### Doppler effect

is the apparent change in the frequency of a wave source due to the relative motion between the wave source and observer.

Let $$f^\prime$$ = observed frequency

$$f$$ = actual wave frequency

$$v$$ = speed/velocity of waves in medium (fixed by medium and its properties)

$$u_s$$ = speed/velocity of wave source

$$u_0$$ = speed/velocity of observer

For a moving source - stationary observer

(change in observed wavelength of source)

$$f^\prime = f \left(\frac{v}{v\pm u_s}\right)$$

Use $$\pm \rightarrow -$$, $$f^\prime > f, \lambda^\prime < \lambda $$ (decrease in wavelength), if wave source moves towards observer

Use $$\pm \rightarrow +$$, $$f^\prime < f, \lambda^\prime > \lambda$$ (increase in wavelength), if wave source moves away from observer

For a moving observer - stationary source

(change in relative speed of waves)

$$f^\prime = f \left(\frac{v \pm u_0}{v}\right)$$

Use $$\pm \rightarrow +$$, $$f^\prime > f$$, increase in relative speed/velocity of waves ($$v + u_0$$), if observer moves towards wave source.

Use $$\pm \rightarrow -$$, $$f^\prime < f$$, decrease in relative speed/velocity of waves ($$v - u_0$$), if observer moves away from wave source.

For electromagnetic waves

$$\Delta f = \frac{v}{c} f$$

where v is the speed of the wave source and c is the speed of electromagnetic waves in a vacuum.