Resonance Tube and the End Correction
A student uses a vertical resonance tube with a water reservoir to measure the speed of sound. A 512 Hz tuning fork is held above the open end, and the water level is lowered until resonance is heard. The first strong resonance occurs when the air column is 16.4 cm long; the second occurs at 50.0 cm. The room temperature is 22°C, and the accepted speed of sound at that temperature is 344 m/s. The student first calculates λ = 4L from the 16.4 cm length and obtains a frequency that disagrees with the tuning fork by more than 2%. The tube’s diameter is 3.2 cm, and the water level is adjusted slowly. The meterstick’s zero end is worn, so lengths may be shifted by about 0.2 cm. The student also notices that the resonance becomes weaker and broader at the second position. The tuning fork is stamped 512 Hz, but its actual frequency is not independently verified.
1. Determine the wavelength, frequency, and end correction from the two resonance positions. Explain why the student’s one-length calculation is biased.
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For a tube closed at one end, the distance between successive resonances is (\lambda/2). Thus
(\lambda=2(50.0-16.4)=67.2\ \mathrm{cm}=0.672\ \mathrm{m}).
Using the accepted speed,
(f=\frac{v}{\lambda}=\frac{344}{0.672}\approx512\ \mathrm{Hz}),
which agrees with the tuning fork. The end correction is
(e=\frac{\lambda}{4}-L_1=16.8-16.4=0.4\ \mathrm{cm}).
Using (L_1) alone gives (\lambda=4(16.4)=65.6\ \mathrm{cm}) and (f=344/0.656\approx524\ \mathrm{Hz}), about 2.4% high. The one-length method assumes the pressure antinode is exactly at the open end, but it actually lies slightly outside; the difference method cancels this end correction.
2. The student claims the 0.4 cm end correction is negligible because it is smaller than the 0.2 cm ruler uncertainty. Evaluate this claim for both the one-length method and the two-resonance method.
Read the worked answer
The claim is not valid for the one-length method. A 0.4 cm end correction shifts the quarter-wavelength by about 2.4%, which changes the inferred frequency by about 12 Hz. The 0.2 cm ruler uncertainty is comparable to or smaller than that systematic shift, so ignoring the end correction produces a biased result. For the two-resonance method, the end correction cancels because the difference between resonances is exactly (\lambda/2). The ruler uncertainty still affects the difference: a 0.4 cm uncertainty in the 33.6 cm separation is about 1.2%, or roughly 6 Hz in frequency. The two-resonance result is therefore more reliable, and the student should report the difference-based frequency with an uncertainty of several hertz.