Configuration to be examined

The configuration we look at for a normal shock is shown below:

A gas enters a shock (the shock is shown in grey) from the left with a speed u₁, and a thermodynamic state given through a pressure p₁ and density . It emerges from the shock with a different velocity u₂ and in a different thermodynamic state p₂, .

Before starting to grind math, let's note that our results are not just meaningful for shocks at rest in a duct flow. If we simply add a velocity to the picture (i.e., look at the shock in a moving coordinate frame,) we get the picture of a moving shock. For example, below I have added a velocity v_s=-u₁ to the picture to get a shock moving with speed v_s through a fluid that is at rest in front of the shock:

And if you look at a much more general shock such as, for example, the bow shock in front of the space shuttle nose shown below, our results will directly apply to the piece of normal shock that I drew a rectangle around:

In fact it will apply better, since there are no duct walls surrounding the shock element to create boundary layers to mess up things.

Even more interestingly, if we view our simple shock in a moving coordinate frame that moves vertically, we will see a shock under an angle with the incoming flow, just like the piece of shock I drew a parallelogram around in the picture above:

The bottom line is that although the configuration we look at is very simple, the results are very powerful and give the behavior of any thin shock, moving and/or oblique, when the gas has a simple thermodynamic state before and behind the shock.