Resonance Tube: When the Tuning Fork Marking Is the Weak Link
A student measures the speed of sound using a resonance tube and a tuning fork marked 512 Hz. The water level is lowered until a loud resonance is heard. First resonance: 0.158 m; second: 0.486 m; third: 0.815 m, measured from the top of the tube. Room temperature is 18.0 °C. Using v = 2f(L2 − L1), the student obtains 336 m/s, but a data book gives about 342 m/s at 18 °C. The student proposes to 'correct' the result by adding 6 m/s. A second fork marked 512 Hz gives resonances at 0.155 m and 0.477 m. The tube diameter is 0.032 m. The student asks whether the first result is wrong, whether the end correction explains the gap, and what to report.
1. Calculate the wavelength, speed and end correction from the first data. Predict the third resonance and assess internal consistency.
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λ = 2(L2 − L1) = 2(0.486 − 0.158) = 0.656 m. With f = 512 Hz, v = fλ = 512 × 0.656 = 336 m s⁻¹. The end correction is e = (L2 − 3L1)/2 = (0.486 − 0.474)/2 = 0.006 m. The third resonance should be L3 = 5λ/4 − e = 5(0.656)/4 − 0.006 = 0.814 m; the observed 0.815 m agrees within 1 mm. Judgement. The end correction is only 6 mm and cancels from L2 − L1, so it cannot explain a 6 m/s difference. The internal consistency is good; the discrepancy is systematic, not random.
2. The second fork gives a different resonance separation. Calculate its implied frequency or speed and decide whether the marked values can both be correct.
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Second fork: λ = 2(0.477 − 0.155) = 0.644 m. If the speed is 336 m/s, its implied frequency is f = 336/0.644 = 522 Hz, not 512 Hz. Alternatively, if the fork really is 512 Hz, the implied speed is 512 × 0.644 = 330 m s⁻¹. The two '512 Hz' forks therefore cannot both be exactly 512 Hz; their implied frequencies differ by about 1.9%. Temperature changes in a tuning fork are far too small to explain this. The marked frequency is an assumption, not a measurement, and the discrepancy in v is dominated by it.
3. Should the student add 6 m/s? What is the best estimate and uncertainty to report?
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Adding 6 m/s is unjustified: it forces agreement with a book value and hides the real source. The defensible statement is v = 336 m s⁻¹ if the first fork is exactly 512 Hz; the uncertainty is at least ± 7 m s⁻¹ from the possible frequency error. Calibrate the fork by the beat method against a signal generator or smartphone frequency app, or use a known-frequency source. Also check the thermometer and repeat with a longer tube to increase L2 − L1. Report the measured value with its method and uncertainty, not a 'corrected' value.