Velocity Projection

The Doppler effect depends only on velocity components along the line connecting the source and listener.

We may therefore orthogonally project the source and listener velocities onto the vector $\underline{x}_{sl}=\underline{x}_l-\underline{x}_s$ pointing from the source to the listener.

Projected source velocity:

$$\begin{eqnarray*} \underline{v}_{sl}&=& {\cal P}_{\underline{x}_{sl}}(\underline... ...{x}_l\,\right\Vert\gg\left\Vert\,\underline{x}_s\,\right\Vert$)} \end{eqnarray*}$$