Traditionally, the Doppler formula is derived by calculating
the wavelength measured by the receiver if either the receiver
or emitter is moving, Fig. 1 shows the wave fronts of
a moving emitter. Given the wavelength k and the phase
speed up, the frequency is calculated from f ¼ up=k (see, for
example, Sec. 15–5 of Ref. 4). An alternate approach is to
use coordinate transformations to change from a frame of
reference (coordinate system) in which the emitter is at rest
to one in which the receiver is at rest, and calculate the corresponding
received frequency. This second approach is the
way in which the relativistic Doppler effect for light is calculated
(see, for example, Sec. 10.2 of Ref. 5). Some authors
have shown how to apply the Lorentz transformations to
acoustic signals to obtain the generalized r