Variables:
v: Source's velocity (Mach #)
w: Wavelengths
f: Frequencies
Subscripts:
a: Arrival
d: Departure
s: Source
1: These are the two wavelengths measured from the base frequency and source velocity.
2: Two equations with equal values – we can drop Wb.
3: Everything flipped over, Fb is just for show.
4: Convert the other wavelengths to frequencies as we have that data.
5: Expanding so that …
6: … we can collect the velocity terms on the left side.
7: Factoring that so we can …
… reach the elegant and useful form.
At low frequency shifts, fa – fd is low, and we get the low velocity. High frequency shifts make fa dominate and we converge on fa / fa, or 1, the speed of sound.