### Vector Formulation

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

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 velocity components along the line
connecting the source and listener [349, 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 [451]

In the

*far field*(listener far away), Eq.(5.4) reduces to

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Doppler Simulation via Delay Lines

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Summary of Flanging