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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

$\displaystyle \omega_l = \omega_s \frac{1+\frac{v_{ls}}{c}}{1-\frac{v_{sl}}{c}} \protect$ (6.2)

where $ \omega_s $ is the radian frequency emitted by the source at rest, $ \omega_l $ is the frequency received by the listener, $ v_{ls}$ denotes the speed of the listener relative to the propagation medium in the direction of the source, $ v_{sl}$ denotes the speed of the source relative to the propagation medium in the direction of the listener, and $ c$ denotes sound speed. Note that all quantities in this formula are scalars.

Vector Formulation

Denote the sound-source velocity by $ \underline{v}_s(t)$ where $ t$ is time. Similarly, let $ \underline{v}_l(t)$ denote the velocity of the listener, if any. The position of source and listener are denoted $ \underline{x}_s(t)$ and $ \underline{x}_l(t)$, respectively, where $ \underline{x}\isdef (x_1,x_2,x_3)^T$ is 3D position. We have velocity related to position by

$\displaystyle \underline{v}_s= \frac{d}{dt}\underline{x}_s(t) \qquad \underline{v}_l= \frac{d}{dt}\underline{x}_l(t). \protect$ (6.3)

Consider a Fourier component of the source at frequency $ \omega_s $. We wish to know how this frequency is shifted to $ \omega_l $ 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 $ \underline{x}_{sl}=\underline{x}_l-\underline{x}_s$ pointing from the source to the listener. (See Fig.5.8 for a specific example.)

The orthogonal projection of a vector $ \underline{x}$ onto a vector $ {\underline{y}}$ is given by [451]

$\displaystyle {\cal P}_{\underline{y}}(\underline{x}) = \frac{\left<\underline{...
...derline{x}^T{\underline{y}}}{{\underline{y}}^T{\underline{y}}}{\underline{y}}.
$

Therefore, we can write the projected source velocity as

$\displaystyle \underline{v}_{sl}= {\cal P}_{\underline{x}_{sl}}(\underline{v}_s...
...line{x}_s\,\right\Vert^2}\left(\underline{x}_l-\underline{x}_s\right). \protect$ (6.4)

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

$\displaystyle \underline{v}_{sl} \approx \frac{\left<\underline{v}_s,\underline...
...derline{x}_l\,\right\Vert\gg\left\Vert\,\underline{x}_s\,\right\Vert). \protect$ (6.5)


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