## 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

 (6.2)

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

 (6.3)

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]

Therefore, we can write the projected source velocity as

 (6.4)

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

 (6.5)

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