Resonance tube: end correction or bad resonance?
In a resonance-tube experiment, a 512 Hz tuning fork is held over a tube closed at one end and partly filled with water. The water level is lowered until the sound is loudest. The first resonance is found at an air-column length of 16.0 cm and the second at 49.5 cm. The room temperature is 20.0 °C and the tube diameter is 3.0 cm. One student uses v=4L1f and reports 328 m/s. Another uses the difference between the two resonance lengths and reports a value close to the accepted speed of sound. The first student argues that the second resonance is less reliable because it is weaker and occurs higher in the tube. The lab manual says to correct for an end effect e, with L+e = nλ/4. The lengths are measured with a ruler whose smallest division is 1 mm.
1. Calculate the speed of sound from the first resonance alone and from the difference between resonances. Find the end correction and explain the discrepancy.
Read the worked answer
First method: assuming a pressure antinode exactly at the open end, λ = 4L1 = 4(0.160) = 0.640 m, so v = fλ = 512×0.640 = 328 m/s.
Difference method: the distance between successive resonances is λ/2, so λ = 2(L2 − L1) = 2(0.495 − 0.160) = 0.670 m, and v = 512×0.670 ≈ 343 m/s.
End correction: the first resonance satisfies L1 + e = λ/4 = 0.670/4 = 0.1675 m, so e = 0.1675 − 0.160 = 0.0075 m = 7.5 mm. Equivalently, e = (L2 − 3L1)/2 = (0.495 − 0.480)/2 = 0.0075 m. The first method ignores e and uses an effective length that is too short by 7.5 mm, so it underestimates the wavelength and the speed. The difference method cancels e because both resonances are shifted by the same end correction. The second resonance being weaker affects the precision with which its position is found, not the validity of the λ/2 relation.
2. Which speed should the students report? Compare it with the expected speed at 20.0 °C and estimate the uncertainty from the length measurements.
Read the worked answer
Report the difference-method value, about 343 m/s, because it eliminates the first-order end correction. At 20.0 °C, a standard approximation is v ≈ 331.4 + 0.606T = 331.4 + 0.606(20.0) ≈ 343.5 m/s. The measured 343 m/s agrees within about 0.5 m/s.
For uncertainty, each length reading is limited by the ruler to about ±0.5 mm. The difference L2 − L1 then has uncertainty sqrt(0.5^2 + 0.5^2) ≈ 0.7 mm. Relative uncertainty in the difference is 0.0007/0.335 ≈ 0.0021, giving about 0.7 m/s in v. The tuning-fork frequency may also be off by about 1 Hz, contributing about 0.7 m/s. A reasonable report is v = (343 ± 1) m/s, consistent with the expected value. The 328 m/s result should not be reported as the best value because it contains a known systematic error.
3. Identify one additional systematic effect in this experiment and describe a change that would reduce it.
Read the worked answer
One systematic effect is the temperature gradient along the tube: the speed of sound depends on temperature, and the air near the water surface or near the room walls may not be at exactly 20.0 °C. The resonance condition uses an average speed over the air column, so a gradient shifts the effective wavelength. Measure the temperature at several points along the tube, use the average, and let the apparatus stand until it reaches room temperature. A related effect is the tube-diameter end correction: the empirical e ≈ 0.6r gives about 0.9 cm for a 3.0 cm diameter, compared with the measured 0.75 cm; this does not bias the difference method to first order but affects the absolute resonance positions.
Another improvement is to drive the tube with a calibrated loudspeaker and detect the maxima with a microphone instead of relying on the ear. This reduces the uncertainty in locating the resonance and allows repeated trials. The tuning-fork frequency should also be checked with a frequency meter or oscilloscope, because a 1 Hz error changes v by about 0.7 m/s.