Moving Source and Moving Receiver

This result is just the product of the two above - moving source causes one shift and moving receiver causes another to get:

$$f' = f_0 \frac{1 \mp \frac{v_r}{v_a}}{1 \pm \frac{v_s}{v_a}}$$ (179)

where in both cases relative approach shifts the frequency up and relative recession shifts the frequency down.

I do not recommend memorizing these equations - I don't have them memorized myself. It is very easy to confuse the forms for source and receiver, and the derivations take a few seconds and are likely worth points in and of themselves. If you're going to memorize anything, memorize the derivation (a process I call ``learning'', as opposed to ``memorizing''). In fact, this is excellent advice for 90% of the material you learn in this course!