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Apparently this phenomenon has nothing to do with jets breaking the sound barrier and has something to do with the Prandtl-Glauert singularity as described on Wikipedia. But, the Wikipedia article isn't very detailed and it doesn't explain why the cone arises.
Is there a reason why a "cone", of all the possible shapes, forms around the jets?
It forms a cone because it depends on a shock wave, and the region enclosed by the shock wave appears conical in shape.
See, for example, the apparent cones here:
They are also visible here:
Wikipedia appears to be fairly clear on why vapor cones are related to shock waves. From the introduction to the article about vapor cones:
Atmospheric water then condenses, and thus becomes visible, as air pressure decreases suddenly across shock waves associated with supersonic flow speed.
It's also put nicely on the page for the Prantl-Glauert singularity:
The reason for the condensation cloud that is being observed is that humid air is entering a low-pressure region, which also reduces local density and temperature, sufficiently so to cause condensation. The vapour vanishes as soon as the pressure increases again to ambient levels.
So we simply have to figure out why these shock waves are conical. The answer to that is essentially that the region inside the cone is at a much different pressure than the region outside it. At any given point in space, the boundary cross-section expands uniformly in all directions (hence it is a circle). In the outside region, air molecules are at one pressure; in the inside region, the molecules are at another pressure. The shock wave is the boundary between these two regions. As it travels outward, the molecules it passes through become the same pressure as the other molecules inside the cone.
The cloud disc that slides along an aircraft’s fuselage in acceleration across Mach I might be explained by the physics of an ultrasound field generated by the Doppler Effect.
Sound, mechanical and aerodynamic, generated by an aircraft accelerating towards Mach I, is confined within a sound cone, the apex of which is at a point at a decreasing distance in front of the nose. At Mach I, the aircraft nose is the cone’s apex. Aircraft-generated sound energy in the sound cone travels at the speed of sound – laterally, perpendicular to the line of flight, and anteriorly where it builds up in intensity in front of the leading edges.
As air speed increases, the sound energy in the anterior sound cone displays rising sound frequency with the wavelength decreasing proportionately (Doppler Effect). Reducing the aircraft to point size, the aerodynamic and aircraft mechanical sounds are confined within a perfect cone. At Mach I, the angle of the cone margins to the line of flight would be 45 degrees (the sound radiating laterally from the line of flight the same distance as the plane moves forwards), creating a 90-degree angle between the cone margins.
Cloud disc formation requires moisture as ice-crystal mist, water mist or rain drops (ultrasound “atomizes” water). Doppler-induced ultrasound propels (sweeps) suspended particles (including droplets) forwards away from the ultrasound source (Liebermann L.N. 1949. The second viscosity of liquids. Phys. Rev. 75, 1415-1422), which is the aircraft; this creates a disc-shaped cloud, its rounded margins defined by the shape of the cross-section of the sound cone. At Mach I, the cloud disc touches the aircraft nose. While accelerating beyond Mach I, , the cloud disc slides along the fuselage (Figure 1), leaving aircraft sound behind.
Condensation of atmospheric water vapour due to the shock wave. Air molecules can travel with the limit of speeed of sound (343 m/s), hence molecules hitting the body of an aircraft travelling at sppeds lower than that move away. Once the aircraft crosses the speed of sound, the molecules are pushed with a speed faster than they can travel. This causes a shock wave on the air around the aircraft. Under certain circumstances of the dew point and relative humidity, the shock wave causes the water vapour molecules to condence and form the vapour cone.
We have to understand there are two phenomena occurring. One is what creates the vapor. The other is what shapes the vapor into a cone. The vapor is created in this case by the localized pressure dropping below the dew point in the air immediately around the structure moving through it. The cone is formed by the high and low pressure waves propagating from the structure as it moves through air.
