Parameter $\frac{V}{V_m}$ .

$$
\frac {T}{T _ {t}} = 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} [ \mathrm{adiab,perf} ]\tag{70}
$$

$$
\frac {p}{p _ {t}} = \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {\frac {\gamma}{\gamma - 1}} = \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {\frac {7}{2}} [ \text { isen,   perf } ]\tag{71}
$$

$$
\frac {\rho}{\rho_ {t}} = \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {\frac {1}{\gamma - 1}} = \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {\frac {5}{2}} [ \text { isen,   perf } ]\tag{72}
$$

$$
\frac {a}{a _ {t}} = \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {\frac {1}{2}} [ \mathrm{adiab,perf} ]\tag{73}
$$

$$
\begin{array}{r l} \frac {q}{p} & = \frac {\gamma}{\gamma - 1} \left(\frac {V}{V _ {m}}\right) ^ {2} \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {- 1} \\ & = \frac {7}{2} \left(\frac {V}{V _ {m}}\right) ^ {2} \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {- 1} [ \text { adiab,   perf } ] \end{array}\tag{74}
$$

$$
\frac {q}{p _ {t}} = \frac {\gamma}{\gamma - 1} \left(\frac {V}{V _ {m}}\right) ^ {2} \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {\frac {1}{\gamma - 1}}
$$

$$
= \frac {7}{2} \left(\frac {V}{V _ {m}}\right) ^ {2} \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {\frac {5}{2}}\tag{75}
$$

$$
M ^ {2} = \frac {2}{\gamma + 1} \left(\frac {V}{V _ {m}}\right) ^ {2} \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {- 1}
$$

$$
= \frac {5}{6} \left(\frac {V}{V _ {m}}\right) ^ {2} \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {- 1} [ \text { adiab,   perf } ]\tag{76}
$$

$$
\left(\frac {V}{a _ {t}}\right) ^ {2} = \frac {2}{\gamma - 1} \left(\frac {V}{V _ {m}}\right) ^ {2} = 5 \left(\frac {V}{V _ {m}}\right) ^ {2} [ \mathrm{adiab,perf} ]\tag{77}
$$

$$
\left(\frac {V}{a _ {*}}\right) ^ {2} = \frac {\gamma + 1}{\gamma - 1} \left(\frac {V}{V _ {m}}\right) ^ {2} = 6 \left(\frac {V}{V _ {m}}\right) ^ {2} [ \mathrm{adiab,perf} ]\tag{78}
$$

Tables I and II list numerical values of the following ratios with Mach number M as the independent variable:

$$
\frac {p}{p _ {t}}, \frac {\rho}{\rho_ {t}}, \frac {T}{T _ {t}}, \frac {q}{p _ {t}}, \frac {V}{a _ {*}}
$$

## STREAM-TUBE-AREA RELATIONS

If it is assumed that the density and speed are uniform across any section of a given stream tube, then the equation of continuity is

$$
\rho V A = \text { constant } = \rho_ {*} a _ {*} A _ {*}\tag{79}
$$

By combining this and certain of the foregoing equations, the area ratio $A_{*}/A$ can be expressed as a function of any one of the four parameters used above. The final equations are

$$
\begin{array}{r l} \frac {A _ {*}}{A} = & \left(\frac {\gamma + 1}{2}\right) ^ {\frac {\gamma + 1}{2 (\gamma - 1)}} M \left(1 + \frac {\gamma - 1}{2} M ^ {2}\right) ^ {- \frac {\gamma + 1}{2 (\gamma - 1)}} \\ & = \frac {2 1 6}{1 2 5} M \left(1 + \frac {M ^ {2}}{5}\right) ^ {- 3} [ \text { isen,   perf } ] \end{array}\tag{80}
$$

$$
\frac {A _ {*}}{A} = \left(\frac {\gamma + 1}{2}\right) ^ {\frac {1}{\gamma - 1}} \left(\frac {V}{a _ {*}}\right) \left[ 1 - \frac {\gamma - 1}{\gamma + 1} \left(\frac {V}{a _ {*}}\right) ^ {2} \right] ^ {\frac {1}{\gamma - 1}}
$$

$$
= \left(\frac {6}{5}\right) ^ {\frac {5}{2}} \left(\frac {V}{a _ {*}}\right) \left[ 1 - \frac {1}{6} \left(\frac {V}{a _ {*}}\right) ^ {2} \right] ^ {\frac {5}{2}} [ \text { isen,   perf } ]\tag{81}
$$

$$
\begin{array}{r l} \frac {A _ {*}}{A} & = \left(\frac {\gamma + 1}{2}\right) ^ {\frac {\gamma + 1}{2 (\gamma - 1)}} \left(\frac {V}{a _ {t}}\right) \left[ 1 - \frac {\gamma - 1}{2} \left(\frac {V}{a _ {t}}\right) ^ {2} \right] ^ {\frac {1}{\gamma - 1}} \\ & = \frac {2 1 6}{1 2 5} \left(\frac {V}{a _ {t}}\right) \left[ 1 - \frac {1}{5} \left(\frac {V}{a _ {t}}\right) ^ {2} \right] ^ {\frac {5}{2}} [ \text {isen,perf} ] \end{array}\tag{82}
$$

$$
\begin{array}{r l} \frac {A _ {*}}{A} & = \left(\frac {2}{\gamma - 1}\right) ^ {\frac {1}{2}} \left(\frac {\gamma + 1}{2}\right) ^ {\frac {\gamma + 1}{2 (\gamma - 1)}} \left(\frac {V}{V _ {m}}\right) \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {\frac {1}{\gamma - 1}} \\ & = 5 ^ {\frac {1}{2}} \left(\frac {2 1 6}{1 2 5}\right) \left(\frac {V}{V _ {m}}\right) \left[ 1 - \left(\frac {V}{V _ {m}}\right) ^ {2} \right] ^ {\frac {5}{2}} [ \text { isen,   perf } ] \end{array}\tag{83}
$$

Numerical values of $A_{*}/A$ as a function of M are given in tables I and II.

Equation (79) combined with equations (26), (29b), (45), and (46) can be employed to obtain the mass-flow rate per unit area $\rho V$ along a stream tube as a function of Mach number, total temperature, and total pressure. Numerical values can be obtained conveniently from chart 1 where the variation with Mach number of the mass-flow rate per unit cross-sectional area is presented for various total temperatures and a total pressure of 1 pound per square inch absolute.

## SHOCK WAVES

## NORMAL SHOCK WAVES

## BASIC EQUATIONS

The previous relations for isentropic flow are valid on either side of a shock wave, but not across it, because at the shock wave the flow quantities have discontinuities. Jump

$$
\begin{array}{c c c} p _ {1} & p _ {1} \\ T _ {1} & o _ {1} \\ s _ {1} \end{array} \xrightarrow [ M _ {1} ]{u _ {1}} \xrightarrow [ M _ {2} ]{u _ {2}} \begin{array}{c c c} p _ {2} & p _ {2} \\ T _ {2} & o _ {2} \\ s _ {2} \end{array}
$$

FIGURE 1.—Notation for normal shock wave.

conditions for a steady normal shock wave (fig. 1) result from requiring conservation of

mass:

$$
\rho_ {1} u _ {1} = \rho_ {2} u _ {2}\tag{84}
$$

momentum:

$$
p _ {1} + \rho_ {1} u _ {1} ^ {2} = p _ {2} + \rho_ {2} u _ {2} ^ {2}\tag{85}
$$

$$
\text { energy: } ^ {3} \quad \frac {1}{2} u _ {1} ^ {2} + h _ {1} = \frac {1}{2} u _ {2} ^ {2} + h _ {2} \quad [ \mathrm{adiab} ]\tag{86a}
$$