Denote the sound-source velocity by
where
is time. Similarly,
let
denote the velocity of the listener, if any. The
position of source and listener are denoted
and
, respectively, where
is 3D
position. We have velocity related to position by
The Doppler effect depends only on velocity components along the line
connecting the source and listener [352, p. 453]. We may
therefore orthogonally project the source and listener
velocities onto the vector
pointing from the source
to the listener. (See Fig.5.8 for a specific example.)
The orthogonal projection of a vector
onto a vector
is given by [454]
Therefore, we can write the projected source velocity as