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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


Answer

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/[8790]
= 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

 
 
 
© Keith Gibbs 2011