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16.2The Doppler effect for sound

Moving source and stationary observer

To derive a formula relating the emitted and observed frequencies, look at the situation in Figure 16.5.

Figure 16.5: Determining the Doppler frequency.

The source emits sound of frequency f and wavelength λ. The sound has speed v in still air. So, in one second, the source will emit f wavefronts.

The source is moving with speed u s towards the observer. So, in one second, the source will move a distance equal to u s towards the stationary observer.

The movement of the source means that these f wavefronts are all within a distance of v − u s. The stationary observer will, therefore, measure a wavelength λ′ (separation of wavefronts) equal to:

λ′=v−u s f

The frequency f′ measured by the stationary observer is then:

f′=v λ′=v v−u s f

In other words:

f′=f(v v−u s) (Source moving towards stationary observer.)

A similar calculation gives:

f′=f(v v+u s) (Source moving away from stationary observer.)

Exam tip

Notice that the speed of the wave measured by the observer is still v despite the fact the source moves. The speed of a wave is determined by the properties of the medium. This is still air for both source and observer, and both measure the same speed, v.

As the source approaches, the stationary observer thus measures a higher frequency than that emitted by the source. The wavelength measured by the observer will be shorter than the wavelength measured at the source. You can see this clearly from the wavefront diagram of Figure 16.2: the distance between the wavefronts in front of the source is smaller than that in Figure 16.1.

Using the basic wave formula v=f λ=f′λ′ we find

λ′=v f′=v f v v−u s

λ′=v f(1−u s v)

λ′=λ(1−u s v) (Source moving towards stationary observer.)

And similarly:

λ′=λ(1+u s v) (Source moving away from stationary observer.)

CHECK YOURSELF 2

A source of sound approaches a stationary observer at speed 40 m s−1. The speed of sound is 340 m s−1.

What is the ratio wavelength observed wavelength emitted?

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Worked Example 16.2

The siren of a car moving at 28.0 m s−1 emits sound of frequency 1250 Hz. The car is directed towards a stationary observer X and moves away from an observer Y. Calculate the frequency and wavelength of sound observed by X and Y. The speed of sound in still air is 340 m s−1.

Answer

We need to use the formula for a moving source; this is:

f′=f v v±u s

It can be difficult to decide whether the sign in the denominator should be plus or minus. The easy way to figure this out is to realise that if the source is approaching observer X we expect an increase in frequency. This means we need to use the minus sign to make the denominator smaller and so get a larger frequency. So substituting the values for X:

f′=1250×340 340−28=1362≈1360 Hz.

Remember that the speed of the sound is still v. To find the wavelength measured by X we use the relationship between speed, frequency and wavelength:

λ′=340 1362≈0.25 m

For Y:

f′=1250×340 340+28=1155≈1150 Hz and λ′=340 1155≈0.29 m

Stationary source and moving observer

In the case of a stationary source and a moving observer we may argue as follows. First, let us consider the case of the observer moving towards the source. The observer who moves with speed u o with respect to the source may claim that he is at rest and that it is the source that approaches him with speed u o. But the air is also coming towards the observer with speed u o and so the observer will measure a higher wave speed, v + u o.

Exam tip

The speed of sound relative to the observer is no longer v. The medium has changed. For the source the medium is still air. For the observer the medium is moving air.

We can now apply the same equations as for a source moving towards the observer, and so the frequency measured by the observer is:

f′=f v+u o v+u o−u o

f′=f(v+u o v) (Observer moving towards source.)

Similarly:

f′=f(v−u o v) (Observer moving away from source.)

Exam tip

The IB data booklet has the formulae for frequency.

There is no need to remember the formulas for wavelength—just use the relationship:

wavelength=wave speed frequency

Worked Example 16.3

An observer runs towards a stationary source with speed u 0. Show that the wavelength measured by the observer will be the same as that measured by the source (as we expect from the analysis with wavefront diagrams).

Answer

We need to use the formulas for a moving observer; this is:

f′=f(v+u o v)

Then

λ′=v+u o f′

λ′=v+u o f(v+u o v)

λ′=v f=λ.

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CHECK YOURSELF 3

A source moving at 40 m s−1 emitting sound of frequency 500 Hz approaches a stationary observer. The frequency heard by the observer will be f′=500×(340 340−40)=567 Hz.

What frequency would an observer hear if she was moving at 40 m s−1 towards a stationary source of frequency 500 Hz?

Worked Example 16.4

A train sounding a 0.500 kHz horn is moving at a constant speed of 25 m s−1. The train approaches and then moves away from an observer at the train station. Sketch a graph to show how the frequency heard by the observer varies with time. (Assume that the train goes past the observer at t = 10 s and take the speed of sound to be 340 m s−1.)

Answer

As the train approaches the constant frequency observed is:

f′=f(v v−u s)=500×340 340−25=540 Hz

As it moves away, the constant frequency is:

f′=f(v v+u s)=500×340 340+25=466 Hz

This leads to the graph in Figure 16.6, showing the sudden drop in frequency at 10 s.

(This is an idealised graph with sharp corners. A realistic graph would be smooth.)

Figure 16.6: For worked example 16.4.

Worked Example 16.5

A sound wave of frequency 15.0 kHz is emitted towards an approaching car. The wave is reflected from the car and is then received back at the emitter at a frequency of 16.1 kHz.

Calculate the velocity of the car. (Take the speed of sound to be 340 m s−1.)

Answer

The car is approaching the emitter so the frequency it receives is:

f 1=15.0×340+u 340 kHz

where u is the unknown car speed.

The car now acts as an emitter of a wave of this frequency (f 1), and the original emitter will act as the new receiver.

So, the frequency received (16.1 kHz) from the approaching car is:

16.1=(15.0×340+u 340)×340 340−u

⇒16.1 15.0=340+u 340−u

340+u 340−u=1.073

Solving for u (easiest done using the solver of your graphics calculator): u = 12 m s−1.

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TEST YOUR UNDERSTANDING

Take the speed of sound in still air to be 340 m s–1.

8We define the Doppler effect as the shift in observed frequency when there is relative motion between the source and the observer. Could we replace wavelength for frequency in this definition?

9Explain, with the help of diagrams, the Doppler effect. Show clearly the cases of a source that:

amoves towards a stationary observer

bgoes away from a stationary observer, as well the case of a moving observer.

10A source approaches a stationary observer at 40 m s−1 emitting sound of frequency 500 Hz.

aDetermine the frequency the observer measures.

bCalculate the wavelength of the sound as measured by

ithe source

iithe observer.

11A source is moving away from a stationary observer at 32 m s−1 emitting sound of frequency 480 Hz.

aDetermine the frequency the observer measures.

bCalculate the wavelength of the sound as measured by:

ithe source

iithe observer.

12A sound wave of frequency 512 Hz is emitted by a stationary source toward an observer who is moving away at 12 m s−1.

aDetermine the frequency the observer measures.

bCalculate the wavelength of the sound as measured by:

ithe source

iithe observer.

13A sound wave of frequency 628 Hz is emitted by a stationary source toward an observer who is approaching at 25 m s−1.

aDetermine the frequency the observer measures.

bCalculate the wavelength of the sound as measured by:

ithe source

iithe observer.

14A source of sound is directed at an approaching car. The sound is reflected by the car and is received back at the source. Carefully explain what changes in frequency the observer at the source will detect.

15A sound wave of frequency 500 Hz is emitted by a stationary source toward a receding observer. The signal is reflected by the observer and received by the source, where the frequency is measured and found to be 480 Hz. Calculate the speed of the observer.

16A sound wave of frequency 500 Hz is emitted by a moving source toward a stationary observer. The signal is reflected by the observer and received by the source, where the frequency is measured and found to be 512 Hz. Calculate the speed of the source.

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17A train emitting sound of frequency 800 Hz has speed 40 m s−1 when far from a train station. The train decelerates at 2.0 m s−2 as it approaches the train station and stops at t = 20 s at the train station next to the observer.

Sketch a graph to show the variation with time of the frequency heard by an observer at the train station. (See Figure 16.7.)

Figure 16.7: For question 17.

18Consider the general case when both the source and the observer move towards each other. Let u s be the velocity of the source and u o that of the observer. In the frame of reference in which the observer is at rest, the waves appear to move with velocity v + u o and the source appears to move with velocity u s + u o. Show that the frequency received by the observer is:

f o=f s c+v o c−v s.