where v is the speed of sound, vD is the speed of the detector, and vS is the speed of
the source. The signs in the numerator and denominator depend on the directions in
which the source and detector are moving.
The most common mistake when using this equation has to do with whether or not
to use a + or a − sign. Fortunately, there’s an easier way to approach this—use
logic!
If a detector approaches a source or a source approaches a detector, the resulting
frequency would increase. Likewise, if the opposite occurred, the frequency would
decrease. In those terms, we really just need to figure out if we need to make the
fraction greater than 1 (increasing the frequency heard by the detector) or less than
1 (decreasing it).
Let’s illustrate this method with some situations.
Situation 1: You are standing on the street not moving, and an ambulance with its
siren is approaching you at the speed of vS. What is the frequency of the siren to
you?
Solution: We know the velocity of sound (v), vD = 0 (because you are not moving)
and we know vS. Because the ambulance is approaching you, we would expect the
frequency heard by you to increase. Do we add or subtract vS? In this case we want
to make our fractional portion greater than 1 so our resulting fD is greater than the
normal fS. Since vS is in the denominator, we want to subtract vS:
Situation 2: Once the ambulance passes you, what is the frequency of the siren to
you?
Solution: Once the siren passes, it is moving away. You would expect the
frequency heard by you to decrease. Do we add or subtract vS? In this case we want
to make our fractional portion less than 1 so our resulting fD is less than the normal
fS. Increasing the denominator will make the fractional portion less than 1, so we’ll
want to add vS instead of subtracting: