Are mach cone and vapor cone the same or not? I have read numerous articles, claiming that Vapor cone

IS NOT the mark that aircraft is passing "sound barrier", or, in other words it is not the visualization of Mach wave

But I can't neither understand nor accept such explanations.
So, if it is not the Mach wave, then what is it?
Particularly, what is the speed of this front?


Let's forget about a plane, hide it and rotate the picture. We will see some air parmaeters disturbance front, propagating through air and producing the vapor.

Prove that it is not the mach wave.
 A: It has nothing directly to do with the speed of sound. It is about pressure and dew point. Certain aircraft designs cause the localized air pressure to drop below the relative dew point. When that happens vapor will appear. As the part of the plane causing the low pressure area moves through the air the vapor mixes with air that is above the dew point and the vapor disappears. It may be below the speed of sound  or above the speed of sound.
A: They are not the same
The cone only becomes visible during transonic flight. It is important to note that the vehicle is not traveling at supersonic speed when condensation collars (rockets) or condensation cones (aircraft) occur. It is merely close to the speed of sound. Once the aircraft is supersonic, the condensation cone disappears.
And the speed vector in your picture is misleading - disturbances travel in the direction of the airplane.
From the vehicle's viewpoint
Air is accelerated by the displacement effect of the vehicle - it pushes the air aside and around itself, causing the flow to accelerate and to become supersonic relative to the vehicle. This acceleration is directly connected with lower pressure (see this answer for a detailed explanation), and air with high relative humidity will be pushed beyond its dew point, causing the water vapor to condensate. Please note that the onset of condensation is slightly delayed, so the fog is not strictly indicating the region of lowest pressure. Also, the forward cone is not the same as the forward shock front of a lambda shock.
However, the rear boundary of the condensation is indeed caused by a shockwave. The low pressure area is instantly ended by a compression shock, when the supersonic flow around the aircraft is abruptly decelerated to subsonic speed by a compression shock. This raises the local air pressure back to around ambient pressure, so the capacity to absorb water is raised again and the fog evaporates. Again, this happens with a slight delay, so the end of the condensation area does not coincide with the compression shock but follows it.
From the viewpoint of the air
Since the low pressure area is caused by the aircraft, it travels with the aircraft, and so does the condensation effect. The air around the aircraft or rocket is accelerated towards the vehicle by its pressure field, which causes the condensation. Now the air is so fast relative to the vehicle that it is decelerated and recompressed by a straight shock, causing the condensation to disappear again.
Factors
A relatively blunt vehicle traveling at close to Mach 1 and humid, almost saturated air are the factors which help this effect to occur. Note that you can witness similar condensation effects over the wings of airliners in humid air during the landing phase, when flaps are fully deployed. Here, however, the condensation does not end abruptly, since no shock is involved. Also, the disturbances produced by the aircraft do not reach out so far at lower speed, so the condensation happens only close to the aircraft's surface. Bluntness helps: The cockpit causes its own small condensation area shortly past its maximum thickness, where suction due to displacement is highest.
