The Doppler effect

When we derived the wavelength-frequency relationship, v = f λ , we assumed that the observer and source were standing still. Suppose the observer is fixed and the source is moving towards the observer at a speed u and emitting a sound of frequency f so that the time elapsed between two successive crests is 1/f. As shown on the graph below, the distance travelled by the first crest in this time will be v/f.


In the meantime the source will have travelled a distance u/f. Therefore, the distance between crests, also called the wavelength, is v/f – u/f. Thus the perceived frequency (fd) is related to the actual frequency f of the source and the relative speed of the source u by:

fd = fv/(v-u)

This shift in frequency of waves which results from sources or observers moving with respect to the medium is called the Doppler effect. It was first explained by Christian Doppler in 1842. This effect also holds true for light waves. In this case, a change in frequency is perceived as a change in colour. An example is given by the red shift of light from remote galaxies moving away from our galaxy. As for acoustic waves, the Doppler effect can be heardwhen listening to the gradual change of pitch in an ambulance siren as it passes by.

Subsonic source : u < v

An illustration of the Doppler effect is given by the animation below. It shows the successive wavefronts emitted by a source moving at a speed lower than the wavefront speed (or speed of sound). It can be seen that, in front of the source, the wavefronts start to bunch up whereas they stretch further apart behind the source. As a result, a listener located in front of the source will perceive a higher frequency than an observer located behind the source.


Transonic source : u = v

If the source moves at exactly the speed of sound (that is about 340 m/s in air under room conditions, but lower at typical flight altitudes), it can be seen that the wavefronts in front of the source are now all bunched up at the same point, resulting in a shock wave. As a result, an observer in front of the source will detect nothing until the source arrives. The energy in the sound waves cannot get away from the source, and this is why aeroplanes undergo extreme forces as they pass through the ‘sound barrier’.


Supersonic source : u > v

If the source is now travelling faster than the speed of sound, the sound source may pass by a fixed observer before the observer actually hears the sound it creates. Then, as the shock wave passes by, the observer will perceive a “bang”-like sound. Such an intense pressure front can be created by supersonic jet aircrafts flying at Mach 1. As observed in the animation below, it is the source that is now leading the wavefront.


Jet pilots flying at Mach 1 report that there is a noticeable ‘wall’ or ‘barrier’ which must be penetrated before achieving supersonic speeds. This is due to the intense pressure front, and flying within this pressure front produces a very turbulent and bouncy ride. The figure below shows a jet fighter at the exact instant it goes supersonic. Notice the smaller shockwave behind the pilot’s cockpit.


(U. S. Navy photo by Ensign John Gay)

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