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Doppler Effect

Let’s start with a simple example in which a bug is at the centre of curve water puddle. It travels with producing vibration in the water by its legs. This disturbance is also travelled from a point in all directions and bug is moved in concentric circle. Let there are two observers on both sides of concentric circle. The frequency of disturbance at both the points is same as the frequency of bug’s disturbance. But if the bug is moving towards one observer at right side and producing disturbance near to that observer and far to the opposite side of the observer.  Thus each disturbance travels with smaller distance and less time to reach that specific observer to right side. This time the frequency of arrival disturbance is greater than the frequency of produced disturbance.

This effect is the Doppler Effect which determines that when the waves is moving with respect to given observer. Here we are explaining a completed description of Doppler Effect and its mathematical equation derivation with some problems.

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What is the Doppler Effect?

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Doppler Effect is a wave phenomenon, it holds not only for sound waves but also for electromagnetic waves such as microwaves, radio waves and visible light.
Doppler Effect can be defined as :
The Apparent change in the frequency of sound when the source, the observer and the medium are in a relative motion is called Doppler Effect.
For the waves which require a medium for their propagation, the apparent frequency depends on three factors :
  1. Velocity of the source
  2. Velocity of the observer
  3. Velocity of the medium or wind.

For example,
Let us consider an van moving with a siren and there are two observers, one which is moving towards the van and the other is moving away from the van. We could observe that the observer who is moving towards the van is experiencing High frequency and the observer who is moving away the van is experiencing low frequency.

Doppler Effect

When a van blowing its whistle approaches him, the pitch of the whistle appears to rise and suddenly appears to drop as the van moves away from him.

Doppler Effect takes place both in sound and light. If sound is considered, it is based on whether the source or observer or both are in motion, while if light is considered, it is based on the fact that whether the distance between the source and observer is decreasing or increasing.

Doppler Effect Sound

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In sound waves, there are some following conditions observed in Doppler effect. This effect depends upon whether the source is in motion or the observer. The velocity of medium affects the Apparent Frequency.

Conditions observed in Doppler effect  for sound waves:
  1. When the source of sound is in motion and the observer is at rest then the Doppler Effect is observed. 
  2. When the observer is in motion and the source at rest then the Doppler Effect is observed.
  3. When the source and the observer are moving with unequal velocities then the Doppler Effect is observed.
  4. When the source and the observer and intervening medium are in motion then the Doppler Effect is observed.
Limitations of Doppler effect in sound:
  1. The Velocity of source of sound must be less than that of the Velocity of sound i.e. Vs < V.
  2. The Velocity of observer must be less than the velocity of sound i.e. Vo< V.
  3. If the velocity of sound of source is greater than that of Velocity of sound then due to shock waves the wave front gets distorted, consequently the change in frequency will not be observed by the observer.
Conditions when Doppler Effect does not take place in Sound waves:
  1. When the source of sound and the observer both are at rest then the Doppler Effect in sound is not observed.
  2. When the source and the observer both are moving with the same velocity in same direction then the Doppler Effect is not observed.
  3. When the source and the observer are moving mutually in perpendicular directions then the Doppler Effect is not observed.
  4. When the medium only is moving then the Doppler Effect is not observed.
  5. When the distance between the source and the observer is constant then the Doppler Effect is not observed.

Doppler Effect Light

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The Velocity of light in free space is independent of the motion of source or observer and its universal constant given as, 
C = 3 $\times$ 108 m/sThus, the Doppler Effect in light depends only upon the relative motion of the light source and the observer and it does not matter which one is moving. When a light source and an observer are approaching each other with a velocity V, then the apparent frequency of light will be
V’ = V
if V << c
V’ = V (1 – $\frac{v}{c}$)
Here, V = Velocity of the body,
c = Velocity of light.
which can be written as:  
V' - V = - V $\frac{v}{c}$

Let Apparent frequency is given by, V' - V = $\delta$ V
Hence, the above equation can be written as:
$\delta$ V = -V $\frac{V}{c}$

Case a:
When the Light source is going away from the earth i.e., if Apparent frequency is less than real frequency then, 
V’ < V and $\delta$ $\lambda$ = lambda or $\lambda$‘ > $\lambda$ 
i.e. $\lambda$ is increased or spectral line will shift towards the red end of the spectrum. This is known as Red shift.

Case b:
When the light source is coming nearer to earth i.e., if Apparent frequency is greater than real frequency then,
V’ > V and $\delta \lambda$ = - $\lambda$
Wavelength appears to be decreasing i.e. the spectral line in electromagnetic spectrum gets displaced towards violet end, Hence it is known as Violet shift.
The Velocity of medium does not contribute to this effect frequency.

Doppler Effect Equation

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Apparent frequency is the frequency observed or heard by the observer when both Source and observer are moving:

Some of the conditions and related formula of the Doppler Effect is mentioned below:

(i) When both Source and observer moves towards each other :

The Apparent frequency is given by
n’ = n $\frac{v + v_{0}}{v – v_{s}}$
and Apparent wavelength $\lambda$' is
$\lambda$’ = $\lambda$ $\frac{v – v_{s}}{v + v_{0}}$
Here, Apparent frequency is greater than Real frequency i.e., n’ > n and $\lambda$‘ < $\lambda$.

(ii) When both Source and observer move away from each other :

The Apparent frequency is given by
n’ = n $\frac{v – v_{0}}{v + v_{s}}$
then, Apparent wavelength $\lambda$' is
$\lambda$’ = $\lambda$ $\frac{v + v_{s}}{v – v_{0}}$
Here, Apparent frequency is n’ < n and $\lambda$’ > $\lambda$

(iii) When the Source is approaching the Stationary observer :

The Apparent frequency is given by:
n’ = n $\frac{v – v_{o}}{v - v_{s}}$then Apparent wavelength $\lambda$' is
$\lambda$’ = $\lambda$ $\frac{v - v_{s}}{v – v_{o}}$

(iv) When the Observer is approaching the Stationary source :

The Apparent frequency is given by
n’ = n $\frac{v + v_{o}}{v + v_{s}}$
then, Apparent wavelength $\lambda$' is 
$\lambda$’ = $\lambda$ $\frac{v + v_{s}}{v + v_{o}}$

(v) When Observer crosses the Source:
Then Apparent frequency of source is given by
n’ = n $\frac{v + v_{o}}{v}$
and Apparent frequency of observer n'' is
n’’ = n $\frac{v + v_{o}}{v}$
$\delta$ n = n’ – n’’
= 2 n $\frac{v_{o}}{v}$

(vi) When moving source crosses a Observer :

The Apparent frequency of source n' is given by
n’ = n $\frac{v}{v + v_{o}}$and Apparent frequency n'' is
n’’ = n $\frac{v}{v + v_{s}}$then
$\delta$ n = 2 n $\frac{v_{s}}{v}$
Here,
vs = Velocity of the Source,
vo = Velocity of the Observer,
v = Velocity of sound or light in medium,
n = Real frequency,
n' = Apparent frequency.

The above formulas are also called Doppler Formula.

Relativistic Doppler Effect

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Relativistic Doppler Effect describes the changes of frequency of light wave when the source and observer are in relative motion. This can be explained using special theory of relativity. It is different from non relativistic Doppler Effect and includes the parameter of time dilation. The medium of propagation is not involved as a reference point.

The Relativistic Doppler effect is given by
T = $\frac{1}{1 – b}$ fsHere b = $\frac{v}{c}$ 

The velocity of observer as considered in terms of speed of light.
Due to this time dilation, the time measured by the observer would be
To = $\frac{t}{y}$
So, the Doppler shift is given as,y = $\frac{1}{\sqrt{1 – b^{2}}}$

Doppler Effect Simulation

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Simulation is the imitation of the operation of a real world process or system over time. The basic goal in the simulation of the Doppler Effect is the simulation if the frequency shift caused by the source or observer. It can be simulated using java applet.

Doppler Effect Example

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The Examples of Doppler effect are:
  1. Pitch is that feeling which produces a sensation of sound being sharp or thick.
  2. The velocity of aeroplane in air.
  3. The apparent change in frequency n of the electromagnetic signal, sent from a rocket approaching the moon, being received by rocket driver reflected by moon is an example of Doppler Effect.

Doppler Effect Problems

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These are the few problems on Doppler Effect in light and sound which are given below:

Solved Example

Question:
A Submarine travels through water at a speed of 8.00 m/s, emitting a sonar wave at a frequency of 1400 Hz. The speed of sound in the water is 530 m/s. A second submarine is located such that both submarines are traveling directly towards each other. The second submarine is moving at 6.00 m/s.
(A) What is the frequency detected by an observer riding on second submarine as the first submarine approaches it?
(B) The submarine barely miss each other and pass. What frequency is detected by an observer riding on second submarine as the subs recede from it?
(C) When both the submarine approaches each other, some of the sound from first submarine reflects from second submarine and returns to it. If the sound were to be detected by an first submarine, what is its frequency?

Solution:
 
Given: Frequency f = 1400 Hz,
Velocity of Source, Vs = 8 m/s,
Velocity of Screen, Vo = 6 m/s.
(A) The Apparent frequency is given by,
  f ' = $\frac{v + v_{0}}{v - v_{s}}$
  f ' = $\frac{530 m/s + (+6.00)m/s}{530 m/s - (+8.00 m/s)}$ (1400 Hz)
  f ' = 1437.54 Hz.

(B) To find Doppler-shifted frequency heard by the observer in second submarine:
      f ' = $\frac{v + v_{0}}{v - v_{s}}$
      f ' = $\frac{530 m/s + 6 m/s}{530 m/s - (-8 m/s)}$ (1400 Hz)
      f ' = 1394.79 Hz.

(C)The Sound of apparent frequency 1436.5 Hz found in part (A) is reflected from a moving source (sub B) and detected by a moving observer
     f = $\frac{530 m/s + (+ 8 m/s)}{530 m/s - (+ 9 m/s)}$ (1416 Hz)
     f = 1432 Hz.
 

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