Question

A speaker is designed for wide dispersion for a high frequency sound. What should the diameter of the circular opening be for a speaker where the desired diffraction angle is 11° and a 9100 Hz sound is generated? The speed of sound is 343 m/s.

Answer 1

To solve this problem we will apply the concepts related to wavelength, as well as Rayleigh's Criterion or Optical resolution, the optical limit due to diffraction can be calculated empirically from the following relationship,

$sin\theta = 1.22\frac{\lambda}{d}$

Here,

$\lambda$ = Wavelength

d= Diameter of aperture

$\theta$ = Angular resolution or diffraction angle

Our values are given as,

$\theta = 11\°$

The frequency of the sound is $f = 9100 Hz$

The speed of the sound is $v = 343 m/s$

The wavelength of the sound is

$\lambda = \frac{v}{f}$

Here,

v = Velocity of the wave

f = Frequency

Replacing,

$\lambda = \frac{(343 m/s)}{(9100 Hz)}$

$\lambda = 0.0377 m$

The diffraction condition is then,

$sin\theta = 1.22\frac{\lambda}{d}$

Replacing,

$sin(11\°) = 1.22\frac{(0.0377 m)}{(d)}$

d = 0.24 m

Therefore the diameter should be 0.24m