This week NASA released two new images of the shock waves surrounding T-38C jets in free flight. They’re the result of NASA’s new adaptations of the schlieren photography technique, which has let scientists visualize shock waves (in the lab, at least) for more than a century. To celebrate, I thought it would be fun to demonstrate some of the data engineers can extract from images like the one above. So I’m going to show you how to calculate how fast this plane was flying!

Shock waves depend a lot on geometry. This is not too surprising, really, since shock waves are nature’s way of quickly turning the air because there’s an object in the way. This leads to a very powerful observation, though: the angle of a shock wave depends on the geometry of the object and the Mach number of the flow. (The Mach number is the ratio of an object’s speed to the local speed of sound, so an object moving at Mach 1 is moving at the speed of sound.)

The reverse observation is also true: if we can measure the angle of a shock wave from a known geometry, then we can calculate the Mach number. Now, I don’t have any special information about the geometry of a T-38, so most of the shock waves in this picture can’t tell me much quantitatively.

But, it turns out, I don’t need to know anything about the geometry of the plane to figure out its Mach number. That’s because that very first shock wave over on the right is coming off a sharp probe mounted over the airplane’s nose. The probe is sharp enough, in fact, that I can treat it as though it’s a tiny point disturbance. That means that rightmost shock wave is a special kind of shock known as a Mach wave, and its geometry depends ** solely **on the Mach number. It’s a pretty simple equation, too:

So, all I have to do is fire up some software like GIMP or ImageJ and estimate the angle of that first shock wave.

I came up with an estimate of about **77 degrees** for the shock wave angle, which gives **Mach 1.026** for the plane’s speed. Keep in mind that a) I’m using a grainy photo; and b) I have no information about the plane’s orientation relative to the camera. Nevertheless, NASA’s caption reports that this plane was moving at Mach 1.05 in the picture. My quick and dirty estimate is only off by 2%!

Of course, engineers are interested in a lot more than estimating an aircraft’s speed from these photos. With a little more geometry information, they can gather a lot of useful data from these images. One of the goals for the new photography technique is to help study new aircraft designs that generate weaker shock waves and quieter sonic booms. (Original images: NASA)