The Doppler Effect - Maple Programming Help

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The Doppler Effect

Main Concept

The Doppler Effect refers to a change in the observed frequency of waves emitted by a moving source. This effect can be heard when a vehicle sounding a siren or horn approaches, passes, and drives away from you: when compared to the actual frequency being emitted, the observed frequency of the sound is higher as the vehicle approaches and lower as it recedes.

 

Why Does This Happen?

When the source of the waves is moving toward the observer, each successive wavefront is emitted from a position slightly closer to the observer than the previous one, thereby reducing the distance between successive wavefronts. The waves are bunched together, and the time between the arrival of successive wavefronts at the observer is similarly reduced, that is, the frequency is increased.

 

When the source of the waves is moving away from the observer, each successive wavefront is emitted from a position slightly further away from the observer than the previous one, thereby increasing the distance between successive wavefronts. The waves are spread out, and the time between the arrival of successive wavefronts at the observer is similarly increased, that is, the frequency is decreased.

 

For a source moving along the same line that connects the source to the observer, the relationship between the observed frequency (f) and the emitted frequency (f0) is given by:

f = c+vrc+vsf0

where

c is the speed of the waves relative to the medium

vr is the speed of the receiver relative to the medium (positive if the receiver is moving closer to the source, and negative otherwise)

vs is the speed of the source relative to the medium (positive if the source is moving away from the receiver, negative otherwise)

f0 is the actual frequency at which the waves are being emitted

 

Using Sound as an Example

A Mach number (M) refers to the ratio of the speed of an object to the speed of sound in a particular fluid medium: M = vsc where vs is the speed of the source relative to the medium and c is the speed of sound in the medium.

 

Subsonic Speeds,  M < 1

In this case, let's say the source of the sound is an emergency vehicle, such as an ambulance.

When the vehicle is stationary, all observers will hear the siren at the same pitch. The sound waves are being emitted at a constant frequency (f0) and the wavefronts are propagating away from the source at the constant speed of sound (c), which means that the waves will reach all observers with the same frequency (equal to the actual frequency at which they are being emitted).

When the vehicle is moving, observers in front of it will hear the siren at a higher pitch because the sound waves are bunching together in front of the vehicle and thus reach these observers at a higher frequency. Meanwhile, observers behind the vehicle will hear the siren at a lower pitch because the sound waves are spreading out behind the vehicle and thus reach these observers at a lower frequency.

 

Breaking the Sound Barrier,  M &equals; 1

In this case, let's say the source of the sound is a jet.

If the jet was to fly exactly at the speed of sound (which is approximately 343.2 m/s in dry air at 20°C), the wavefronts ahead of the jet would all bunch together at a single point, which means that observers in front it will hear nothing until the jet arrives and passes overhead.

 

Supersonic Speeds,  M &gt; 1

In this case, the source of the sound is also a jet.

If the jet was to fly faster than the speed of sound, it would actually lead the advancing wavefronts and pass by the observer before they hear the sound it creates. Behind the jet, the sound waves would create a cone-like formation known as a Mach cone, and the intense air pressure which builds up on the leading edges of this cone would cause a shock wave known as a sonic boom just after the jet passes.

 

In this model, explore how sound waves change as the jet travels at subsonic or supersonic speeds. Use the slider to adjust the Mach ratio. Click the radio buttons to choose the frequency at which the sound waves are emitted by the source. Select "Add an Observer" and then click or drag on the plot to place an observer and watch how the frequency they observe changes depending on whether they are in front of or behind the source.

 

Ratio of the velocity of the jet, vs , to the speed of sound, c :

vsc =

 

Emitted Frequency, f0&equals;

 

   

 

 

Observed Frequency, f =

 

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