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 [#!Chowning71!#], and it is an important component of the ``Leslie effect'' (described in §).

As derived in elementary physics texts, the Doppler shift is given by

$$\omega_l = \omega_s \frac{1+\frac{v_{ls}}{c}}{1-\frac{v_{sl}}{c}} \protect$$ (6)

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 $\underline{x}\mathrel{\stackrel{\Delta}{=}}(x_1,x_2,x_3)^T$ is 3D position. We have velocity related to position by

$$\underline{v}_s= \frac{d}{dt}\underline{x}_s(t) \qquad \underline{v}_l= \frac{d}{dt}\underline{x}_l(t). \protect$$ (7)

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.

Velocity Projection

The Doppler effect depends only on velocity components along the line connecting the source and listener [#!Pierce!#, 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.1.1 for a specific example.)

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

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

Therefore, we can write the projected source velocity as

$$\underline{v}_{sl}= {\cal P}_{\underline{x}_{sl}}(\underline{v}_s... ...line{x}_s\,\right\Vert^2}\left(\underline{x}_l-\underline{x}_s\right). \protect$$ (8)

In the far field (listener far away), Eq.$\,$(8) reduces to

$$\underline{v}_{sl} \approx \frac{\left<\underline{v}_s,\underline... ...derline{x}_l\,\right\Vert\gg\left\Vert\,\underline{x}_s\,\right\Vert). \protect$$ (9)