A speaker has a diameter of 0.303 m. (a) Assuming that the speed of sound is 343 m/s, find the diffraction angle è for a 1.80-kHz tone. (b) What speaker diameter D should be used to generate a 3.72-kHz tone whose diffraction angle is as wide as that for the 1.80-kHz tone in part (a)?
For circular opening sin(theta) = 1.22 (lambda)/D
To get lambda use v = f (lambda) i.e.
343 = (lambda)/1800.
To find the diffraction angle (θ) for a given frequency and speaker diameter, you can use the formula:
θ = λ / D
Where:
- θ is the diffraction angle
- λ is the wavelength of the sound wave
- D is the diameter of the speaker
(a) For a 1.80 kHz tone:
Given: frequency f = 1.80 kHz = 1800 Hz
speed of sound v = 343 m/s
First, find the wavelength (λ) using the formula:
λ = v / f
Substituting the given values:
λ = 343 m/s / 1800 Hz = 0.1906 m
Now, substitute λ and D into the equation to find θ:
θ = λ / D = 0.1906 m / 0.303 m ≈ 0.628 radians
(b) To find the speaker diameter D for a 3.72 kHz tone with the same diffraction angle as the 1.80 kHz tone:
Given: frequency f = 3.72 kHz = 3720 Hz
θ = 0.628 radians
First, find the wavelength (λ) using the formula:
λ = v / f (same as in part (a))
λ = 343 m/s / 3720 Hz = 0.0925 m
Now, substitute λ and θ into the equation to find D:
θ = λ / D
Rearranging the equation:
D = λ / θ = 0.0925 m / 0.628 ≈ 0.147 m
Therefore, the speaker diameter D that should be used to generate a 3.72 kHz tone with the same diffraction angle is approximately 0.147 m.
(a) To find the diffraction angle θ for a 1.80-kHz tone, we need to use the formula for diffraction:
θ = λ / D
where θ is the diffraction angle, λ is the wavelength of the sound wave, and D is the diameter of the speaker.
The wavelength of a sound wave can be found using the formula:
λ = v / f
where v is the speed of sound (343 m/s) and f is the frequency of the sound wave (1.80 kHz = 1800 Hz).
Let's calculate the wavelength first:
λ = 343 / 1800 = 0.1906 m
Now, we can substitute the values into the diffraction formula to find the diffraction angle:
θ = 0.1906 / 0.303 = 0.628 radians
(b) To find the diameter D of the speaker needed to generate a 3.72-kHz tone with the same diffraction angle as the 1.80-kHz tone, we can rearrange the formula for diffraction as follows:
D = λ / θ
We already know the diffraction angle θ (0.628 radians) and the frequency f (3.72 kHz = 3720 Hz).
Let's calculate the wavelength first:
λ = v / f
λ = 343 / 3720 = 0.0922 m
Now, we can substitute the values into the formula for diameter:
D = 0.0922 / 0.628 = 0.147 m
Therefore, the speaker diameter, D, needed to generate a 3.72-kHz tone with the same diffraction angle as the 1.80-kHz tone is approximately 0.147 m.