Doppler Effect Derivation

The Doppler effect is defined as the change in frequency or the wavelength of a wave with respect to an observer who is moving relative to the wave source. This phenomenon was described by the Austrian physicist Christian Doppler in 1842. 

The Doppler effect finds applications in sirens used in emergency vehicles that have a varying pitch in order to reach the observer. It is used in radars to measure the velocity of detected objects. The derivation of the Doppler effect is given below. Let us learn the Doppler effect derivation with respect to moving source and stationary observer where the wave travels with the source and moving observer and a stationary source.

Step-By-Step Derivation of Doppler Effect

In order to derive the Doppler effect, there are two situations that needs to considered, and they are:

Moving Source and Stationary Observer Where the Wave Travels with the Source

\(\begin{array}{l}c=\frac{\lambda _{s}}{T} \,\, \textup{(wave velocity)}\end{array} \)

Where,

c: wave velocity

λs: wavelength of the source

T: time taken by the wave

\(\begin{array}{l}T=\frac{\lambda _{s}}{c}\,\, \textup{(after solving for T)}\end{array} \)

\(\begin{array}{l}d=v_{s}T \,\, \textup{(representation of distance between source and stationary observer)}\end{array} \)

Where,

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

d: distance covered by the source

\(\begin{array}{l}\lambda _{0}=\lambda _{s}-d \,\, \textup{(observed wavelength)}\end{array} \)

\(\begin{array}{l}T=\frac{\lambda _{s}}{c}\end{array} \)

\(\begin{array}{l}d=\frac{v_{s}\lambda _{s}}{c} \,\, \textup{(substituting for T and using the equation of d)}\end{array} \)

\(\begin{array}{l}\lambda _{0}=\lambda _{s}-\frac{v_{s}\lambda _{s}}{c} \,\, \textup{(substituting for d)}\end{array} \)

\(\begin{array}{l}\lambda _{0}=\lambda _{s}(1-\frac{v_{s}}{c}) \,\, \textup{(factoring)}\end{array} \)

\(\begin{array}{l}\lambda _{0}=\lambda _{s}(\frac{c-v_{s}}{c})\end{array} \)

\(\begin{array}{l}\Delta \lambda =\lambda _{s}-\lambda _{0}\end{array} \)

\(\begin{array}{l}\lambda _{0}=\lambda _{s}-d\end{array} \)

\(\begin{array}{l}\Delta \lambda =\lambda _{s}-(\lambda _{s}-d)\end{array} \)

\(\begin{array}{l}\Delta \lambda =(\lambda _{s}-\frac{v_{s}\lambda _{s}}{c})\end{array} \)

\(\begin{array}{l}\Delta \lambda =(\frac{v_{s}\lambda _{s}}{c})\end{array} \)

\(\begin{array}{l}∴ \lambda _{0}=\frac{\lambda _{s}(c-v_{s})}{c}\end{array} \)

\(\begin{array}{l}\Delta \lambda =\frac{\lambda _{s}v_{s}}{c}\end{array} \)

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Moving Observer and a Stationary Source

\(\begin{array}{l}f_{0}=\frac{c-v_{0}}{\lambda _{s}}\end{array} \)

Where,

f₀: observed frequency

v₀: observer velocity

\(\begin{array}{l}f_{0}=\frac{c}{\lambda _{0}}\end{array} \)

\(\begin{array}{l}∴ \frac{c}{\lambda _{0}}=\frac{c-v_{0}}{\lambda _{s}}\end{array} \)

\(\begin{array}{l}\frac{\lambda _{0}}{c}=\frac{\lambda _{s}}{(c-v_{0})}\end{array} \)

\(\begin{array}{l}\lambda _{0}=\frac{\lambda _{s}c}{(c-v_{0})}\end{array} \)

\(\begin{array}{l}\lambda _{0}=\frac{\lambda _{s}}{(\frac{c-v_{0}}{c})}\end{array} \)

\(\begin{array}{l}\lambda _{0}=\frac{\lambda _{s}c}{c-v_{0}} \,\, \textup{(multiplying c)}\end{array} \)

\(\begin{array}{l}\lambda _{0}=\frac{\lambda _{s}}{1-\frac{v_{0}}{c}}\end{array} \)

\(\begin{array}{l}\Delta \lambda =\lambda _{s}-\lambda _{0} \,\, \textup{(change in wavelength)}\end{array} \)

\(\begin{array}{l}\Delta \lambda =\lambda _{s}-\frac{\lambda _{s}c}{c-v_{0}}\,\,(\textup{substituting for}\,\, λ_0)\end{array} \)

\(\begin{array}{l}\Delta \lambda =\frac{(\lambda _{s}(c-v_{0})-\lambda _{s}c)}{c-v_{0}}\end{array} \)

\(\begin{array}{l}\Delta \lambda =-\frac{\lambda _{s}v_{0}}{c-v_{0}}\end{array} \)

\(\begin{array}{l}∴ \lambda _{0}=\frac{\lambda _{s}c}{c-v_{0}}\end{array} \)

\(\begin{array}{l}\Delta \lambda =\frac{-\lambda _{s}v_{0}}{c-v_{0}}\end{array} \)

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