A 3.00-kHz tone is being produced by a speaker with a diameter of 0.185 m. The air temperature changes from 0 to 29° C. Assuming air to be an ideal gas, find the change in the diffraction angle θ.
To find the change in the diffraction angle θ, we need to use the formula for diffraction angle of sound waves:
θ = λ / (π × D)
where:
- θ is the diffraction angle
- λ is the wavelength of the sound wave
- D is the diameter of the speaker
First, let's find the initial wavelength (λ1) of the sound wave at 0°C. We can use the formula for the speed of sound in air:
v = 331.5 + 0.6T
where:
- v is the speed of sound
- T is the air temperature in °C
At 0°C, the speed of sound is:
v1 = 331.5 + 0.6(0)
= 331.5 m/s
The wavelength at 0°C can be calculated using the formula for the wavelength of a wave:
v1 = f × λ1
where:
- f is the frequency of the sound wave
Rearranging the equation, we can solve for λ1:
λ1 = v1 / f
Now, let's find the final wavelength (λ2) of the sound wave at 29°C. Following the same steps as before but using the new temperature:
v2 = 331.5 + 0.6(29)
= 340.9 m/s
Using the same formula:
λ2 = v2 / f
The change in the diffraction angle is given by:
Δθ = θ2 - θ1
Substituting the values into the formula:
Δθ = (λ2 / (π × D)) - (λ1 / (π × D))
Now, we can calculate the change in the diffraction angle θ by substituting the given values into the formula.
To find the change in the diffraction angle θ, we need to use the formula for diffraction of sound waves:
sin(θ) = λ / (d + λ)
Where:
- θ is the diffraction angle
- λ is the wavelength of the sound wave
- d is the diameter of the speaker
First, let's find the initial wavelength λ₁ of the sound wave at 0°C. We can use the formula for the speed of sound in air:
v = 331.4 + 0.6 * T
Where:
- v is the speed of sound in m/s
- T is the air temperature in °C
At 0°C, the speed of sound is:
v₁ = 331.4 + 0.6 * 0 = 331.4 m/s
To find the initial wavelength, we can use the formula:
λ₁ = v₁ / f
Where:
- λ₁ is the initial wavelength
- v₁ is the initial speed of sound
- f is the frequency of the sound wave
The given frequency is 3.00 kHz, which can be converted to Hz:
f = 3.00 kHz * 1000 Hz/kHz = 3000 Hz
Substituting the values, we get:
λ₁ = 331.4 m/s / 3000 Hz = 0.1105 m
Now, let's find the final wavelength λ₂ of the sound wave at 29°C. Using the same formula for the speed of sound, we can find the final speed of sound:
v₂ = 331.4 + 0.6 * 29 = 348.0 m/s
Applying the formula for wavelength at the new temperature, we have:
λ₂ = v₂ / f = 348.0 m/s / 3000 Hz = 0.1160 m
Finally, we can find the change in the diffraction angle θ by subtracting the initial wavelength from the final wavelength and using the formula:
Δθ = arcsin((λ₂ - λ₁) / (d + λ₂))
Substituting the known values, we have:
Δθ = arcsin((0.1160 m - 0.1105 m) / (0.185 m + 0.1160 m))
Calculating this expression will give the change in the diffraction angle Δθ.
Theta = 1.22*(wavelength)/D
Apply it at each temperature and take the difference.