# 11.4.3.2: Relationship Between the Mach Number and Cross Section Area

The equations used in the solution are energy (47), second law (??), state (37), mass (34). Note, equation (41) isn't the solution but demonstration of certain properties of the pressure profile. The relationship between temperature and the cross section area can be obtained by utilizing the relationship between the pressure and temperature (45) and the relationship of pressure with cross section area (41). First stage equation (47) is combined with equation (45) and becomes

\[

{(k - 1) \over k}\, { dP \over P } = - { (k -1) \, M \,dM

\over 1 + \dfrac{ k - 1 }{ 2 } M^{2} }

\label{gd:iso:eq:press-M} \tag{48}

\]

Combining equation (48) with equation (??) yields

\[

{1 \over k} {\dfrac {\rho \,U^{2}} { A} \, \dfrac{dA }{ 1 - M^2} \over P }

= - \dfrac{ M \, dM }{ 1 + \dfrac{ k - 1 }{ 2 } M^{2} }

\label{gd:iso:eq:M-A-0} \tag{49}

\]

The following identify, \( \rho \, U ^{2} = k\,M\, P\) can be proved as

\[

k\, M^2\, P =

k \overbrace{ U^{2} \over c^2}^{M^2} \overbrace{\rho R T}^{P} =

k { U ^{2} \over k\,R\,T} \overbrace{\rho\, R \, T}^{P}

= \rho U ^{2}

\label{gd:iso:eq:rU-kMP} \tag{50}

\]

Using the identity in equation (50) changes equation (49) into

\[

\dfrac{dA }{ A} = { M^2 -1 \over

M \left( 1 + \dfrac{k-1 }{ 2} M^2

\right)} dM

\label{gd:iso:eq:M-A} \tag{51}

\]

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.