Moving Receiver

Now imagine that the source of waves at frequency $f_0$ is stationary but the receiver is moving towards the source. The source is thus surrounded by spherical wavefronts a distance $\lambda_0 = v_a T$ apart. At $t = 0$ the receiver crosses one of them. At a time $T'$ later, it has moved a distance $d = v_r T'$ in the direction of the source, and the wave from the source has moved a distance $D = v_a T'$ toward the receiver, and the receiver encounters the next wave front. That is:

$$\lambda_0 = d + D$$ (173)

$$= v_r T' + v_a T'$$ (174)

$$= (v_r + v_a) T'$$ (175)

$$v_a T = (v_r + v_a) T'$$ (176)

We use $f_0 = 1/T$, $f' = 1/T'$ (where $T'$ is the apparent time between wavefronts to the receiver) and rearrange this into:

$$f' = f_0 (1 + \frac{v_r}{v_a})$$ (177)

Again, if the receiver is moving away from the source, everything is the same but the sign of $v_r$, so one gets:

$$f' = f_0 (1 - \frac{v_r}{v_a})$$ (178)