no difficulties arise over the initiation process, but 
that later the piston is gradually breught to rest, as shown in ig. 17-1. 
