239 
Sec. 13 -28- Eq. 89-96 
If the initial values of P and C are given, the solution of this equation 
satisfying these initial conditions is the desired function P = (7? ye 
For an ideal gas this procedure Gives the adiabatic law 
P Rie = k, a constant, (89) 
where 0 = c,/c,,. It should be noted that for a given mterial point, i.e. 
for a given point moving with the gas, k will remain a constant only while 
no discontinuity occurs. It will change on the passage of a shock wave. 
Furthermore, in the general case k may be different for different material. 
points. 
13. Riemann's Form of the Fundamental Equations 
Whenever there is a portion of the x-t plane in which no discontinuities 
occur and in which all the material is on the same adiabatic, it is possible 
to transform the fundamental differential equations of Sec, 2 to another 
form of considerable value which is due to Riemann. Under the restrictions 
given, the argument of the last section shows that P is a definite function 
of fF alone, Thén also c, the velocity of sound, is a function of ? alone. 
Introduce a mathematical quantity 
POivenughip vend : 0 
(xe J a Cc ‘¢ /( ) (9 ) 
ai : 
where c = (aP/a ? )2 is the velocity of small amplitude sound waves under 
the given conditions, (3 is the initial density. Then 
220 _ 2 9? rape _1 dy (52) 
zx ( of CAP? xt oP. dx 2 7 
aire aw S cA . (92) 
c me. ops 
Since a4. = © +U 9_ , the equations of conservation of mass and 
dt, pt ox 
momentum, Iq. (3) and (4), can be expressed as 
OW ow 3 
Tree re U 
aot Se = Giatoes (93) 
ou au ow 
. +U EL bees c ae . (94) 
Addition and subtraction of these equations yields the new pair: 
P) a) 
os Sp QUIS Cp ee ) es 
{3 ( Nein PaO: (95) 
: : 
= 6) (96) 
ee + (U-c $}w-0 
