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
0.24
To find the diameter of the circular opening for the speaker, we need to use the concept of diffraction and the equation:
d = (λ * D) / (2 * sin(θ))
Where:
- d is the diameter of the opening
- λ is the wavelength of the sound wave
- D is the distance between the speaker and the listener
- θ is the diffraction angle
First, let's calculate the wavelength of the 9100 Hz sound wave using the formula:
λ = v / f
Where:
- λ is the wavelength
- v is the speed of sound (343 m/s in this case)
- f is the frequency of the sound wave (9100 Hz)
Substituting the values:
λ = 343 m/s / 9100 Hz
λ ≈ 0.0377 meters
Now, we can calculate the diameter of the opening:
d = (0.0377 meters * D) / (2 * sin(11°))
Note that we need the value of D, the distance between the speaker and the listener, in order to obtain the diameter.
Let's assume a specific value for D, for example, D = 1 meter:
d = (0.0377 meters * 1 meter) / (2 * sin(11°))
d ≈ 0.189 meters
Therefore, if the distance between the speaker and the listener is 1 meter, the diameter of the circular opening for the speaker should be approximately 0.189 meters (or 18.9 centimeters).