What would the density profile of this shock wave look like, roughly?

The YouTube video titled Shock Wave Formation in Transonic Flight shows a hand-held video of a visible shock wave extending from the top of a jet engine near the front. The shock wave is visible by its refraction of features on the ground (muddy river bed) cause by the large changes in air density over short distances.

I would like to simulate the optics of this strictly for fun and general interest. If one made a simple plot of the density as a function of position crossing the shock wave, roughly what shape would have?

I've plotted six simple examples of shapes, perhaps one is appropriate, but perhaps something else is better.

below: GIF made from this video around 00:31.

below: some example guesses of the density behavior across the part of the shock wave seen to refract ground features in the video.

below: Description from this video.

Published on Oct 18, 2012

Formation of a shock wave around the engine cowl of a Boeing 767 at high speed.

The aircraft itself isn't going supersonic, but it's going fast enough (probably around Mach 0.8) that the local airflow around the engine cowl is supersonic, in a small area. This is called "transonic," and results in the formation of a shock wave at a right angle to the surface of the plane. The air in front of the shock is supersonic, and the air behind it is subsonic.

The shock wave is visible because of the abrupt change in the density of the air. The change in density causes a change in the refractive index of the air, and so the scenery behind it is optically distorted. This is similar to how a spoon in a glass of water looks "bent" at the water's surface, due to the difference between the refractive indices of water and air. It helps that this is seen nearly edge-on... if I had been a couple of rows further forward or back, it might not have been visible.

• Not an answer, but this document shows an abrupt transition.
– mins
Commented Jul 9, 2017 at 12:47
• @mins very nice! So pretty much something like either the first or the third curve in my example guesses. As the video caption explains, it's like a simple transition from one index of refraction to another. I've just got to generate it in 3D (make a cone around the engine axis) and see what happens. I can always add a transition thickness of a millimeter or centimeter and see if that changes the appearance.Thanks!
– uhoh
Commented Jul 9, 2017 at 12:48
• Another pressure distribution graph (middle) at the end of this article.
– mins
Commented Jul 9, 2017 at 13:05