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I've been thinking about this question for a while and I didn't get to any conclusion yet.

When an aircraft flies faster than the speed of sound (specifically when it achieves it) a cloud of condensed water becomes visible. Like in the figure below:

enter image description here

I know that as the speed of the object increases to the sonic velocity (the local velocity of sound waves), these sound waves begin to pile up in front of the object, so the localised air pressure and air temperature around the object drops. If the temperature drops below the saturation temperature, a cloud forms.

Considering that after the cloud formation the aircraft still accelerates, will this effect happen again as the aircraft achieves a multiple of the sonic velocity? For instance the aircraft achieves the velocities $2v_s$, ..., $n v_s$, where $v_s$ is the sonic velocity .

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The Prandtl–Glauert effect vapour-cone will be there for all sufficiently fast (transonic) velocities. After a compression-shock bow wave, the moist air expands suddenly to below ambient pressure and cools below the dewpoint. The aircraft speed does not have to be supersonic. The cloud appears a bit below supersonic and, as does the compression shock, persists for all higher velocities.

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