Denote the sound-source *velocity* by
where
is 3D position and
is time. Similarly,
let
denote the velocity of the listener, if any. The
*position* of source and listener are denoted
and
, respectively. We have velocity related to position by

Consider a Fourier component of the source at frequency . We wish to know how this frequency is shifted to at the listener due to the Doppler effect.

The Doppler effect depends only on the relative motion between the
source and listener [12, p. 453]. We must therefore
*orthogonally project* the source and listener velocities onto the vector
pointing from the source to the listener. (See
Fig. for a specific example.)

The *orthogonal projection* of a vector
onto a vector
is given by [15]

Therefore, we can write the projected source velocity as

In the

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