A two end open pipe organ's sixth harmonic is produced by a tuning fork with a frequency of 440 Hz. If it is 17 degrees Celsius, what is the length of the tube?
http://media.paisley.ac.uk/~davison/labpage/tube/tube.html
To find the length of the tube, we need to use the formula for the fundamental frequency of a pipe that is open at both ends:
f = (n * v) / (2 * L)
where:
- f is the frequency of the harmonic (in this case, the 6th harmonic)
- n is the harmonic number (in this case, 6)
- v is the speed of sound in air (approximately 343 m/s at 20 degrees Celsius)
- L is the length of the tube
However, before we can use this formula, we need to account for the change in the speed of sound with temperature. The speed of sound in air decreases as the temperature decreases. We can use the following equation to adjust the speed of sound for a given temperature:
v = v₀ * √(T₀ / T)
where:
- v is the adjusted speed of sound
- v₀ is the speed of sound at a reference temperature T₀
- T is the actual temperature
In this case, T₀ is 20 degrees Celsius and T is 17 degrees Celsius. We can use the speed of sound at 20 degrees Celsius to find v₀.
Now, let's calculate the adjusted speed of sound at 17 degrees Celsius:
v₀ = 343 m/s (speed of sound at 20 degrees Celsius)
T₀ = 20 degrees Celsius (reference temperature)
T = 17 degrees Celsius (actual temperature)
v = v₀ * √(T₀ / T)
v = 343 m/s * √(20 / 17)
v ≈ 343 m/s * √(1.176)
v ≈ 343 m/s * 1.084
v ≈ 372.312 m/s
Now we have the adjusted speed of sound (v). We can substitute the values into the formula for the 6th harmonic:
f = (n * v) / (2 * L)
440 Hz = (6 * 372.312 m/s) / (2 * L)
Now, we can solve for the length of the tube (L):
L = (6 * 372.312 m/s) / (2 * 440 Hz)
L ≈ 3.997 m
Therefore, the length of the tube is approximately 3.997 meters.