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 75 degrees at a frequency of 9100 Hz
Speed of sound in air is 343 m/s
If sin theta = 1.22( wavelength /diameter)
Wavelength = speed / frequency
This gives:
Wavelength = 343/9100 = .0377m
Then:
Diameter = 1.22 ( wavelength )/ sin theta
This gives:
Diameter = .048 m...my answer!!!!!!
Answer
If we treat the speaker like a simple circular aperture then there will be a diffracted angle of
75 degrees on EACH side of the centre line. This gives a TOTAL SPREAD of 150 degrees. If
the question means a TOTAL spread of 75 degrees then the answer will be different.
Anyway – assuming that it is 75 degrees 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.