Abstract
The Doppler effect is a phenomenon inherent to source motion, which introduces a variable propagation time between the source and a listening point. In the case of a vibrating piston, this is responsible for distortion of the radiated sound pressure. This moving-boundary phenomenon is part of the nonlinear effects involved in loudspeaker radiation. The present paper investigates the significance of this distortion, usually considered as neglectible, and addresses its correction. First, the direct problem is solved by: (a) converting the (Lagrangian) position of the moving source into its equivalent (Eulerian) velocity field at a fixed position; (b) deriving the acoustic pressure radiated from this velocity field. A series solution of (a) is derived and time-domain simulations of (b) are built from the truncated series combined with a baffled piston radiation model. Simulations show that Doppler distortion can be significant for realistic loudspeaker diaphragm motion with a wide spectral content. Second, the inverse (anti-Doppler) problem is examined, that is, the derivation of a piston displacement that generates a targeted Eulerian velocity field. The corrected piston velocity solution proves to be an uncentered signal, leading to a diverging displacement. In order to remove this practical problem, a centered approximation is preferred, based on modified inverse Volterra kernels. The anti-Doppler algorithm is reliable in the audio range.
Highlights
When a loudspeaker diaphragm vibrates, its displacement modulates the propagation time of the generated acoustic waves, between its surface and a listening point
The distortion due to the Doppler effect is evaluated through the computation of the Eulerian source (22) and the on-axis pressure field (1) at z = 1 m
In the case of loudspeaker diaphragm motion, this effect is often considered as neglectable [6]
Summary
When a loudspeaker diaphragm vibrates, its displacement modulates the propagation time of the generated acoustic waves, between its surface and a listening point This moving-boundary phenomenon, similar to the Doppler effect, is usually considered to be negligible in practice, based on distortion measurements [1,2,3,4]. Distortion increases for more complex signals, due to a significant intermodulation between the low-frequency content (large displacements) and the higher-frequency content (large accelerations). It has been shown [5, 6] that this phenomenon can be a dominant source of intermodulation distortion (compared to other sources such as force factor, suspension, etc.) for full-range speakers at high frequencies. Various models have been proposed to describe this phenomenon: (i) van Wulfften Palthe [1] adopted a pulsating sphere as a loudspeaker radiation approximation and performed calculation of intermodulation distortion, considering both the moving-boundary effect (Doppler)
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