Derivation of the Doppler Effect
There are two circumstances that must be taken into account to derive the Doppler effect:
Case 1: Moving Source and Stationary Observer
c = λs/T ————— (1)
Where “c” is the wave velocity,“λs” is the wavelength of the source, and
“T” is the time taken by the wave.
From equation (1),
T = λs/c ————— (2)
Let the distance between the source and the stationary observer be “d”, and the velocity with which the source is moving towards a stationary observer be “vs“.
d = vsT ————— (3)
By substituting the value of T in equation (3), we get,
d = vsλs/c ————— (4)
Now, the observed wavelength (λ0) will be equal to the difference between the wavelength of the source and the distance covered by the source, i.e.,
λ0 = λs – d ————— (5)
From equations (4) and (5) we get,
λ0 = λs – vsλs/c
λ0 = λs(1 – vs/c)
λ0 = λs[(c – vs)/c]
Make a note that the sign of the source velocity (vs) changes as the source moves away from the observer, but everything else in this formula remains the same.
Δλ (change in wavelength) = λs − λ0
We know that,
λ0 = λs – d
So, Δλ = λs − (λs − d)
Δλ = d = vsλs/c
∴ λ0 = λs(c−vs)/c
Δλ = vsλs/c
Case 2: Moving Observer and Stationary Source
f0 = (c − v0)/λs ————— (a)
Where “f0” is the observed frequency, and
“v0” is the velocity of the observer.
f0 = c/λ0 ————— (b)
From equations (a) and (b)
(c − v0)/λs = c/λ0
λs/ (c − v0) = λ0/c
λ0 = λsc/(c−v0)
λ0 = λs/ [(c−v0)/c]
λ0=λs/ [1− (v0/c)]
Δλ (change in wavelength) = λs − λ0
Now, by substituting the value of λ0, we get
Δλ = λs − [λsc/(c−v0)]
Δλ = (λs(c − v0) − λsc)/ (c − v0)
Δλ = −λsv0/ (c − v0)
∴ λ0 = λsc/(c−v0)
Δλ = −λsv0/ (c − v0)
Difference between Doppler Effect and Doppler Shift
Doppler effect or Doppler shift phenomenon was described in 1842 by an Austrian physicist, Christian Doppler, and it is named after him. The Doppler effect or Doppler shift is a change (increase or decrease) in the frequency of a wave as the source and the observer move (towards or away from) each other relative to the medium. Based on the direction of the source and the observer and the magnitudes of their velocities, the observed frequency can be less or more than the source frequency. For example, the pitch of the sound of an ambulance siren changes as it passes us. It happens because of the relative velocity between the source and the observer. When the ambulance is approaching us, the relative velocity is negative, and the relative velocity is positive when it is moving away. So, that is the reason behind the difference in the pitch of the sound of the ambulance siren while it is approaching and while it is moving away. The Doppler effect applies to all types of waves, including sound and light. The Doppler effect is responsible for the Blue Shift or Red Shift phenomenon observed in light waves. The Doppler effect is used in various fields such as radar, astronomy, satellite communication and navigation, medical imaging, etc.
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