Doppler Effect and Sonic Booms
Introduction


The speed v at which a sound wave travels through air is related to its frequency f and wavelength according to the following equation:



This equation also links the speed, frequency, and wavelength of ripples spreading out over the surface of a pond.



A boat travelling across a lake also produces ripples. However, this source of waves is moving and the shape of the wave created by the boat's bow is more complicated than the simple case of ripples on a pond.



In this unit we will look at how relative motion between a source of waves and the observer leads to apparent changes in the effective frequency and wavelength of the waves from the source.

Doppler effect
A siren on a fire engine produces a constant
frequency
The wave frequency f is the number of complete waves passing any point each second. Frequency is measured in hertz, Hz.
frequency
. Provided the fire engine is moving at a steady speed, its passengers will hear a constant tone. However, when standing beside a road and listening to the siren, the frequency produced by the approaching fire engine will be different from the frequency produced by the same fire engine as it gets further away. The noise coming from the fire engine's siren appears different to an observer at the roadside than it does to the fire engine's occupants. Someone at the road side will hear a rising and falling tone even though the pitch of the created sound is constant.

Similarly, if an approaching car sounds its horn at a constant frequency, the driver of the car will hear this constant pitch. However someone stationary outside the car will hear a higher pitch as the car approaches and a lower pitch as it passes and disappears into the distance. This phenomenon was called the Doppler effect after the Austrian mathematician, Christian Doppler, who first explained it in 1842.

Figure 1.   Notice the spacing of the waves when the fire engine is moving.
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Fig.1 illustrates what we now call the Doppler effect. Complete the following statements to summarize the behaviour of the waves.

  • When the vehicle is stationary, the waves emitted in all directions are spaced. The time between waves reaching the observer will be no matter where he stands.

    When the vehicle is moving, the waves in front are together than those .

    As the vehicle approaches the observer, the time between the arrival of successive waves is than when it is stationary. This means the waves are arriving more often and the frequency is .

    As it moves away from the observer, the waves are arriving less often than in the stationary case and the is reduced.
  • Click here to mark the question


We can explain the Doppler effect as follows:

Figure 2.   Click to reveal the effect of the car moving.
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When the sound source is stationary the distance between successive waves, d = . The time interval between waves being emitted by the source is the periodic time, T.
When the source in Fig.2 is moving, it travels a distance ds towards the observer in the time it takes to emit 4 successive wavefronts. If its velocity is vs, then ds = vs × 4T.
The apparent length of the 4 waves is given by .
Since ds = vs × 4T,

or

But the periodic time, T, can be expressed as , where v is the speed of the sound waves through the medium
     Therefore,
 



When a sound source is moving towards an observer with a speed vs, the
wavelength
The wavelength is the distance from one point on a wave to the identical point on the next wave. This can be stated as the distance from a crest on a wave to the next crest on the wave.
wavelength
is reduced by a factor depending on the speed of the source:



The faster the source, the greater the reduction in the wavelength.

The sound waves emitted by a car horn move through air at a constant speed, v, whether the car is moving or not. The effective wavelength for sound from the moving car is less than for the stationary one, so the frequency is higher for the moving car than for the stationary one. We can calculate the change in frequency for the approaching car.

The effective wavelength, , for the waves from an approaching car is given by
 
The effective frequency, f', for the waves is given by
 
Therefore,
 
But
 
     Therefore,
 

Complete the following summary for the effect of a sound source moving towards an observer.

  • For a sound source approaching an observer, the bottom line of this equation will be than 1 so the effective frequency at the observer will be than the frequency emitted by the stationary source.
  • Click here to mark the question


We can use similar reasoning to show that a source moving away from an observer at a speed vs will have an effective frequency lower than the stationary source.

When the source is moving away from the observer the effective wavelength, which we will again call , is given by .
     By a similar analysis to that used earlier, we can show that the effective frequency, f' is given by
 

Complete the following summary for the effect of a sound source moving away from an observer.

  • For a sound source moving away from an observer, the bottom line of this equation will be than 1 so the effective frequency at the observer will be than the frequency emitted by the stationary source.
  • Click here to mark the question


In these examples, we have looked at situations where the observer is stationary and the sound source is moving. However, the Doppler effect can also be observed when the sound source is stationary and the observer is moving.

Figure 3.   Consider how often the moving object encounters waves from the sound source.
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Complete the following statements to summarize the sound wave interaction with the moving observer.

  • The source is emitting sound waves of frequency f and the interval between them is one periodic time, T. When the observer and the source are both then f waves per second arrive at the observer and the time interval between them is T seconds.

    When the observer is moving towards the stationary sound source, the time interval between sound pulses being detected is than T, so the effective frequency is than f.

    When the observer is moving away from the sound source sound pulses are encountered often so the effective frequency is than the frequency emitted by the source.
  • Click here to mark the question


We have now considered the situations where the source is moving relative to a stationary observer and where the observer is moving relative to a stationary source. A third possibility is that both are moving. If the source is moving with a speed vs and the observer with a speed vo then the effective frequency that an observer will hear is given by:



This single equation covers all cases where the source and /or the observer are moving. The upper mathematical signs apply if the source and observer are getting closer together. The lower signs apply if the source and observer are getting further apart. This equation confirms that, if the relative motion is towards each other, the effective frequency is higher; while, if the source and observer are separating, the effective frequency is reduced.

Sonic booms
Aircraft travelling faster than the speed of sound are said to have achieved supersonic speeds. Such speeds are denoted by Mach numbers after the Austrian physicist Ernst Mach. The Mach number is a multiple of the speed of sound in the surrounding air. For example, in the high atmosphere where the speed of sound in air is 310 ms−1, Mach 3 means that an object is travelling at 930 ms−1.

When an object is moving at less than the speed of sound the wavefronts produced stack up one behind the other at the front, but they do not actually overlap.

Figure 4.   Sound energy 'stacking up' in front of the plane.
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When an object's speed approaches the speed of sound the wavefronts emitted in the forward direction pile up one on top of the other directly in front of it. This collection of energized air molecules acts as a barrier, called the sound barrier. In order to travel faster, the plane's engines must produce extra
thrust
Thrust is the name given to the force propelling an object in a specific direction.
thrust
to pass through this sound barrier. Once the plane has passed through the sound barrier and is travelling faster than the speed of sound it will continuously emit a sonic boom while its speed remains above the speed of sound.

Figure 5.   Wavefronts overlap at the edges producing interference effects.
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When the plane is travelling faster than the speed of sound it is actually 'outrunning' the sound wavefronts it produces. These wavefronts form the shape of a cone behind the plane. Along the edges of this cone the wavefronts overlap and interfere to form a single shock wave of very large
amplitude
The amplitude of a wave is the distance between the peak of the crest and the undisturbed position.
amplitude
. When this shock wave passes an observer on the ground it is heard as a very loud low frequency 'rumble' called a sonic boom. This can have sufficient
energy
A system has energy when it has the capacity to do work. The scientific unit of energy is the joule.
energy
to break windows or cause other structural damage when the plane is flying at supersonic speeds at low altitudes.

The sonic boom follows the plane for all the supersonic parts of its journey. While flying at supersonic speeds at high altitude, planes have to maintain a minimum separation so as not to get caught up in one another's shock waves.

Redshift
In 1927 Edwin Hubble published the first observational evidence for the expansion of the universe. This work was based on observations of the Doppler effect on light emitted by stars in distant galaxies.

In our earlier analysis of the Doppler effect for sound waves, we used the fact that sound waves travel through a medium with a certain speed. In this case, there is no medium. However, the overall effect is still the same. If a star moves towards us, its detected frequency is higher and its wavelength lower than those actually emitted by the star.

The equations already determined for the Doppler effect in sound also apply to nearby galaxies whose speeds are very much smaller than the speed of light. However, fast-moving distant galaxies whose speeds approach the speed of light are affected by a phenomenon called time dilation and this gives rise to a slightly different Doppler equation. According to the theory of relativity, the relativistic Doppler effect formula for a distant star can be expressed as:



where v is the star's
velocity
An object's velocity states both the speed and direction of motion relative to a fixed reference point.
velocity
and c the speed of light.

When an object is moving away from the observer its velocity v is positive, so the apparent frequency f' is lower than that emitted by the star itself. If the star emits visible light, its colour will appear to be shifted towards the lower frequencies at the red end of the spectrum. This effect is known as redshift.

The theory of relativity also gives us an equation describing the change in wavelength for the Doppler effect on moving stars :



where is the wavelength of the light emitted by a star and v can be thought of as the
relative velocity
The velocity of a moving object as viewed from a stationary or moving point is called the relative velocity.
relative velocity
between the star and the observer. If the star is moving away from the observer v is positive and so > . The light emitted by a star moving away from an observer will have a longer wavelength than expected.

Complete the following statements

  • When a source is moving away from an observer its velocity will be positive. If it is emitting light of a particular wavelength, the above equation shows that its wavelength appears to the observer. If the star emits visible light, its colour will appear to be shifted towards the longer wavelengths at the red end of the spectrum. This effect is known as .
  • Click here to mark the question
The amount of the shift depends upon the velocity of the source. When the star is moving with a speed v, the fractional change in wavelength Δ can be expressed as:
When the star's speed is very much less than the speed of light the change in wavelength Δ can be expressed as:
    



Astronomers have discovered that all the distant galaxies appear to be moving away from the earth. By considering the size of their redshifts, researchers have been able to estimate their speeds. Hubble found that distant galaxies moved faster. He even went on to propose that a galaxy's velocity is directly proportional to its distance from the earth:



The constant of proportionality in this equation is called the Hubble constant H, so Hubble's law is expressed as:



The most distant galaxies have been estimated to have speeds approaching the speed of light.

The simplest explanation of these observations is that the universe is expanding. At first sight, it seems as though all the distant galaxies are moving away from us. However, this does not mean we are at the centre of the expanding universe. Remember, all these velocities are relative. From the point of view of an observer in a distant galaxy, we (and the other galaxies) are moving away from them.

Speed cameras and other uses of the Doppler effect
The speed of cars passing a stationary speed camera is determined using the Doppler effect.
 
Figure 6.   Speed cameras use the Doppler effect.
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The transmitter in the camera emits short bursts of microwave radiation. When these waves are reflected from a passing vehicle, the frequency of the reflected signal is slightly different from that of the transmitted signal. This frequency difference occurs because the moving vehicle appears to be the source of the reflected signal. Since the source is moving, the frequency of the reflected signal will be different from that of the transmitted signal. The difference in frequency depends on the speed of the motion. It can be shown that the vehicle's speed is given by:



where c is the speed of
electromagnetic
Electromagnetic waves, such as light, are made up from oscillating electric and magnetic fields. Because of this, they are self-propagating and can travel through a vacuum. All types of electromagnetic wave travel at the same speed in a vacuum, 3 × 108 ms−1.
electromagnetic
waves, f the frequency of the
microwaves
Microwaves are electromagnetic waves with wavelengths in the range 1 mm to 0.1 m.
microwaves
, and Δf the frequency difference between the transmitted and reflected signals.

The Doppler effect can also be used to measure the speed at which blood flows through veins or arteries. In this technique a beam of ultrasound is directed towards flowing blood. When the beam is reflected off the moving blood its frequency is altered. The difference between the emitted and reflected frequencies can be used to calculate the speed of blood flow.

Summary


Relative motion between a source of waves and the observer leads to apparent changes in the effective frequency and wavelength of the waves from the source. This is called the Doppler effect.

The Doppler effect can be used to measure speed in cases such as moving cars or blood flow.

When a sound source is moving towards an observer with a speed vs, the wavelength is reduced by a factor depending on the speed of the source:



The faster the source, the greater the reduction in the wavelength.

The Doppler effect occurs in all situations where there is relative motion between a sound source and an observer. The following equation covers all cases where the source and / or the observer are moving:



The upper mathematical signs apply if the source and observer are getting closer together. The lower signs apply if the source and observer are getting further apart.

Sonic booms are large amplitude shock waves produced by planes flying faster than the speed of sound.

The Doppler effect applies to light as well as sound waves but, for light, uses slightly different equations. Redshifted light from distant galaxies provides evidence for the theory of the expanding universe.

Exercises
1. The frequency of the sound emitted by the siren of a fire engine is 2.00 kHz when the vehicle is at rest. Answer the following questions to 2 decimal places. (You may assume that the speed of sound is 330 ms−1.)

  • What frequency would you hear if the fire engine were moving towards you at 33 ms−1?
    kHz

    What frequency would you hear if the fire engine were moving away from you at 10 ms−1?
    kHz
  • Click here to mark the question
2. A starship is travelling towards a star at 3 × 107 ms−1. Viewed by an observer at rest relative to the star, the star appears yellow in colour (with a predominant wavelength of 580 nm). What is the predominant wavelength of the starlight viewed by an observer on the starship?
  • nm   (to the nearest whole number)
  • Click here to mark the question
3. A galaxy is observed to have a fractional redshift of 0.01. If Hubble's constant is 55 kms−1MPc−1 (where 1 Megaparsec (MPc) = 3.26 × 106 light-years), how far away is the galaxy?
  • million light-years   (to the nearest whole number)
  • Click here to mark the question
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