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I am trying to design a parabolic speaker to allow an audio signal to be emitted in only a single direction. For a single acoustic tone with wavelength $w$, what should the diameter of the reflector dish be? Wikipedia states that the diameter should be at least twice the wavelength, but does not explain why.

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The beamwidth of a parabolic dish is on the order of $BW \approx 70\dfrac{\lambda}{D}$ measured in degrees. Depending on the illuminator (source) the factor "$70$" may vary between $60$ and $100$, or so. The shape of the illumination (buzzword: "tapering") itself determines the size and number of sidelobes that may cause uncontrolled multiple reflections/reverberations.

You have to define what the longest wavelength (lowest frequency) of interest to you is because that will determine your maximum spatial extent of the signal at that frequency.

Note though that since the spectrum of music can be several octaves wide a parabolic reflector will show marked variation in its beamwidth across the frequencies as a result of which if you are not on or very near the axis you may hear spectral filtering distortion commensurate with your off-axis angle.

For the quoted beamwidth formula to hold you need a few wavelengths wide diameter, otherwise the wave from the source just diffracts strongly around the perimeter and you get very little collimation.

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  • $\begingroup$ Thanks for this! I'll sketch a diagram in the morning just to consolidate, then double check here if that's okay. $\endgroup$ – Harry Stuart Apr 28 at 15:04
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Three main reasons:

  1. reduce diffraction/interference related effects (occur when $\lambda$~D)

  2. increase directionality (beamwidth $\propto \frac\lambda D$)

  3. increase gain ($\propto \left(\frac D \lambda \right)^2$)

where $\lambda$ is sound's wavelength and D the dish's aperture.

As to why the minimum limit of $D/\lambda$ is suggested to be $2$: I think its a rule of thumb heuristic motivated by practical engineering. In real world applications, the ratio would in fact be higher and its minimum would be decided by the min-max of supported bandwidth.

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