# 12.2.2.4: Given Two Angles, \(\delta\) and \(\theta\)

It is sometimes useful to obtain a relationship where the two angles are known. The first upstream Mach number, \(M_1\) is

Mach Number Angles Relationship

\[

\label{2Dgd:eq:OM1}

{M_1}^2 = \dfrac{ 2 \,( \cot \theta + \tan \delta ) }

{ \sin 2 \theta - (\tan \delta)\, ( k + \cos 2 \theta) } \tag{59}

\]

The reduced pressure difference is

\[

\dfrac{2\,(P_2 - P_1) }{ \rho\, U^2} =

\dfrac{2 \,\sin\theta \,\sin \delta }{ \cos(\theta - \delta)}

\label{2Dgd:eq:OreducedPressure} \tag{60}

\]

The reduced density is

\[

\dfrac{\rho_ 2 -\rho_1 }{ \rho_2} =

\dfrac{\sin \delta }{ \sin \theta\, \cos (\theta -\delta)}

\label{2Dgd:eq:OreducedDensity} \tag{61}

\]

For a large upstream Mach number \(M_1\) and a small shock angle (yet not approaching zero), \(\theta\), the deflection angle, \(\delta\) must also be small as well. Equation (51) can be simplified into

\[

\theta \cong {k +1 \over 2} \delta

\label{2Dgd:eq:OlargeM1theta} \tag{62}

\]

The results are consistent with the initial assumption which shows that it was an appropriate assumption.

*Fig. 12.9 Color-schlieren image of a two dimensional flow over a wedge. The total deflection angel (two sides) is \(20^\circ\) and upper and lower Mach angel are \(\sim 28^\circ\) and \(\sim 30^\circ\), respectively. The image show the end–effects as it has thick (not sharp transition) compare to shock over a cone. The image was taken by Dr.~Gary Settles at Gas Dynamics laboratory, Penn State * University.

### 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.