Both Source and Observer Moving

When both the source and the observer are simultaneously in motion along the line joining them, the two corrections combine into the fully general Doppler formula introduced above, $f' = f_0\,\frac{v+v_o}{v-v_s}$ with each of $v_s$ and $v_o$ carrying its own independent sign according to the convention already given: $v_s$ positive when the source moves toward the observer, negative when it moves away; $v_o$ positive when the observer moves toward the source, negative when it moves away. Four combinations follow directly from this one formula. Both approaching each other (source toward observer, observer toward source): both $v_s$ and $v_o$ are positive, so both the numerator grows and the denominator shrinks, giving the largest possible rise in apparent frequency for given speeds. Both receding from each other: both $v_s$ and $v_o$ are negative, so both the numerator shrinks and the denominator grows, giving the largest possible fall in apparent frequency. Source approaching while observer recedes (they are moving in the same sense, source chasing observer, source behind): $v_s$ positive, $v_o$ negative, and the two effects work in opposite directions, partially cancelling. Source receding while observer approaches: $v_s$ negative, $v_o$ positive, again partially cancelling but in the opposite sense. It is exactly this general, both-moving formula that governs the familiar rapid downward pitch "swoop" heard when a fast train passes very close to a listener who is also moving (for instance standing on a moving platform trolley, or in another t …