## What is the Doppler Effect?

The Doppler effect is the perception of a change in frequency of sound, light or other waveforms according to the positions of the source and the observer of that waveform. Specifically, it is because of the increasing or decreasing **distance **that creates the perception that the waveforms are getting longer or shorter and, by extension, the frequency of that waveform is getting larger or smaller, respectively. The Doppler effect can only take place when either the source or observer are moving or both are moving.

This effect is more of a general observation, but it can be useful when studying the distance between Earth and stars within our and distant galaxies. This is because a star’s heat excites its atoms and, by extension, its electrons, forcing them to move. This motion causes stars to emit electromagnetic waves and, if those stars are moving away or toward us due to gravitational attraction or the universes’ expansion, those waveforms will appear to be changing in frequency with respect to the Earth.

## Doppler Effect Equation

The Doppler effect is dependent on many factors. Previously, we mentioned the most important – whether either the observer or the source is moving. Each of these scenarios creates one equation, and, therefore, there can be a maximum of eight slightly different equations. However, we can construct a general equation if we understand the main variables that must be involved in the Doppler effect. These include: (1) the frequency from the source, (2) the perceived frequency according to position, (3) the velocity of the observer, (4) the velocity of the source and (5) the velocity of the waveform. Our general equation, which includes these variables is as follows:

$f\u2018=f\left(\frac{v\pm {v}_{0}}{v\pm {v}_{s}}\right)$where *f’* is equivalent to the perceived frequency, *f* is the frequency from the source, *v* is the velocity of the waveform, *v _{0}* is the velocity of the observer and

*v*is the velocity of the source.

_{s}The medium with which a waveform reacts will impact its velocity. Sound, for example, is a *mechanical waveform*, meaning that it is created at the atomic level through the vibration of atoms. Therefore, if there are no atoms, sound cannot be transmitted from one place to another. But even if there were atoms, they can behave differently according to external factors, which causes the sound waves to also change behavior. For example, if sound is traveling through the air, and that air is hot, then the sound waves will travel far more quickly because the atoms within the air have a natural disposition to decrease the amount of energy within them, which is solved by vibrating more quickly. Therefore, if the query calls for the perceived frequency of a sound waveform, you can accurately represent the velocity of sound in its environment with a known temperature with the equation:

where 331 m/s is the base speed of sound, 0.6 is a constant representing the increase of the speed of sound in meters per second for every degree Celsius, and *T* is the temperature of the air in Celsius.

Light follows a different set of rules from those of sound. Since light is composed of photons, which are particles that have behavior like that of electromagnetic waveforms, light has the capacity to travel either alone or through atoms. However, the speed of light can still be impacted according to the medium it is forced to go through. The only difference between light and sound is that light’s velocity is also impacted by the medium’s *index of refraction*, which is the degree to which a substance slows light waves down. The equation for the velocity of light in a medium is

where *c* is the base velocity of light and *n* is the index of refraction. In air, the index of refraction is 1, but in water, that index increases to 1.3.

### Specific Doppler Effect Equations

With the general equations established, we can start to fill out the specific cases. It is less important that you memorize these specific cases and their complementary equations than it is to focus on the reasons why they exist and using your reasoning to adapt the general equation to the situation.

First and foremost, if the observer or source are not moving then that velocity variable, either or , is equal to zero. That said, for the Doppler effect to take place, at least one must be moving toward or away from the other. In the case where only the source or observer are moving, we have four equations:

- When the source is moving toward a stationary observer.
- In this case,
*v*must be zero. The velocity of the waveform does not change; however, the source has moved closer to the observer by the time they perceive the first waveform (either hearing it, in the case of sound or seeing it in the case of light). This means that the amount of time that the second waveform has to travel to get to the unmoving observer is less than that of the first waveform. When time decreases, the perceived frequency (_{0}*f’*) increases. Mathematically, the only way that happens in the Doppler equation is with a decreasing denominator, and so we choose the minus sign here. The equation is given as:

$f\u2018=f\left(\frac{v}{v\u2013{v}_{s}}\right)$

- In this case,
- When the source is moving away from a stationary observer.
*v*must still be zero in this case. The difference is that the source is moving away from a stationary observer. That means that the frequency that the observer hears a waveform decreases, which can only happen if the equation’s denominator increases. The resultant form of the Doppler equation is:_{0}

$f\u2018=f\left(\frac{v}{v\u2013{v}_{s}}\right)$

- When the observer is moving toward a stationary source.
- This time, the velocity of the source,
*v*, is equal to zero. The logic behind the equation used is similar to the first scenario; we add the velocity of the observer to the velocity of the waveform because the frequency because the frequency must increase as an observer and source grow closer. This equation used is:_{s}

$f\u2018=f\left(\frac{v+{v}_{0}}{v}\right)$

- This time, the velocity of the source,
- When the observer is moving away from a stationary source.
*v*is equal to zero. The frequency decreases when the source and observer grow closer, meaning the numerator of the equation must be decreasing. That is represented as:_{s}

$f\u2018=f\left(\frac{v\u2013{v}_{0}}{v}\right)$

The Doppler effect can also occur if both the source and the observer are moving. There are another four cases here as well.

- When the observer and source are moving toward each other.
- This is the most rapid increase in perceived frequency. To reflect this, the denominator must decrease while the numerator increases. In other words, we combine the “toward” cases above to create a more unified equation:

$f\u2018=f\left(\frac{v+{v}_{0}}{v\u2013{v}_{s}}\right)$

- This is the most rapid increase in perceived frequency. To reflect this, the denominator must decrease while the numerator increases. In other words, we combine the “toward” cases above to create a more unified equation:
- When the observer and source are moving away from each other.
- In this instance, this is the fastest decrease in perceived frequency. To represent that, we combine the “away” cases to create a unified equation. This is given as:

$f\u2018=f\left(\frac{v\u2013{v}_{0}}{v+{v}_{s}}\right)$

- In this instance, this is the fastest decrease in perceived frequency. To represent that, we combine the “away” cases to create a unified equation. This is given as:
- When the observer and source are moving in the same direction, yet the observer is moving toward the source at a faster velocity.
- This one is a little more complex to conceptualize. If the source is moving away from the observer, but the observer is moving toward it at a faster velocity, then that means the frequency would actually slowly increase. In other words, the observer is moving toward a source that is moving away. The Doppler general equation would, then, have a numerator that increases faster than the denominator is decreasing. The equation in this case is given as:

$f\u2018=f\left(\frac{v+{v}_{0}}{v+{v}_{s}}\right)$

- This one is a little more complex to conceptualize. If the source is moving away from the observer, but the observer is moving toward it at a faster velocity, then that means the frequency would actually slowly increase. In other words, the observer is moving toward a source that is moving away. The Doppler general equation would, then, have a numerator that increases faster than the denominator is decreasing. The equation in this case is given as:
- When the observer and source are moving in the same direction, yet the source is moving toward the observer at a faster velocity.
- This is the same as the above, except the frequency is now slowly decreasing. The observer is not moving fast enough, while the source is moving toward the observer at a faster speed. Now, the concept flips – the source moves toward instead of away, while the observer tries to move away. Mathematically, the equation is given as:

$f\u2018=f\left(\frac{v\u2013{v}_{0}}{v\u2013{v}_{s}}\right)$

- This is the same as the above, except the frequency is now slowly decreasing. The observer is not moving fast enough, while the source is moving toward the observer at a faster speed. Now, the concept flips – the source moves toward instead of away, while the observer tries to move away. Mathematically, the equation is given as:

## Doppler Effect Example

This effect is most notably observed through sound, particularly if you live in an urban environment. Imagine if you were sitting in a vehicle and stuck in traffic. Suddenly, you hear an ambulance’s sirens propagating through the air. This is the case in which you are a stationary observer of the waveform and the source of the waveform is moving toward you at a certain velocity. The Doppler effect would become apparent to you immediately; the siren would sound different the closer to you it grew. That is because the *pitch* of the sound changes with the frequency of its waveform. We know that the frequency increases as the ambulance moves toward your car. This also represents an increase in pitch, which you, quickly, become aware of when the ambulance is right next to you; the sound of the siren is incredibly sharp. After that point, when the ambulance is moving away from you the sharp sound begins to wane. That is because we enter the next case where the source is moving away from you – the stationary observer. In this case, we know that the frequency decreases, meaning the pitch decreases. When the ambulance is far enough away that you can barely hear its siren, the sound is soft because the frequency and pitch are at their minimum.

## Quiz