Resonance Tube: Which Speed-of-Sound Calculation Should Be Trusted?
Students determine the speed of sound using a resonance tube, a 512 Hz tuning fork and a metre rule. The tube has an internal diameter of 3.0 cm. Group 1 finds the first resonance at 15.8 cm and the second at 49.2 cm. Group 2, using the same fork, finds 16.4 cm and 50.4 cm. Group 1 calculates v = 2(L2 - L1)f = 342 m/s. Group 2 calculates v = 4L1f = 336 m/s and v = 4L2f/3 = 344 m/s, then averages these to 340 m/s. The room temperature is 21 °C, for which the accepted speed is about 344 m/s. A student in Group 2 says their method is better because it uses two measurements. Another student says Group 1's value must be wrong because the first resonance should be at a quarter wavelength, not at a difference. The teacher must decide what feedback to give and how to assess the practical write-up.
1. Evaluate the three calculations. Which is physically best justified, and what does the end correction do to each?
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For a tube closed at one end, the antinode lies slightly beyond the open end. If the end correction is e, then L1 + e = λ/4 and L2 + e = 3λ/4. Subtracting gives L2 - L1 = λ/2, so v = 2(L2 - L1)f. Group 1's value is 2 × 0.334 × 512 = 342 m/s, within about 0.6% of the accepted value. The end correction cancels in the difference method, which is why it is the best-justified calculation.
Group 2's individual formulas do not cancel e: 4L1f = v - 4ef and 4L2f/3 = v - (4/3)ef, so both are biased low by different amounts. Averaging them does not remove the bias because the errors are not independent. The fact that their L2 calculation happens to give 344 m/s is due to measurement scatter, not a sound method. Their 340 m/s is close, but the reasoning is weaker and should not receive full method marks.
2. What feedback should the teacher give to Group 2, and how should the write-up be marked?
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The teacher should acknowledge that taking two readings can improve precision, but explain that precision does not remove a systematic error shared by both readings. Ask Group 2 to derive L1 and L2 in terms of λ and e, then show algebraically that L2 - L1 = λ/2. This demonstrates why Group 1's method is valid without simply telling them they are wrong.
The write-up should be marked on method as well as final value. Award marks for identifying the end correction, using a valid combination of L1 and L2, calculating a value, and commenting on uncertainty. Penalise averaging two biased values without justification, and require a statement that the difference method cancels the end correction. The final numerical answer alone should carry relatively few marks.
3. Design a quick whole-class check to see whether students understand why the difference method cancels the end correction.
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Give pairs a table of two hypothetical tubes, for example tube A with L1 = 16.0 cm and L2 = 50.0 cm, and tube B with L1 = 15.0 cm and L2 = 49.0 cm. Ask: 'Without calculating e, which tube gives the same speed of sound? Calculate v by the difference method and by the 4L1f method.' Students should find that the difference method gives the same speed for both tubes if the wavelength is the same, while 4L1f gives different values because the end correction differs. Then ask them to write one sentence explaining why the difference method is insensitive to e. Collect responses as an exit ticket and use two contrasting answers for whole-class discussion.