Rotating Horn Doppler Shift, Cont'd

Substituting into the Doppler formula with the listener velocity $v_l$ set to zero yields

$$\omega_l = \frac{\omega_s }{1+r_s\omega_m\sin(\omega_m t)/c} \approx \omega_s \left[1-\frac{r_s\omega_m}{c}\sin(\omega_m t)\right], \protect$$

where the approximation is valid for small Doppler shifts.

Thus, in the far field, a rotating horn causes an approximately sinusoidal multiplicative frequency shift:

Doppler shift = horn length $r_s$ times horn angular velocity $\omega_m$ divided by sound speed $c$

Note that $r_s\omega_m$ is the tangential speed of the assumed point of horn radiation.