### Rotating Horn Simulation

The heart of the Leslie effect is a rotating horn loudspeaker. The rotating horn from a Model 600 Leslie can be seen mounted on a microphone stand in Fig.5.7. Two horns are apparent, but one is a dummy, serving mainly to cancel the centrifugal force of the other during rotation. The Model 44W horn is identical to that of the Model 600, and evidently standard across all Leslie models [189]. For a circularly rotating horn, the source position can be approximated as

where is the circular radius and is angular velocity. This expression ignores any

*directionality*of the horn radiation, and approximates the horn as an omnidirectional radiator located at the same radius for all frequencies. In the Leslie, a

*conical diffuser*is inserted into the end of the horn in order to make the radiation pattern closer to uniform [189], so the omnidirectional assumption is reasonably accurate.

^{6.10}

By Eq.(5.3), the source velocity for the circularly rotating horn is

Note that the source velocity vector is always orthogonal to the source position vector, as indicated in Fig.5.8.

Since and are orthogonal, the projected source velocity Eq.(5.4) simplifies to

Arbitrarily choosing (see Fig.5.8), and substituting Eq.(5.8) and Eq.(5.9) into Eq.(5.10) yields

In the far field, this reduces simply to

Substituting into the Doppler expression Eq.(5.2) with the listener velocity set to zero yields

where the approximation is valid for small Doppler shifts. Thus, in the far field, a rotating horn causes an approximately

*sinusoidal*multiplicative frequency shift, with the amplitude given by horn length times horn angular velocity divided by sound speed . Note that is the

*tangential speed*of the assumed point of horn radiation.

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Rotating Woofer-Port and Cabinet

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System Block Diagram