Doppler Effect

If a sound source moves with the speed $v$ towards an observer, the wavelength of the emitted sound changes according to $$\lambda' = \lambda - \Delta \lambda$$ Die difference $\Delta \lambda$ in the wavelength is given as the fraction of $v$ and $c$ multiplied with the wavelength: $$\lambda' = \lambda - \frac{v}{c}\lambda$$ The observer then measures the frequency $$f' = \frac{c}{\lambda'} = \frac{c}{\lambda - \frac{v}{c}\lambda}$$ With the help of $c=f\lambda$ this can be written as $$f' = \frac{f}{1-\frac{v}{c}}$$ If the source moves away from the observer, the negative speed $v$ has to be used. This leads to the following formula for the change in frequency for a moving sound source $$\boxed{f_\mathrm{S}' = \frac{f}{1\mp\frac{v}{c}}}$$ If the position of the source remains constant and the observer moves with the speed $v$ instead towards the source, the following frequency can be heard: $$f' = \frac{v + c}{\lambda}$$ Using again the formula $c = f\lambda$ leads to $$\boxed{f_\mathrm{O}' = f \left(1\pm\frac{v}{c}\right)}$$ Again, the negative speed has to be used for observers moving away from the source. If the source and the observer move towards each other, both equations can be combined: $$\boxed{f_\mathrm{OS}' = f\frac{c+v_\mathrm{O}}{c-v_\mathrm{S}}}$$ For a motion in opposite directions, the signs in front of $v$ have to be interchanged.