# National Aeronautics and Space Administration

## Doppler Shift

### Doppler Shift

Johann Christian Doppler
1803-1853

Have you noticed that when an emergency vehicle with its siren blaring passes you that the tone that you hear changes in pitch? This is an example of what is called the Doppler shift, and it is an effect that is associated with any wave phenomena (such as sound waves). It is the same effect that Hubble used to measure the velocities of distant galaxies. This is how it works:

Consider first a case where the fire truck is at rest in the station driveway waiting for the firemen to board the firetruck. The siren is already on, and a listener some distance away to the right will perceive the siren at the same frequency as which it is emitted. Note that another stationary person on the left side of the truck would hear the same tone also.

The stationary listener on the right hears the same 400 Hz tone emitted by the fire truck.

Now consider how this situation changes when the truck is moving towards the stationary observer with a constant velocity, v.

The frequency of the fire engine's siren as heard by a person on the firetruck has not changed! However, the waves in the direction of the truck bunch up as the fire truck is catching up to its own sound waves. This means that the pressure variations, which are represented by the sine waves, impinge upon the eardrum of the stationary observer at an increased frequency. The stationary observer to the right therefore perceives a higher tone. Notice also that the waves traveling towards the rear of the fire truck seem are spread out as the siren is moving away from its own sound as it travels with velocity v to the right. This would cause a stationary observer to the left of the truck to perceive a decrease in the frequency of the of the siren.

For a source moving to the right, a stationary observer to the right would perceive a higher tone and one to the left would perceive a lower tone.

 Play me a song about the Doppler Shift! (text) Listen

The non-relativistic Doppler shifted frequency of an object moving with speed v with respect to a stationary observer, is:

and the Doppler shifted wavelength can be shown to be:

In these two equations, co is the speed of the wave in a stationary medium (in this case the speed of sound), and the velocity is the radial component of the velocity (the part in a straight line from the observer). Both these formulas are non-relativistic approximations that are true as long as the velocity of the moving object is much less than the speed of light. As a convention, the velocity is positive if the source is moving away from us and negative if the source is moving towards the observer. Thus:

• if the source is moving away (positive velocity) the observed frequency is lower and the observed wavelength is greater (redshifted).
• if the source is moving towards (negative velocity) the observed frequency is higher and the wavelength is shorter (blueshifted).

Without much additional thought, as I know you have been thinking for a while now, you could easily convince yourself that the doppler shift will occur under any of the following circumstances:

• The source is approaching a stationary observer.
• The observer is approaching a stationary source.
• The source and the observer are moving towards one another.
• The source is moving away from a stationary observer.
• The observer is moving away from a stationary source.
• The source and the observer are both moving away from each other.
• The source and the observer are moving in the same direction at different speeds.

You should also be able to easily convince yourself that the shift will yield an increase in the perceived frequency whenever the source and the observer are approaching one another, and a decrease in the perceived frequency whenever the source and the observer are moving away from each other.

So....here's the big question. How does this affect the spectra of distant objects in the Universe, you might ask? Does light doppler shift? You probably remember the spectrum of visible light as ROYGBIV. So if the doppler shift works also for light then it must be possible to move so quickly towards a red traffic light that it would appear green to you! You might find it clever to use this argument if you get stopped for running a red light. However, the police officer might in turn charge you not for running the red but rather speeding. You might find it fun on your own to calculate your fine for traveling the speed necessary to make the light appear green to you, if the fine is \$1 per mile per hour over the limit and you are traveling in a 25 mph zone. (You can do this problem in the Doppler Quiz which is linked to this site.)

Well, we guess you have discovered by now that light (or any part of the electromagnetic spectrum) can be shifted up in frequency or down depending upon your relative motion. In fact, if your recession velocity is great enough away from a visible light source, you could in theory warm yourself as you would be able to shift to the infrared or heat area of the electromagnetic spectrum.

Now take a moment to consider the diagram that follows:

You will recognize it as the same as the fire truck approaching the stationary observer except now the source is emitting light instead of sound. Notice that the right region where a perceived increase in the frequency is noticed is referred to as "blueshifted", and the region which would appear to be of a lower frequency to an observer on the left is referred to as "redshifted." And, it is important to note that the equations which were derived for sound will work equally well for moving light sources provided the light sources are not moving near the speed of light. (We would have to take relativistic effects into account at speeds approaching the speed of light.)

 A service of the High Energy Astrophysics Science Archive Research Center (HEASARC), Dr. Alan Smale (Director), within the Astrophysics Science Division (ASD) at NASA/GSFC