Rotating Horn Doppler Shift

Relevant geometry for a rotating horn

Source velocity projected onto source-listener path:

$$\underline{v}_{sl}= {\cal P}_{\underline{x}_{sl}}(\underline{v}_s) = \frac{\left<\underline{v}_s,\underline{x}_l\right>}{\left\Vert\,\underline{x}_l-\underline{x}_s\,\right\Vert^2}\left(\underline{x}_l-\underline{x}_s\right). \protect$$

Choosing $\underline{x}_l=(r_l,0)$ yields

$$\underline{v}_{sl}= \frac{-r_l r_s\omega_m\sin(\omega_m t)}{r_l^2 + 2r_l r_s\cos(\omega_m t)+r_s^2} \left[\begin{array}{c} r_l-r_s\cos(\omega_m t) \\ [2pt] -r_s\sin(\omega_m)t \end{array}\right]. \protect$$

Far field approximation:

$$\underline{v}_{sl}\approx -r_s\omega_m\sin(\omega_m t) \left[\begin{array}{c} 1 \\ [2pt] 0 \end{array}\right]. \protect$$