What is the longest wavelength for a standing wave, in the same length tube, that is open at one end and closed at the other?
The length of the tube is 1/4 wavelength. If the tube is 12furlongs, then the wavelength is 48furlongs.
To determine the longest wavelength for a standing wave in a tube that is open at one end and closed at the other, we can use the formula for the resonant frequencies of closed-closed or open-open tubes.
For a tube open at one end and closed at the other, the resonant frequencies are given by the equation:
fn = (2n-1) * v / 4L
Where:
fn = the resonant frequency of the nth harmonic
n = the harmonic number (1, 2, 3, ...)
v = the speed of sound in the medium (air in this case)
L = the length of the tube
For the longest wavelength, we need to find the frequency with the smallest harmonic number. Since the harmonic number (n) is an integer and starts at 1, the smallest harmonic number will be 1.
Using n = 1 in the equation, we get:
f1 = (2*1-1) * v / 4L
f1 = v / 4L
To find the corresponding wavelength, we can rearrange the formula:
λ = v / f
Substituting in f1, we get:
λ1 = v / f1
λ1 = v / (v / 4L)
λ1 = 4L
Therefore, the longest wavelength for a standing wave in a tube that is open at one end and closed at the other is equal to 4 times the length of the tube (4L).