Often we have noticed that the siren sound coming from a police vehicle or ambulance increases when it comes closer to us and decreases when it moves away from us.

**DOPPLER
EFFECT**

Often
we have noticed that the siren sound coming from a police vehicle or ambulance
increases when it comes closer to us and decreases when it moves away from us.
When we stand near any passing train the train whistle initially increases and
then it will decrease. This is known as Doppler Effect, named after Christian
Doppler (1803 â€“ 1853). Suppose a source produces sound with some frequency, we
call it the as source frequency *f _{s}*.
If the source and an observer are at a fixed distance then the observer
observes the sound with frequency

What
happens if either source or an observer or both move?. Certainly, *f _{o}* â‰

**(a) Source and Observer: **We can consider** **either the source or observer in motion or both are in motion.
Further we can treat the motion to be along the line joining the source and the
observer, or inclined at an angle Î¸ to this line.

**(b) Medium: **We can treat the medium to** **be stationary or the direction of
motion of the medium is along or opposite to the direction of propagation of
sound.

**(c) Speed of Sound: **We can also consider** **the case where speed of the source or
an observer is greater or lesser than the speed of sound.

In
the following section, we make the following assumptions: the medium is
stationary, and motion is along the line joining the source and the observer,
and the speeds of the source and the observer are both less than the speed of
sound in that medium.

We
consider three cases:

(i)
Source in motion and Observer is at rest.

(a) Source moves towards
observer

(b) Source moves away from
the observer

(ii) Observer in motion and Source is at rest.

(a) Observer moves towards
Source

(b) Observer receding away
from the Source

(iii)
Both are in motion

(a) Source and Observer
approach each other

(b) Source and Observer
recede from
each other

(c) Source chases Observer

(d) Observer chases Source

*Stationary
observer ***and stationary source means the observer and source are both at rest with respect to medium
respectively**

Suppose
a source S moves to the right (as shown in Figure 11.46) with a velocity *v _{s}* and let the frequency of
the sound waves produced by the source be

Let
*Î»* be the wavelength of the source S
as measured by the observer when S is at position *x*_{1} and *Î»Í´* be
wavelength of the source observed by the observer when S moves to position *x*_{2}. Then the change in
wavelength is âˆ†*Î» = Î»*âˆ’ *Î»Í´ = v _{s}t*, where

On
substituting equation (11.84) in equation (11.83), we get

Since
frequency is inversely proportional to wavelength, we have

Since,
v_{s}/v << 1, we use the binomial expansion and retaining only
first order in v_{s}/v , we get

Since
the velocity here of the source is opposite in direction when compared to case
(a), therefore, changing the sign of the velocity of the source in the above
case i.e, by substituting (*v _{s}*â†’ âˆ’

Let
us assume that the observer O moves towards the source S with velocity *v _{o}*. The source S is at rest
and the velocity of sound waves (with respect to the medium) produced by the
source is

If
the observer O is moving away (receding away) from the source S, then velocity *v*_{0} and *v *moves in the same direction. Therefore, their
relative velocity is *v _{r}* =

Let
*v _{s}* and

At
that instant of time, the true observer approaches the dummy from the other
side. Since the source (true source) comes in a direction opposite to true
observer, the dummy (source) is treated as stationary source for the true
observer at that instant. Hence, apparent frequency when the true observer
approaches the stationary source (dummy source), from equation (11.89) is

Since
this is true for any arbitrary time, therefore, comparing equation (11.91) and
equation (11.92), we get

Hence,
the apparent frequency as seen by the observer is

Here,
we can derive the result as in the previous case. Instead of a detailed
calculation, by inspection from Figure 11.49, we notice that the velocity of
the source and the observer each point in opposite directions with respect to
the case in (a) and hence, we substitute (*v _{s}*
â†’ âˆ’

Only
the observerâ€™s velocity is oppositely directed when compared to case (a).
Therefore, substituting (*v _{0}*
â†’ âˆ’

Only
the source velocity is oppositely directed when compared to case (a).
Therefore, substituting *v _{s}*
â†’ âˆ’

Doppler effect has many applications. Specifically Doppler effect in light has many applications in astronomy.

As an example, while observing the spectra from distant objects
like stars or galaxies, it is possible to determine the velocities at which
distant objects like stars or galaxies move towards or away from Earth. If the
spectral lines of the star are found to shift towards red end of the spectrum
(called as red shift) then the star is receding away from the Earth. Similarly,
if the spectral lines of the star are found to shift towards the blue end of
the spectrum (called as blue shift) then the star is approaching Earth.

Let
âˆ†*Î» *be the Doppler shift. Then âˆ†*Î»= v*/*c*
*Î», *where *v* is the velocity of the star. It may be noted that Doppler shift
measures only the radial component (along the line of sight) of the relative
velocity *v*.

A
sound of frequency 1500 Hz is emitted by a source which moves away from an
observer and moves towards a cliff at a speed of 6 ms^{-1}.

(a)
Calculate the frequency of the sound which is coming directly from the source.

(b)
Compute the frequency of sound heard by the observer reflected off the cliff.
Assume the speed of sound in air is 330 mâ€†s^{-1}

(a)
Source is moving away and observer is stationary, therefore, the frequency of
sound heard directly from source is

(b)
Sound is reflected from the cliff and reaches observer, therefore,

An
observer observes two moving trains, one reaching the station and other leaving
the station with equal speeds of 8 mâ€†s^{âˆ’1}. If each train sounds its
whistles with frequency 240â€†Hz, then calculate the number of beats heard by the
observer.

Observer
is stationary

(i)
Source (train) is moving towards an observer:

Apparent
frequency due to train arriving station is

(ii)
Source (train) is moving away form an observer:

Apparent
frequency due to train leaving station is

So
the number of beats = | *f*_{in}
-*f*_{out}| = (246-234) = 12

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11th Physics : UNIT 11 : Waves : Doppler Effect |

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