Doppler Effect
The Doppler effect causes the pitch of a sound source to appear to rise or fall due to motion of the source and/or listener relative to each other. You have probably heard the pitch of a horn drop lower as it passes by (e.g., from a moving train). As a pitched sound-source moves toward you, the pitch you hear is raised; as it moves away from you, the pitch is lowered. The Doppler effect has been used to enhance the realism of simulated moving sound sources for compositional purposes [80], and it is an important component of the ``Leslie effect'' (described in §5.9).
As derived in elementary physics texts, the Doppler shift is given by
where





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


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.

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Doppler Simulation
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