A two end open pipe organ's 5th harmonic produces a frequency of 260 Hz. If it is 23 degrees Celsius, what is the length of the tube? (2 answers)
Fn=n(v/2L)
Fn=260 Hz
n=5
v=332+0.6(23)=345.8 m/s
I got 3.325 meters as the length, but it says there's two answers. Why is there two answers and what's the other answer?
Is it possible that the length of the tube is 1/2 wavelength, and one wavelength?
I don't understand why the length would be 1/2 the wavelength and one wavelength. I thought that was only for the fourth and second harmonics.
To determine why there are two possible answers for the length of the tube, we need to consider the concept of harmonics in a pipe organ.
In an open pipe organ, when a pipe is open at both ends, the fundamental frequency (also known as the first harmonic) is the frequency produced when the entire length of the pipe vibrates. The n-th harmonic, where n is an integer, is a frequency produced when the pipe vibrates in n equal segments.
In this case, we are given that the 5th harmonic produces a frequency of 260 Hz. Using the formula Fn = n(v/2L), where Fn is the frequency, n is the harmonic number, v is the speed of sound in air, and L is the length of the pipe, we can solve for L.
Given:
Fn = 260 Hz
n = 5
v = 332 + 0.6(23) = 345.8 m/s (speed of sound in air at 23 degrees Celsius)
Now, let's solve the equation for L:
Fn = n(v/2L)
260 Hz = 5(345.8 m/s)/(2L)
Re-arranging the equation to solve for L:
L = 5(345.8 m/s)/(2 * 260 Hz)
L = (5 * 345.8 m/s) / (2 * 260 Hz)
L ≈ 3.325 meters
So, you correctly calculated the length of the tube to be approximately 3.325 meters.
However, there can be two possible solutions for the length of the tube because the pipe can vibrate in two different ways to produce the 5th harmonic. The first solution corresponds to the entire pipe vibrating as one segment, and the second solution corresponds to the pipe vibrating in two equal segments.
For the first solution, the length of the tube is approximately 3.325 meters. But for the second solution, the length of the tube would be half of that, which is approximately 1.663 meters.
Therefore, the two possible answers for the length of the tube in the given scenario are approximately 3.325 meters and 1.663 meters.