The frequency of the fundamental note produced by a pipe closed at one end and open at the other end is given by the formula:
f = (n * v) / (4 * L)
where:
- f is the frequency of the fundamental note
- n is the harmonic number (for the fundamental frequency, n = 1)
- v is the speed of sound in air (approximately 343 m/s at room temperature)
- L is the length of the pipe
For a pipe closed at one end and open at the other end, the fundamental frequency is the first harmonic, so n = 1.
We can use the formula to find the length of the pipe for each of the two resonant frequencies, and then solve for the frequency of the fundamental note.
For the frequency of 100 Hz:
100 Hz = (1 * 343 m/s) / (4 * L)
L = 8.575 m
For the frequency of 140 Hz:
140 Hz = (1 * 343 m/s) / (4 * L)
L = 6.125 m
The length of the pipe must be such that it resonates at both 100 Hz and 140 Hz, but not at any frequency in between. One possible length that satisfies this condition is the half-wavelength of the fundamental frequency, which is:
L = (1/2) * (v / f) = (1/2) * (343 m/s / 100 Hz) = 1.715 m
We can now use this length to find the frequency of the fundamental note:
f = (1 * 343 m/s) / (4 * 1.715 m) = 50 Hz
Therefore, the frequency of the fundamental note produced by the pipe is 50 Hz.