 # Doppler Effect Derivation

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 the year 1842.

It finds applications in sirens used in emergency vehicles that have a varying pitch in order to reach the observer, in radars to measure the velocity of detected objects. The derivation of Doppler Effect is given below.

## 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

$$c=\frac{\lambda _{s}}{T}$$ (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 source and stationary observer)

Where,

vs: velocity with which source is moving towards 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)

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

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

$$\Delta \lambda =\lambda _{s}-\lambda _{0}$$

$$\lambda _{0}=\lambda _{s}-d$$

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

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

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

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

$$\Delta \lambda =\frac{\lambda _{s}v_{s}}{c}$$

### Moving observer and a stationary source

$$f_{0}=\frac{c-v_{0}}{\lambda _{s}}$$

Where,

f0: observed frequency

v0: observer velocity

$$f_{0}=\frac{c}{\lambda _{0}}$$

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

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

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

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

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

$$\lambda _{0}=\frac{\lambda _{s}}{1-\frac{v_{0}}{c}}$$

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

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

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

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

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

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

Above are the steps of Doppler effect derivation. To know more, stay tuned with BYJU’S.

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