Sound

Doppler Effect

The Doppler effect refers to an alteration in a sound’s observed frequency. Furthermore, this alteration happens because of either an observer or the source. Furthermore, one can easily notice this effect for a moving observer and a stationary source.

Introduction to Doppler Effect

Change can take place in a sound that a listener hears in case the listener and the sound’s source move relative to each other. This is what is known as the Doppler Effect.

As the listener and the source move closer to each other, the frequency heard will become higher in comparison to the frequency of the emitted sound. In contrast, as the listener and the source move away from each other, the frequency heard will become lower in comparison to the frequency of the source’s sound.

How Do We Measure Doppler Effect?

Suppose that in the centre of circular water puddle is a happy bug. Furthermore, the bug periodically shakes its legs to create disturbances that travel through the water. Moreover, these disturbances would travel outward from the point of origin in all directions.

Each disturbance travels in the same medium.  Consequently, they would all travel at the same speed in every direction. Moreover, the pattern whose production takes place by the bug’s shaking would be in the form of a series of concentric circles.

The circles would reach the water puddle’s edges while the frequency would remain the same. An observer at point A (the left edge of the puddle) would witness the disturbances that strike the edge of the puddle, while the frequency remains the same. This in turn would be observed by an observer at point B (at the right edge of the puddle).

An important point to note here is that the frequency at which disturbances make it to the edge of the puddle would be the same as the frequency at which the production of disturbances takes place by the bug. Suppose a production of disturbances by the bug takes place at a frequency of 2 per second, then the observer would find them approaching at 2 per second frequency.

Now suppose that our bug moves right across the water puddle, thereby creating disturbances at 2 disturbances per second frequency. Furthermore, since the movement of the bug is towards the right, the origination of each consecutive disturbance is from a position that is farther from observer A and closer to observer B. Subsequently, each consecutive disturbance has to travel a distance that is shorter before making it to observer B and thus requires less amount of time to make it to observer B.

As can be seen, observer B makes an observance that the frequency of arrival of the disturbances is higher in comparison to the frequency at which the production of disturbances takes place. On the other hand, each consecutive disturbance would travel for a further distance before making it to observer A. For this particular reason, the observance made by observer A would be of a frequency of arrival that is less in comparison to the frequency at which the production of disturbances takes place.

The net effect of the bug’s motion (the source of waves) would be such that the observer, towards whom the movement of the bug takes place, makes observance of a frequency higher than 2 disturbances/second.  Furthermore, the observer that is away, from whom the movement of the bug takes place, makes an observance of a frequency lesser than 2 disturbances/second. Most importantly, this is known as the Doppler Effect.

Formula of Doppler Effect

For Doppler Effect formula, experts usually write the unit of sound frequency as Hertz (), where one Hertz happens to be a cycle per second ().

Sound frequency that a listener heard

= speed of sound + listener velocity/speed of sound + source velocity (sound frequency emitted by source)

\(f_{L}= \frac{v+v_{L}}{v+v_{S}}f_{S}\)

fL = frequency of sound that a listener hears (, or )

v = speed of sound that is present in the medium (m/s)

vL = listener’s velocity (m/s)

vs = velocity of the source’s sound (m/s)

fs = frequency of sound that the source emits (, or )

Derivation of the Formula of Doppler Effect

The procedure of Doppler Effect derivation is as follows

c=\(\frac{\lambda _{s}}{T} (wave\ velocity)\)

Where,

c: wave velocity

λs: wavelength of the source

T: time taken by the wave

T=\(\frac{\lambda _{s}}{c} (after\ solving\ for\ T)\)

d=\(v_{s}T (representation\ of\ distance\ between\ the\ source\ and\ stationary\ observer)\)

Where,

vs: velocity with which source is moving towards a stationary observer

d: distance covered by the source

\(\lambda _{0}=\lambda _{s}-d (observed\ wavelength)\)

T=\(\frac{\lambda _{s}}{c} (after\ solving\ for\ T)\)

d=\(v_{s}T (representation\ of\ distance\ between\ the\ source\ and\ stationary\ observer)\)

Where,

vs: velocity with which source is moving towards a stationary observer

d: distance covered by the source

\(\lambda _{0}=\lambda _{s}-d (observed\ wavelength)\)

T=\(\frac{\lambda _{s}}{c}\)

d=\(\frac{v_{s}\lambda _{s}}{c} (substituting\ for\ T\ and\ using\ the\ equation\ of\ d)\)

\(\lambda _{0}=\lambda _{s}-\frac{v_{s}\lambda _{s}}{c} (substituting\ for\ d)\)

Moreover, \(\lambda _{0}=\lambda _{s}(1-\frac{v_{s}}{c}) (factoring)\)

Furthermore, \(\lambda _{0}=\lambda _{s}(\frac{c-v_{s}}{c})\)

\(\Delta \lambda =\lambda _{s}-\lambda _{0}\)

Furthermore,\(\lambda _{0}=\lambda _{s}-d\)

Moreover, \(\Delta \lambda =\lambda _{s}-(\lambda _{s}-d)\)

\(\Delta \lambda =(\lambda _{s}-\frac{v_{s}\lambda _{s}}{c})\)

\(\Delta \lambda =(\frac{v_{s}\lambda _{s}}{c})\)

∴ \(\lambda _{0}=\frac{\lambda _{s}(c-v_{s})}{c}\)

\(\Delta \lambda =\frac{\lambda _{s}v_{s}}{c}\)

Moving observer and a stationary sourced

f0: observed frequency

v0: observer velocity

\(f_{0}=\frac{c}{\lambda _{0}}\)

∴ \(\frac{c}{\lambda _{0}}=\frac{c-v_{0}}{\lambda _{s}}\)

\(\frac{\lambda _{0}}{c}=\frac{\lambda _{s}}{(c-v_{0})}\)

\(\lambda _{0}=\frac{\lambda _{s}c}{(c-v_{0})}\)

\(\lambda _{0}=\frac{\lambda _{s}}{(\frac{c-v_{0}}{c})}\)

Furthermore, \(\lambda _{0}=\frac{\lambda _{s}c}{c-v_{0}} (multiplying\ c)\)

Moreover,\(\lambda _{0}=\frac{\lambda _{s}}{1-\frac{v_{0}}{c}}\)

\(\Delta \lambda =\lambda _{s}-\lambda _{0} (change\ in\ wavelength)\)

\(\Delta \lambda =\lambda _{s}-\frac{\lambda _{s}c}{c-v_{0}} (substituting\ for\ λ0)\)

Furthemore, \(\Delta \lambda =\frac{(\lambda _{s}(c-v_{0})-\lambda _{s}c)}{c-v_{0}}\)

Moreover,\(\Delta \lambda =-\frac{\lambda _{s}v_{0}}{c-v_{0}}\)

∴ \(\lambda _{0}=\frac{\lambda _{s}c}{c-v_{0}}\)

\(\Delta \lambda =\frac{-\lambda _{s}v_{0}}{c-v_{0}}\)

FAQs for Doppler Effect

Question 1: What is meant by Doppler Effect?

Answer 1: The Doppler Effect is an alteration which takes place in a sound’s observed frequency. Moreover, the reason for this alteration is either an observer or the source. Furthermore, it is easy to notice this effect for a moving observer and a stationary source.

Question 2: Give a real-life example of Doppler Effect?

Answer 2: A real-life example of Doppler Effect can be of a police car or emergency vehicle that travels towards an individual standing on a road. As the car approaches with its siren sound, the siren sound’s pitch is high. Afterwards, when the car passes by, the siren sound’s pitch is low.

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