   # Diffraction by a loudspeaker

QUESTION: A speaker is designed for wide dispersion for a certain frequency.
What should be the diameter of the circular opening in a speaker?

Diffraction angle is 75o at a frequency of 9100 Hz
Speed of sound in air is 343 m/s If sin(diffracted angle) = 1.22( wavelength /diameter)
Wavelength = speed / frequency
This gives: Wavelength = 343/9100 = 0.0377m
Then: Diameter = 1.22 ( wavelength )/ sin(diffracted angle)
This gives: Diameter = 0.048 m

If we treat the speaker like a simple circular aperture then there will be a diffracted angle of 75o on EACH side of the centre line. This gives a TOTAL SPREAD of 150o. If the question means a TOTAL spread of 75o then the answer will be different.

Anyway – assuming that it is 75o EACH side of the centre line then, using the standard formula for the diffraction at a circular opening:

Diameter of speaker x sin (diffracted angle) =1.22 x wavelength

So: Diameter of speaker = 1.22 x Speed of sound/[frequency x sin(diffracted angle)]
= 1.22 x 343/[9100xsin(75)]
= 1.22 x 343/
= 1.22 x 0.039
= 0.48 m

As you can see from the calculation if we had used the formula for a single SLIT and not a CIRCULAR aperture we would have got 0.039m