The Physics Philes, lesson 115: The Doppler Shift

We’ve been talking, over the last several weeks, about sound and sound waves and how those waves interact with each other. Implicit in those discussions is the assumption that everyone is at rest relative to one another. That’s not always how it works.

We know, of course, that we can hear things eve if they are moving. It would be a pretty weird world if we couldn’t. But that doesn’t mean that that motion doesn’t have an effect on what we hear. The classic example of this is an ambulance siren. If you are standing still and an ambulance rushes past, you’ll notice that the siren sounds different as it approaches, passes, and moves away from you.

So…what gives?

It’s called the Doppler effect, first described by 19th century Austrian physicist Christian Doppler. In general terms, when the source of the sound and/or the listener are moving relative to each other, the frequency of the sound the listener hears is not the same as the frequency of the sound at the source. It has to do with the relationship between the velocities of the source and the listener and the amount the frequency shifts. The frequency shifts will look a little different depending on if the listener is moving relative to the source or if both the listener and the source are moving relative to the other.

First let’s think about a listener moving towards the stationary source of sound. The sound wave is moving toward the listener at a certain velocity (relative to the medium) and the wavelength can be described as the velocity of the wave over the frequency at the source. The listener is traveling toward the waves with a certain speed. That means the speed of the wave relative to the listener is actually the speed of sound relative to the medium added to the velocity of the listener. So, if the frequency relationship is velocity of the wave divided by wavelength, we can find the frequency of the sound heard by the listener by adding the velocity of the listener to the wave speed:

Screen shot 2014-09-14 at 10.48.55 AM

Where fL is the frequency heard by the listener, fS is the frequency of the wave at the source, vL is the velocity of the listener, and v is the velocity of the wave.

From this equation we can see that when the listener is moving toward the source (that is, when vL > 0), the moving listener is going to hear a higher pitch than a listener at the source. If the listener is moving away, that pitch is lower. This tracks pretty well with our ambulance example. As the ambulance gets closer to a listener standing on the sidewalk, the sound is higher. As it moves away, the sound the siren makes is lower.

The situation gets slightly more tricky when both the source and the listener are moving relative to each other. The wave speed in the medium is still the same as it always was (that is determined by the medium, not the motion of the source). However, if the source is moving, the frequency is no longer represented by the velocity over the frequency at the source. Let me explain.

We know that the period of a wave is the inverse of the frequency, in this case, the frequency at the source. So the time it takes for the wave to travel a distance is the velocity times the period, or the velocity divided by the frequency at the source. If the source is moving, it moves a distance equal to the velocity of the source divided by the frequency of the source. The wavelength, then is determined by the relative displacement of the source and wave. This is going to be different in the front of the wave than it is behind it. To find the wavelength in front of the source, we need to subtract the distances, like this:

Screen shot 2014-09-14 at 11.11.17 AM

The wavelength behind the moving source is found by adding the distances:

Screen shot 2014-09-14 at 11.13.06 AM

The waves are either compressed or stretched, depending on the motion of the source. And to find the frequency of the sound heard by the moving listener who is not at the source, we add the velocities of the wave and divide it by the wavelength of the waves behind the source, or:

Screen shot 2014-09-14 at 11.18.29 AM

We can use this equation in any situation in which the source and listener are moving.

So far we’ve been talking about the Doppler shift of waves through a medium, particularly sound waves. However, waves that don’t need a medium – electromagnetic waves – are also subject to Doppler shift. However, we can’t just take the equations we’ve derived here and use them to find the speed of light relative to a moving observer, for example. You see, the Doppler shift we’ve been discussing has its roots in classical mechanics. To find the frequency of a wave going an appreciable fraction of the speed of light, we need to use special relativity. We’ll delve further into that next time.

Featured image credit: Till Krech via Flickr

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Mindy is an attorney and Managing Editor of Teen Skepchick. She hates the law and loves stars. You can follow her on Twitter and on Google+.

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