Normal shock tables

In aerodynamics, the normal shock tables are a series of tabulated data listing the various properties before and after the occurrence of a normal shock wave.^[1] With a given upstream Mach number, the post-shock Mach number can be calculated along with the pressure, density, temperature, and stagnation pressure ratios. Such tables are useful since the equations used to calculate the properties after a normal shock are cumbersome.

The tables below have been calculated using a heat capacity ratio, ${\displaystyle \gamma }$, equal to 1.4. The upstream Mach number, ${\displaystyle M_{1}}$, begins at 1 and ends at 5. Although the tables could be extended over any range of Mach numbers, stopping at Mach 5 is typical since assuming ${\displaystyle \gamma }$ to be 1.4 over the entire Mach number range leads to errors over 10% beyond Mach 5.

Given an upstream Mach number, ${\displaystyle M_{1}}$, and the ratio of specific heats, ${\displaystyle \gamma }$, the post normal shock Mach number, ${\displaystyle M_{2}}$, can be calculated using the equation below.

${\displaystyle M_{2}={\sqrt {\frac {M_{1}^{2}\left(\gamma -1\right)+2}{2\gamma M_{1}^{2}-\left(\gamma -1\right)}}}}$

The next equation shows the relationship between the post normal shock pressure, ${\displaystyle p_{2}}$, and the upstream ambient pressure, ${\displaystyle p_{1}}$.

${\displaystyle {\frac {p_{2}}{p_{1}}}={\frac {2\gamma M_{1}^{2}}{\gamma +1}}-{\frac {\gamma -1}{\gamma +1}}}$

The relationship between the post normal shock density, ${\displaystyle \rho _{2}}$, and the upstream ambient density, ${\displaystyle \rho _{1}}$ is shown next in the tables.

${\displaystyle {\frac {\rho _{2}}{\rho _{1}}}={\frac {\left(\gamma +1\right)M_{1}^{2}}{\left(\gamma -1\right)M_{1}^{2}+2}}}$

Next, the equation below shows the relationship between the post normal shock temperature, ${\displaystyle T_{2}}$, and the upstream ambient temperature, ${\displaystyle T_{1}}$.

${\displaystyle {\frac {T_{2}}{T_{1}}}={\frac {\left(1+{\frac {\gamma -1}{2}}M_{1}^{2}\right)\left({\frac {2\gamma }{\gamma -1}}M_{1}^{2}-1\right)}{M_{1}^{2}\left({\frac {2\gamma }{\gamma -1}}+{\frac {\gamma -1}{2}}\right)}}}$

Finally, the ratio of stagnation pressures is shown below where ${\displaystyle p_{01}}$ is the upstream stagnation pressure and ${\displaystyle p_{02}}$ occurs after the normal shock. The ratio of stagnation temperatures remains constant across a normal shock since the process is adiabatic.

${\displaystyle {\frac {p_{02}}{p_{01}}}=\left({\frac {{\frac {\gamma +1}{2}}M_{1}^{2}}{1+{\frac {\gamma -1}{2}}M_{1}^{2}}}\right)^{\frac {\gamma }{\gamma -1}}\left({\frac {1}{{\frac {2\gamma }{\gamma +1}}M_{1}^{2}-{\frac {\gamma -1}{\gamma +1}}}}\right)^{\frac {1}{\gamma -1}}}$

Note that before and after the shock the isentropic relations are valid and connect static and total quantities. That means, ${\displaystyle p_{total}\neq p_{static}+p_{dynamic}}$ (comes from Bernoulli, assumes incompressible flow) because the flow is for Mach numbers greater than unity always compressible.

• Normal shock
• Mach number
• Compressible flow

• University of Cincinnati shock relations calculator
• Parkin Research Normal shock calculator