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# Illustration 18.4: Doppler Effect

In this Illustration we consider what happens when the source of sound is moving either toward or away from a detector at rest (position is given in meters and time is given in milliseconds). At the same time we can consider what happens when the detector is moving toward a sound source at rest. Restart.

What we notice from everyday experience is that if the source of the sound is moving toward us, the frequency we hear increases. If it is moving away, the frequency we hear drops. If we are moving toward a source, the frequency we hear increases and if we are moving away the frequency we hear drops. The reason for the difference between when the observer is moving and the source is moving is due to how the detected sound waves change in each case.

Animation 1 depicts what happens when the source of the sound wave and the detector of the sound wave are both stationary. Notice that the wavelength of the sound wave is 1.7 m and its period is 0.5 ms, which corresponds to a frequency of 200 Hz.

When the observer is moving, as in Animation 2, the sound waves emitted from the source are undisturbed. The wavelength does not change as observed from the moving observer. He/she just comes across more/less wave fronts per time ([vt ± vDt]/λt) when moving toward/away from the source, and consequently sees a change in frequency.

For the case in which the source is moving, shown in Animation 3, the frequency (time in between wave fronts) and wavelength change. The wave fronts are emitted much closer together/farther apart (λ' = vT -/+ vST = [v -/+ vS]/f) as the source is moving toward/away from us. Animation 4 represents the sound wave of a source moving according to a linear restoring force (simple harmonic motion).

We may write both these cases together, with vS as the velocity of the source and vD as the velocity of the observer or detector, as

f ' = f [v ± vD]/[v -/+ vS].

Hence when the source is stationary and the observer/detector is moving f' = f [v ± vD]/v, and when the detector/observer is stationary and the source is moving f' = f v/[v -/+ vS]. Here the upper signs indicate a velocity towards and the lower signs represent a velocity away.

When the source is moving at the speed of sound, the emitted wave travels forward at the same speed as the source. The sound waves build up, causing a sonic boom as shown in Animation 5. For things like supersonic airplanes, we get a double boom-a boom from the front of the plane and a boom from the back. The resulting waves pile up, all on top of each other, and create at first a huge increase in pressure and then a huge decrease in pressure before the return to normal atmospheric pressure.

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