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Stopping the video of the recent test of the raptor rocket engine, one can see a standing wave forming in the exhaust gases: https://youtu.be/MAAzbjG_Duc?t=23

enter image description here

I was wondering what might cause a standing wave pattern to appear there (if it isn't something completely different).

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They are called shock diamonds.

They are caused by the surrounding air pressure being slightly more than "perfect" for the current operating conditions. As the air expands in the nozzle, it exits it to find itself constrained by the airflow around it. This forms a sort of "barrier".

The lines you see are shock waves in the fuel. Since the fuel is supersonic it will form shock waves. This bounce off the barrier and the air expands behind it. This process repeats several times, until the exhaust reaches ambient conditions.

Note that a "perfect" nozzle will not (???) form these patterns. You do not see them in aerospike engines, for instance:

enter image description here

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