Doppler Effect Derivation

Doppler Effect Definition

In 1842, Austrian physicist Christian Doppler discovered that frequency of wavelengths tends to change with the movement of wave source in relation to an observer. If that sounds too complex, in convenient terms you can ask what is the Doppler Effect simple explanation?

Doppler Effect is the increase or decrease in light, sound or other waves when the source and observer move towards or away from each other. This effect gives rise to not just a crucial theory of physics but also helps in mathematical calculation of waves and their frequencies. Before proceeding to Doppler Effect derivation, let us learn more about it through some examples.

Example 1

Suppose a frog sits in the middle of a lake. It is moving its leg in a way to cause ripples or waves on this water’s surface. These waves arise from this frog’s position and move outward toward the edges of this lake in concentric circles. Two observers, ‘A’ and ‘B’ are standing at the left and right sides of this lake, respectively. 

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Figure 1.0 The circles in this image represent the waves moving outward from the frog’s position.

At the position above, both observers will find that the waves reach them at similar frequencies, considering that the frog is equidistant from them. However, the frequency of waves, observable to ‘A’ and ‘B’ will start differing as soon as the frog moves toward observer B.

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Figure 1.1 The waveform changes for both observers with the movement of the frog (source of the waves).

At this position, wave frequency for observer B is higher than it is for observer A. B experiences higher frequency because the wave source moves toward it. Similarly, ‘A’ observes lower wave frequency as the wave source moves away from it. This is what the Doppler Effect defines.

Doppler Effect Derivation Class 11 for Moving Source and Stationary Observer

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Figure 2.0 Wave source moving toward an observer.

Wave velocity c = λS/T

• In this equation, λS defines the source’s wavelength.

• c is the wave velocity

• T is the time for the wave to move one wavelength distance.

For the Doppler Effect derivation, we can say that

T = λS/c (eq.1)

Now, consider that the source is moving with velocity ‘vS’ towards the stationary observer. In T time, this source can travel d distance,

Thus, d = vST (eq.2)

Suppose the source moves in direction x, and due to the shortening wavelength, λO is the wavelength reaching the observer.

λO = λS – d (eq.3)

Now, substitute the value of T from eq.1 into eq.2.

d = vSλS/c

Substitute this value of d into eq.3

λO = λS – (vSλS/c)

λO = λS (1 – vS/c)

Keep in mind that the sign of vS changes as the source moves away from the observer. Everything else in this formula remains the same.

Example 2

Consider that observer A is riding a bike and moving away from a stationary ambulance whose siren is switched on. At first, pitch and sound of its siren are different to observer A when he is closer to the sound’s source. As he moves away, sound and pitch changes, thanks to the Doppler Effect.

Notice that in this example, source of wave remains stationary, but an observer moves away from it.

Doppler Effect Derivation for Moving Observer and Stationary Source

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Determining observed frequency is easy since it is the combination of observer velocity and wave velocity divided by actual wavelength. 

fO = (c – vO)/λS

fO refers to the frequency observed, and vO is the velocity of the observer.

But, fO = c/λO

Thus, c/λO = (c – vO)/λS

Now, reciprocate both sides, we get

λO/c = λS/(c – vO)

Multiplying by c,

λO = λS/[(c − vO)/c]

Thus, λO = λSc/(c − vO)

If you need further assistance, consult our derivation of Doppler Effect equations pdf, available online. You can also attend our live classes online and get ahead on your preparations. Now you can even download our Vedantu app for further convenience.