216 
Sec. } aye Tq. 10-12 
Dis Uj) + Vy f (Po - Py )/(Vy - Vo), (10) 
| FMR Te a 
U, =U, + V (P, “ Ps (V5, - Vp) A (11) 
Bp Ss Ty = = (Py + Po) (Vz - Vo) ¥ (12) 
Knowledge of the equation of state and heat capacity enables one to 
calculate EH, - Ey as a function of Pp and Vp 80 that Hq. (12) becomes a rela- 
tion between Po and Vo (for given Pi, V,). This is called the Rankine? 
Hugoniot” relation. A typical curve of this kind is shown in Fig. 3-1. It 
Should be noted that this curve 
Fig. 3-1 
wU 
ge ee 
V., 
Aa 
is not the same as either the adiabatio or isothermal P-V curves. 
Since there are three equations involving the quantities, Pj, Vj, Uj, 
D, Po, Vo, Up, a knowledge of Pi, Vy, U, and one of the others enables all 
to be computed. 
A snecial type of discontinuity satisfying the above equations is worth 
notins;. it is one in which D=U;=U5, Po=P), but Vivo and To#T,- Such a 
discontinuity (which may be a boundary between different kinds of gases, for 
example) would not be stable for long if heat conduction and diffusion were 
considered, but is of practical importance in the short time intervals cf 
interest here. 
The solution of a given problem is therefore determined if values of 
P, V and U can be found which satisfy the differential equations (3,,6) 
wherever the properties are continuous, fit the relations (10-12) at all 
discontinuities, and are such that U equals U of the pistons, if any, closing 
the ends of the tube. 
4, Simple Shock Waves in an Ideal Gas 
a. General considerations. A simple and yet important application of 
the basic equations is to the problem of a shock wave produced in a perfece 
ges ina tube by a piston suddenly accelerated to a constant velocity w. ‘Th 
