Page 116 - 48Fundamentals of Compressible Fluid Mechanics

78 CHAPTER 5. NORMAL SHOCK

is substituted
the equation is remains the same. Thus, one
Equation (5.19) is a symmetrical equation in the sense that if
solution is substituted by
by
and
(5.20)

It can be noticed that equation (5.19) is biquadratic. According the Gauss Bi-

quadratic Reciprocity Theorem this kind of equation has a real solution in a certain
3
range which be discussed later. The solution can be obtained by rewriting equa-
tion (5.19) as polynomial (fourth order). It also possible to cross multiply equation

(5.19) and divided it by

(5.21)

Equation (5.21) becomes

(5.22)
%

The first solution (5.20) is the trivial solution in which the two sides are identical

%

%
and no shock wave occurs. Clearly in this case, the pressure and the temperature
. The
second solution is the case where the shock wave occurs.
The pressure ratio between the two sides can be now as a function of
from both sides of non–existent shock are the same i.e.
. Utilizing equation (5.16) and equation
(5.22) provides the pressure ratio as only function of upstream Mach number as
only single Mach number, for example,

(5.23)

The density and upstream Mach number relationship can be obtained in the same

fashioned to became

(5.24)

Utilizing the fact that the pressure ratio is a function of the upstream Mach num-

, provides additional way to obtain additional useful relationship. And the
temperature ratio as a function of pressure ratio is transfered into
ber,

(5.25)

%

%

%
3 Ireland, K. and Rosen, M. ”Cubic and Biquadratic Reciprocity.” Ch. 9 in A Classical Introduction to

%

Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 108-137, 1990.

"