Dopper effect

A wave source travels at velocity $v_s$ while it emits waves at frequency $f_s$. The waves propagate at a velocity $c$. The wavefronts are closer together in front of the source and they are farther apart behind the source. An observer at the red dot travels at a velocity $v_o$ and observes a frequency of $f_o$. All of the wavefronts in the simulation are traveling at the same speed so where the wavefronts are closer together, the observer hears a higher frequency and where the wavefronts are farther apart, an observer hears a lower frequency. This is the Doppler effect.

If the source and the observer are approaching each other, $f_o > f_s$, and if the source and the observer are moving away from each other, $f_o < f_s$. If source speed, the observer speed, and the wave speed are all much less than the speed of light, and assuming that both the source and the observer are moving at constant velocities, the frequency at the observer is given by,

Here $x_o$ is the position of the observer, $x_s$ is the position of the source, and both $v_o$ and $v_s$ can be positive or negative.

If the waves that are emitted are light waves, then Einstein's theory of special relativity must be used and only the relative motion is important. The Doppler shift is given by,

Where $v$ is the relative bewteen the source and the observer. If they are approaching each other, $v < 0$, and if they are moving away from each other, $v > 0$. In this case, $c$ is the speed of light.