The speed of sound in air varies with the temperature and humidity of the air. In dry air at 20 degrees Celsius, the speed of sound is approximately 343 m/sec. A closed tube resonates with a node at each end for a sound wave in air. What is the length of tube you would choose if you were trying to have a 256 Hz wave resonate in the tube? Would the tube allow a128 Hz wave to resonate? Would the tube allow a 512 Hz wave to resonate?
To determine the length of the tube required for a specific frequency wave to resonate, we can use the formula for the wavelength of a sound wave:
λ = v / f
Where:
λ is the wavelength of the wave
v is the speed of sound in air
f is the frequency of the wave
Given that the speed of sound in dry air at 20 degrees Celsius is approximately 343 m/sec, we can calculate the wavelength corresponding to a 256 Hz wave using the formula above.
λ = 343 m/sec / 256 Hz
λ ≈ 1.339 meters
Since a closed tube resonates with a node at each end, the length of the tube should be an integer multiple of half the wavelength (λ/2) to ensure that a standing wave is formed. Therefore, to have a 256 Hz wave resonate in the tube, we need to calculate the length that corresponds to half the wavelength:
Length = λ / 2
Length ≈ 1.339 meters / 2
Length ≈ 0.6695 meters
Therefore, the length of the tube required for a 256 Hz wave to resonate would be approximately 0.6695 meters.
Now, let's determine if the tube would allow a 128 Hz wave to resonate. We can follow the same procedure:
λ = 343 m/sec / 128 Hz
λ ≈ 2.680 meters
Length = λ / 2
Length ≈ 2.680 meters / 2
Length ≈ 1.340 meters
Therefore, the tube would allow a 128 Hz wave to resonate if it is approximately 1.340 meters long.
Similarly, let's determine if the tube would allow a 512 Hz wave to resonate:
λ = 343 m/sec / 512 Hz
λ ≈ 0.670 meters
Length = λ / 2
Length ≈ 0.670 meters / 2
Length ≈ 0.335 meters
Therefore, the tube would allow a 512 Hz wave to resonate if it is approximately 0.335 meters long.