### 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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Doppler Simulation via Delay Lines
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Summary of Flanging