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If the observer is moving away from the source ( [link] ), the observed frequency can be found:

λ s = v T o v o T o v T s = ( v v o ) T o v ( 1 f s ) = ( v v o ) ( 1 f o ) f o = f s ( v v o v ) .
Picture is a drawing of a stationary source that emits sound waves with a constant frequency, with a constant wavelength moving at the speed of sound. Observer X moves away from the source with a constant speed.
A stationary source emits a sound wave with a constant frequency f s , with a constant wavelength λ s moving at the speed of sound v . Observer Y moves away from the source with a constant speed v o , and the figure shows initial and final position of the observer Y . Observer Y observes a frequency lower than the source frequency. The solid lines show the position of the waves at t = 0 . The dotted lines show the position of the waves at t = T o .

The equations for an observer moving toward or away from a stationary source can be combined into one equation:

f o = f s ( v ± v o v ) ,

where f o is the observed frequency, f s is the source frequency, v w is the speed of sound, v o is the speed of the observer, the top sign is for the observer approaching the source and the bottom sign is for the observer departing from the source.

[link] and [link] can be summarized in one equation (the top sign is for approaching) and is further illustrated in [link] :

f o = f s ( v ± v o v v s ) ,
Doppler shift
f o = f s ( v ± v o v v s )
Stationary observer Observer moving towards source Observer moving away from source
Stationary source f o = f s f o = f s ( v + v o v ) f o = f s ( v v o v )
Source moving towards observer f o = f s ( v v v s ) f o = f s ( v + v o v v s ) f o = f s ( v v o v v s )
Source moving away from observer f o = f s ( v v + v s ) f o = f s ( v + v o v + v s ) f o = f s ( v v o v + v s )

where f o is the observed frequency, f s is the source frequency, v w is the speed of sound, v o is the speed of the observer, v s is the speed of the source, the top sign is for approaching and the bottom sign is for departing.

The Doppler effect involves motion and a video will help visualize the effects of a moving observer or source. This video shows a moving source and a stationary observer, and a moving observer and a stationary source. It also discusses the Doppler effect and its application to light.

Calculating a doppler shift

Suppose a train that has a 150-Hz horn is moving at 35.0 m/s in still air on a day when the speed of sound is 340 m/s.

(a) What frequencies are observed by a stationary person at the side of the tracks as the train approaches and after it passes?

(b) What frequency is observed by the train’s engineer traveling on the train?

Strategy

To find the observed frequency in (a), we must use f obs = f s ( v v v s ) because the source is moving. The minus sign is used for the approaching train, and the plus sign for the receding train. In (b), there are two Doppler shifts—one for a moving source and the other for a moving observer.

Solution

  1. Enter known values into f o = f s ( v v v s ) :
    f o = f s ( v v v s ) = ( 150 Hz ) ( 340 m/s 340 m/s 35.0 m/s ) .

    Calculate the frequency observed by a stationary person as the train approaches:
    f o = ( 150 Hz ) ( 1.11 ) = 167 Hz .

    Use the same equation with the plus sign to find the frequency heard by a stationary person as the train recedes:
    f o = f s ( v v + v s ) = ( 150 Hz ) ( 340 m/s 340 m/s + 35.0 m/s ) .

    Calculate the second frequency:
    f o = ( 150 Hz ) ( 0.907 ) = 136 Hz .
  2. Identify knowns:
    • It seems reasonable that the engineer would receive the same frequency as emitted by the horn, because the relative velocity between them is zero.
    • Relative to the medium (air), the speeds are v s = v o = 35.0 m/s .
    • The first Doppler shift is for the moving observer; the second is for the moving source.

    Use the following equation:
    f o = [ f s ( v ± v o v ) ] ( v v v s ) .

    The quantity in the square brackets is the Doppler-shifted frequency due to a moving observer. The factor on the right is the effect of the moving source.
    Because the train engineer is moving in the direction toward the horn, we must use the plus sign for v obs ; however, because the horn is also moving in the direction away from the engineer, we also use the plus sign for v s . But the train is carrying both the engineer and the horn at the same velocity, so v s = v o . As a result, everything but f s cancels, yielding
    f o = f s .
Practice Key Terms 2

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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