Doppler Effect Formula: Equations and Solved Questions

Doppler Effect is a phenomenon that is observed when the source of the waves moves with respect to an observer. It is a characteristic of sound waves in general. The Doppler Effect describes the apparent shift in sound frequency when the observer and the medium are both moving in the same direction. It is determined by three factors: the source's velocity, the medium's velocity, and the observer's velocity. An example of the Doppler Effect is an ambulance crossing you with its siren blaring.

Key Takeaways: Doppler Effect, Doppler Shift, Observed Frequency, Doppler Effect Formula, Satellite communication

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What is Doppler Effect?

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The Doppler Effect, also known as the Doppler shift, is the result of a change in the frequency of sound waves caused by movement. 

The Doppler effect is a phenomenon caused by a moving wave source that causes an apparent upward shift in frequency for observers who are approaching the source and a visible downward change in frequency for observers who are retreating from the source.

• When the source and the observer move relative to each other, the frequency that is observed by the observer (f_a) is different from the actual frequency that is produced by the source (f₀).
• When the source of the sound waves is moving towards the observer, an upward shift in frequency is observed.
• As for the observers from whom the source is receding, a downward shift in frequency will be observed.
• It is crucial to note that the impact isn't caused by a change in the source's frequency. 
• The Doppler effect may be seen in any wave type, including water waves, sound waves, and light waves.
• Doppler effect is known to our encounters with sound waves.

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Doppler Effect Formula

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If the source of the sound and the listener move in relation to each other, the sound heard by the listener changes. When the listener and the source get near enough, the frequency heard by the listener is higher than the sound produced by the source. When the listener and the source moves away from each other, the frequency heard by the listener is lower than the frequency heard by the source. The frequency of the sound is measured in Hertz (Hz) where one Hertz is one cycle per second. (1 Hz = 1s^-1 = 1 cycle/s).

As a result, the formula for the doppler effect is:

f_L = \(\frac {v + vl} {v + vs}\)fs

In the Doppler Effect formula,

• f_L is the frequency of sound that the listener perceives (Hz, or 1/s)
• v is the speed of sound in the medium (in m/s)
• v_L is the listener's velocity (m/s) 
• v_s is the velocity of the sound source (in m/s) 
• f_s is the frequency of sound emitted by the source (Hz, or 1/s) 

a) Source Moving Towards the Observer at Rest

In this case, the velocity of the observer velocity is zero, so v_o is equal to zero. Substituting this into the Doppler effect formula, we get the equation of the Doppler effect when a source moves towards an observer at rest- 

f_L = \(\frac {v} {v - vs}\)fs

b) Source Moving Away from the Observer at Rest

Since the velocity of the observer, in this case, is zero, we can eliminate v_o from the equation. However, the source moves away from the observer, so its velocity becomes negative to indicate the direction. Therefore, the equation now becomes- 

f_L = \(\frac {v} {v - (- vs)}\)fs

c) Observer Moving Towards a Stationary Source

In this case, v_s will become zero, hence we get the following doppler effect formula:

f_L = \(\frac {v\ +\ v_{0}} {v}\)fs

d)Observer Moving Away from a Stationary Source

Since the observer is moving away, the velocity of the observer in this situation becomes negative. So, instead of adding v_o, we can now subtract it, since v_o is negative.

f_L = \(\frac {v\ -\ v_{0}} {v}\)fs

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Doppler Effect Formula – Frequency

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The Doppler shift, also known as the Doppler effect, describes the connection between the perceived frequency and the emitted frequency when the velocity of the source and receiver is slower than the waves’ velocity in the medium. When the receiver and source speeds are smaller than the wave velocity, the formula is as follows:

Observed Frequency: F = (\(1 + \frac {\bigtriangleup v} {c}\))f₀ 

Change in Frequency: Δf = \(\frac {\bigtriangleup v} {cf_0}\) 

Where, 

• Δf = f – f₀, Δv = v_r – v_s, the velocity of the receiver relative to the source;
• C is the wave's amplitude in the medium;
• v_r is the receiver's speed in relation to the medium (positive if the receiver moves towards the source and negative if it moves in the opposite direction).
• v_s denotes the source's velocity in relation to the medium (positive if the source moves away from the receiver and negative if it moves in the opposite direction).
• The frequency observed is denoted by the letter f.
• The frequency transmitted is f₀, and the transmitter and receiver are traveling toward each other.

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Doppler Effect Examples

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Consider the following scenario: You are standing on the sidewalk when an ambulance whizzes by. Do you notice how the siren's tone alters as it travels a certain distance?

It gets louder as it gets closer to you, but there is another aspect of the sound that changes. The pitch of the vehicle increases as it approaches you and decreases as it pulls away. The frequency of the waves, or how many waves travel through an area per unit of time, causes the pitch to shift.

Doppler Effect observed in a moving ambulance 

• In the case of an ambulance, you are standing motionless as the vehicle approaches.
• As sound waves approach you, they compress, causing the frequency to rise, resulting in a higher pitch.
• However, when the vehicle moves away from you, the sound waves spread more apart, lowering the frequency and hence the pitch.

When the wave source travels in relation to the observer, the Doppler effect is noticed. 

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Application of Doppler Effect

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The applications of the doppler effect are – 

• Sirens

The principle behind a siren is that it begins at a pitch higher than its stationary pitch as it moves away from the observer and then returns to a lower pitch as it moves away from the observer. Siren's speed is calculated as follows:

V_radial = vs.cosθ

where θ is the angle formed between the line of sight of the object and the advancing velocity.

Application of Doppler Effect in Sirens 

• Astronomy

Astronomers utilize measurements of electromagnetic waves emitted by stars in distant galaxies to support the idea that the universe is expanding. Furthermore, the Doppler effect will be used to analyze the fine features of stars within galaxies.

• Velocity Profile Measurement

The real-time performance velocity profile of any liquid containing suspended particles, such as dust, emulsions, and gas bubbles, is calculated using an ultrasonic Doppler Velocimeter. 

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Things to Remember

• The Doppler Effect is defined as a shift in wave frequency caused by relative motion between the wave source and the observer. 
• Distinct stars have different absorption lines at different frequencies, however, the Doppler Effect can only be seen when the absorption lines are distant from the stated frequencies.
• The Doppler effect formula can be written as: f_L=\(\frac {v + vl} {v + vs}\)fs
• The Doppler Effect may be used in a variety of ways. Its use can be found in:
  • Sirens
  • Astronomy
  • Radars
  • Medical imaging and blood flow management
  • Flow management
  • Velocity profile management
  • Satellite communication
  • Audio
  • Vibration measurement

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