Have you ever seen an internet video wherein a cone of condensation appears suddenly around the fuselage of a jet breaking the sound barrier, usually a fighter jet? Or witnessed the phenomenon yourself from land or out at sea from a naval ship flyby? This striking vapor cone is caused by a phenomenon referred to as Prandtl-Meyer expansion fans, named after their discoverers, the German engineer Ludwig Prandtl and his student, the mathematician Theodor Meyer.
An F/A-18 Hornet maneuvers in the transonic regime. Note the prominent supersonic expansion fan, technically known as the Prandtl–Meyer expansion fan, best visible in more humid weather
Prandtl-Meyer expansion fans form when an object (in our case a supersonic capable jet) abruptly exceeds the speed of sound. A low pressure region is formed in the vicinity of the expansion fans, which decreases the surrounding air pressure, density, and temperature below the dew point. The result is a visible cone of water vapor.
Illustration of Prandtl-Meyer expansion fans. They are more visible in weather with a small temperature-dewpoint difference
First, a quick aside on the speed of sound. The simplest explanation is as follows: the speed of sound depends almost entirely on the temperature of the air through which the sound waves propagate. At sea level, at a temperature of about 59 degrees fahrenheit the speed of sound is 340.3 meters/second, or 761.2 MPH. Under normal meteorological conditions (absent temperature inversion) the speed of sound decreases as altitude increases, as temperature has an inverse relationship with altitude.
Prandtl-Meyer expansion fans form when abruptly induced supersonic flow bends around a convex corner (as in the illustration above). In this case, the apex formed between the jet's fuselage and the supersonic relative wind incident upon the jet. These fans consist of an infinite number of Mach waves (pressure waves propagating at or above the speed of sound). The physics underlying this phenomenon are rather complex, for more drill-down on the specifics please refer to this writeup by NASA.
A note about the Prandtl-Meyer fan in the preceding illustration: the arrow pointing towards the aircraft may at first seem counter intuitive, as the arrow is pointing towards the jet, not in the jet's direction of travel. Yes, the aircraft is exceeding the speed of sound, but the relative wind is also incident upon the jet at or above the speed of sound. Subsequently as the jet quickly passes the sound barrier, the Mach waves encounter a convex corner, resulting in the Prandt-Meyer fans.
More examples of vapor cones formed from different perspectives, and one from a rocket.
Real world examples of vapor cones formed as a result of Prandtl-Meyer expansion fans, both from the point of view of naval vessels:
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