215 
Sec. 3 -4- Eq. 5-9 
the distance it moves in one second, or PU. Consequently the energy equa- 
tion is 
CS (n+ re w@) = - SD), (5) 
where © has again been omitted from both sides. # is the internal energy 
(chemical and thermal) per unit mss. This equation may be expanded and 
then simplified by combining it with Iq. (3) and (4). The result is 
RHE FL enrate ey 6 
ag PR) (6) 
Eq. (3), (4) and (GS) are the fundamental equations which must be satisfied 
by (A> P, E and U. 
, 
5. Conditions at a Discontinuity 
The differential equations of Sec. 2 govern the situation wherever no 
discontinuities occur, but if there is a discontinuity in Cy P or U the con- 
ditions of conservation limit the values of the density etc. across the dis- 
continuity. 
a. Basic equations. Ina time dt an amount of mass (°y(D - Uz) dt is 
brought up to a moving discontinuity (velocity D) from the right, if the sub- 
script 1 denotes the properties immediately to the ricsht of the discontinuity. 
The cross-sectional area is unity. The mass Po(D - Up) dt is taken away on 
the left in the same time. When dt is made very small so that the layers on 
either side are infinitesimal, the matter brought up on the risht must equal 
that removec on the left so that 
ees eavore ee EN Commae (7) 
even if D is not a constant. 
Similarly, the change in momentum of a very thin slice of mtter of 
mass (1 (D = U;) upon the passage of the discontinuity can be equated to the 
force acting; i.e. 
(1 - WY -)=%-F- a 
Finally the wort: done on the slice by the forces acting must equal 
the increase in energy so 
2 2 
PoUp ~ yy = Py(D - Uy)(Bp - By + 4 U5 - 3 04), (9) 
where 3 is the internal enersy per unit mass. 
b. Another form for basic equations. From the above equations by 
straight algebra one obtains (if V = 1/P ) 
