## 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 is the radian frequency emitted by the source at rest, is the frequency received by the listener, denotes the

*speed*of the listener relative to the propagation medium in the direction of the source, denotes the speed of the source relative to the propagation medium in the direction of the listener, and denotes sound speed. Note that all quantities in this formula are scalars.

### 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

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