This page supports the multimedia chapter The Doppler Effect in the volume Waves and Sound. It gives background information and further details.
This page has details about the shock waves caused by a source travelling faster than the wave speed, Mach numbers, the sonic boom, and cloud nucleation.
The animation below shows a source of spherical wavefronts, each centred on the position of the source when the wavefront was emitted. In this case the speed of the source vs exceeds the speed v of the wave (vs > v). You might like to compare this with a similar animation for the case where vs < v in the support page about the Doppler effect.
Here, because vs > v, we see that the rightmost point of each circle lies to the right of (outside of) the previous wavefront. This is because the source has travelled vsT in time T, compared with only vT for the wavefront. Consequently, the wavefronts quickly intersect. In two dimensions, the common tangent to all the circles is two lines, along which the wavefronts all add up to give a locus of high pressure amplitude: a shock wave.
In three dimensions (where wavefronts are spherical), the shock wave is a cone, as suggested here by the shading. Our diagram shows only one frequency and wavelength. However, if the wave were created by a supersonic airplane, all frequencies and wavelengths would be created and the cone would be continuous.
The last frames of the animation show a right angle triangle with sides vT and vsT, from which can see that the half-angle θ of the cone is given by
sin θ =
v/vs
The ratio vs/v is one of several dimensionless parameters used in the study of fluids: it is called the Mach number. A plane flying at twice the speed of sound is said to be travelling at Mach 2.
Sonic boom and sound barrier
For a plane in horizontal flight, the intersection of this cone with the ground is a parabola. When this line of shock wave passes by, listeners on the ground hear a very loud noise, called a sonic boom. The shock may be great enough to shatter windows or to cause objects to fall to the ground, as suggested in the cartoon below. The shock wave also imposes stresses on the airplane. Sometimes the stresses on the plane as it approaches or passes the speed of sound are called the sound barrier.
The shock wave can produce interesting effects, one of which is shown in the classic photo (US Navy photo taken by John Gay; Wikimedia link).
It seems likely that, at the moment that this photo was taken, some parts of the flow near the plane had just reached the speed of sound*. Clear air is often supersaturated: its humidy can exceed 100%. However, in the absence of objects to nucleate the formation of water (or ice), condensation doesn't occur, and the air remains in an unstable, nonequilibrium state. Unless there is a disturbance. In this case, the sudden over- and under-pressure of the shockwave, which abruptly heat and cool the air respectively, appear to have nucleated water droplets to form a cloud.
* This is not quite the same as the plane reaching the speed of sound with respect to the distant air, because of inhomogeneities in the flow. In some subsonic planes, parts of the mechanism, such as the outer edges of fan blades in the engines, may be travelling at speeds greater than sound.
We should expect that, because the shock wave is conical when vs > v, a plane travelling well above the speed of sound would create a conical cloud in sufficiently supersaturated air. The length of the cone would be determined by the amplitude of the shock wave, with the cloud ending when the amplitude fell below some threshold. If the speed of the plane is only slightly supersonic, θ is only a little less than 90° then the cone is almost flat.
The condensation trails ('con trails') left by commercial airliners are not due to supersonic shock waves. Rather, they are the result of the engine exhaust. The jet fuel burns to give water and CO2, and the water can take the air well over saturation point. At high altitude, the air may be cold enough that the cloud is made from water rather than ice.
