Speed of Sound — Resonance Tube Experiment

Walk into any well-stocked CBSE school laboratory in July and find the resonance-tube apparatus on the back bench: a tall glass cylinder, perhaps a metre long, clamped vertically on a stand, with a narrow rubber tube running from its lower end down to a reservoir of water clamped on a sliding platform. A hand-written label hangs from the metal: "RESONANCE COLUMN — HANDLE WITH CARE." On the shelf above sit half a dozen tuning forks stamped with their frequencies in neat black ink — 256 Hz, 288 Hz, 320 Hz, 384 Hz, 480 Hz, 512 Hz — and a small rubber mallet. There is a thermometer tucked into the stand's base. This apparatus is sixty years old in design and has not been improved since, for a reason: it measures the speed of sound in air to within about 1%, in a single lab period, using nothing but a tuning fork, a tube, some water, and your ears.

The physics is elegant. Strike the fork, hold it over the tube, and slide the reservoir. Each water level sets a different length of air column above the water surface. For most lengths, the fork's vibration couples weakly — you hear the fork, but the tube is silent. But at certain discrete lengths — L_1, L_2, L_3, \ldots — the column's own natural frequency matches the fork, and resonance kicks in: the air column amplifies the sound so strongly that the tube seems to hum on its own, loud enough to be heard across the room. The lengths at which this happens are spaced by exactly half a wavelength of the sound in air. Measure two consecutive lengths with a metre scale, subtract, multiply by 2f — and you have the speed of sound.

This article is the companion to the theoretical articles Sound Waves — Nature and Propagation and Vibrations of Strings and Pipes. Those derive the formulas v = \sqrt{\gamma P/\rho} and f_n = nv/(4L) for a closed pipe. This one is the lab counterpart: the apparatus, the derivation of why two consecutive resonances give v without any correction, the procedure as followed in a CBSE/ISC class-11 practical, the temperature dependence v \propto \sqrt{T} tested on a sweltering Indian afternoon, and the honest error analysis.

The apparatus

A "resonance tube" is really a resonance column — the column of air above the water surface inside the tube is what resonates, not the tube itself. The water's upper surface acts as a rigid wall (at a pressure antinode / displacement node); the open mouth at the top acts as a free pressure-release (at a pressure node / displacement antinode, approximately). A sliding water level lets you vary the column length continuously.

Typical apparatus in an Indian school/college lab:

• Glass tube: vertical, inner diameter 2–5 cm, length about 100 cm, graduated along its length with millimetre markings painted on the outside.
• Water reservoir: an open metal or plastic can, connected to the lower end of the tube by a flexible rubber tube. Slide the reservoir up and down to change the water level in the glass tube (the connected-vessels principle keeps the two levels equal).
• Clamp stand: a sturdy retort stand holding the glass tube vertical. Inexact verticality is a common source of small systematic error.
• Tuning forks: several, with frequencies stamped on them (256 Hz, 320 Hz, 384 Hz, 480 Hz, 512 Hz are common CBSE set values).
• Rubber mallet or a soft pad on a wooden block to strike the fork. Never strike it on a hard surface — the overtones it produces make the resonance fuzzy.
• Thermometer: mercury-in-glass, 0–100 °C, least count 1 °C. Records the air temperature inside the tube.
• Metre scale or vernier-marked scale: to read the air column length from the water surface to the mouth of the tube.
• Plumb-bob (optional): confirms the tube is truly vertical.

Why the water surface acts as a closed end

The water is essentially incompressible on the timescale of sound oscillations (milliseconds). Air molecules arriving at the surface bounce back almost without loss — the acoustic impedance of water is about 3500 times greater than that of air, so very little sound energy crosses into the water. The surface is therefore a near-perfect pressure-antinode / displacement-node boundary, exactly like the closed end of an organ pipe. The top of the tube, opening into the room, is a pressure-node / displacement-antinode boundary — to a good approximation. The resonance-tube system is, acoustically, a pipe closed at one end, with its length adjustable by moving the water up and down.

Why resonance happens — standing waves in a closed pipe

From Vibrations of Strings and Pipes, a pipe of length L closed at one end supports standing waves with a pressure antinode at the closed end and a pressure node at the open end. The allowed mode-shapes squeeze exactly (2n-1) quarter-wavelengths into the length L:

The frequencies of these modes are f_n = (2n-1)\,v/(4L).

Turn this around: with the fork forcing a fixed frequency f (and hence fixing \lambda = v/f), resonance occurs for those lengths L at which the forced frequency matches one of the natural mode frequencies of the column. Solve for L:

So the resonance lengths are \lambda/4,\; 3\lambda/4,\; 5\lambda/4,\; 7\lambda/4,\;\ldots — equally spaced by \lambda/2.

Why: consecutive resonances correspond to adding one full half-wavelength of extra column. Geometrically, each time you lower the water by \lambda/2, you fit one more half-wavelength into the column while keeping the node/antinode pattern at the two ends intact.

The end correction — and why it does not matter

There is a small but important refinement. The pressure node at the open end is not exactly at the rim of the tube. It sits a short distance above it. The reason is physical: an oscillating parcel of air at the mouth of the tube pushes on, and is slightly loaded by, the cylinder of still air just outside the mouth. That outside cylinder — roughly of length 0.6\,R where R is the tube's inner radius — must also oscillate slightly as part of the standing wave. The effective acoustic length is therefore a little longer than the measured column length.

Let e be this end correction, with e \approx 0.6\,R for a uniform circular tube (derived by Rayleigh in 1894 for an unflanged pipe; the closely-related flanged-pipe correction is e \approx 0.82\,R). For the resonance condition, the effective length is L_n + e:

Here is the clean move. Subtract (1) from (2):

Why: the end correction e appears with the same sign in both equations and cancels on subtraction. You do not need to know e — you do not even need to know R — as long as the two resonances are of the same tube with the same mouth geometry. This is what makes the two-resonance method robust.

Multiply both sides by 2f, using v = f\lambda:

Equation (3) is the working formula of the experiment. Measure L_1 and L_2 with a metre scale, multiply their difference by twice the fork's frequency, and read off the speed of sound.

If you want to recover the end correction for its own sake — say, as an extra task in a practical exam — rearrange (1): e = \lambda/4 - L_1 = (L_2 - L_1)/2 - L_1. Compare this with the geometric prediction 0.6\,R for the given tube; the two should agree to about 10%. Discrepancies usually trace to slight non-circularity of the mouth, or to a fork held slightly to one side of the axis.

The procedure

The canonical CBSE / ISC / NCERT procedure runs like this. Treat it as a recipe you will actually follow in the lab next Monday.

Step 1: set up and level. Clamp the glass tube vertically using a spirit level or a plumb-bob against the tube. A tilt of 2–3° can shift L_1 and L_2 by a couple of millimetres each through parallax on the scale.

Step 2: fill the tube. Raise the reservoir so the water rises almost to the top of the glass tube. Watch for air bubbles inside the rubber tube; tap the rubber gently to clear them, because a trapped bubble makes the water level jumpy when you move the reservoir.

Step 3: pick a fork. Start with the highest-frequency fork available (say 512 Hz). Higher f means shorter \lambda, so L_1 and L_2 are shorter and both fit comfortably inside a 1 m tube. A 256 Hz fork gives L_2 \approx 100 cm, which can fall off the scale if the tube is not long enough.

Step 4: record the temperature. Note the air temperature inside the tube (or the laboratory's bulk temperature if they are the same — they usually are within 0.5 °C). You will need this to compare against theory.

Step 5: strike the fork. Hit the fork on a rubber mallet or a soft pad. Never on the corner of a desk or a metal edge — that produces audible overtones (modes other than the fundamental at the fork's stamped frequency) and makes resonance fuzzy.

Step 6: find the first resonance. Hold the vibrating fork horizontally with its prongs about 1 cm above the open mouth of the tube, prongs oriented so the air they push moves up and down the tube's axis. Starting from the water level high (short column), lower the reservoir slowly. The tube will suddenly roar at a certain length. That is L_1. Read the length from the water surface to the mouth of the tube using the scale marked on the glass. Record it.

Tip. The roar is unmistakable when you are close to it. As you approach from one side (water too high), the sound swells sharply. As you continue past it (water too low), it dies equally sharply. Bracket the maximum by going back and forth: mark the water level where the sound is loudest.

Step 7: find the second resonance. Keeping the fork struck (re-strike it as needed — it only rings for 5–10 seconds), continue lowering the reservoir past the first resonance. The sound will die, then swell again as you cross L_2 \approx 3L_1. Record L_2. A third resonance (at L_3 \approx 5L_1) may exist if the tube is long enough; finding it gives an independent check but is not required.

Step 8: repeat. Do the whole measurement five or six times with the same fork, using slightly different approach speeds and from both directions (water rising and water falling). Calculate the mean \bar{L}_1 and \bar{L}_2; these are your best estimates. Compute

Step 9: change the fork. Repeat with at least one other fork of different frequency (say, switch from 512 Hz to 384 Hz). You should get the same v within error — because the speed of sound is a property of the air, not the fork. If the two forks disagree by more than 1%, check the apparatus (verticality, clean tube, no trapped bubbles).

Interactive: find the resonance yourself

The figure below is a simplified idealised version of the experiment. Drag the slider to change the air column length L; the plot shows the amplitude of the air-column oscillation at the fork's frequency, as a function of L. Sharp peaks at L_1 = \lambda/4 - e and L_2 = 3\lambda/4 - e mark the two resonance positions. The tuning fork is set to 512 Hz and the simulation uses v = 348 m/s (a room at about 30 °C); the inner tube radius is R = 1.25 cm, giving an end correction e = 0.75 cm.

Notice how sharp the peaks are. A mistuning of just 1 cm from L_1 already drops the amplitude to about a third of its peak value — this is why the resonance is easy to locate by ear, and why careful measurement can pin L_1 to about \pm 2 mm on a good day.

Worked examples

Sources of error — and how to beat them

Honest error analysis is part of the experiment. Here are the dominant error sources in a CBSE/ISC setup, with their typical magnitudes and the standard fixes.

• Reading L_1 and L_2. A metre scale with 1 mm markings, read with parallax, is good to \pm 1 mm. Five repeats cut the random component by \sqrt{5}, leaving roughly \pm 0.5 mm in the mean. Fractional error in \lambda = 2(L_2 - L_1) \approx 68 cm: about 0.1\%.

• Fork frequency drift. A fork stamped "512 Hz" when new drifts with age; temperature of the fork metal also matters slightly (steel's Young's modulus falls by \sim 0.02\% per °C). After years of use, a real fork in an old lab may be 510–511 Hz. Fractional error: up to 0.4\%. Fix: calibrate the fork against a stroboscope or a calibrated oscillator when available.

• Locating the resonance maximum. The resonance peak has a full width at half-max of about 2 cm for a typical tube with moderate damping. You can locate the peak to about \pm 2 mm by bracketing from both directions. Fractional error: about 0.3\%.

• Humidity. Moist air is lighter than dry air (H₂O is lighter than N₂ or O₂), so humid air has slightly lower \rho and hence slightly higher v. A fully saturated atmosphere at 30 °C gives v about 0.4\% higher than dry air at the same temperature. A Delhi monsoon afternoon at 80% humidity raises the measured v by about 1 m/s over the dry-air formula. This is usually not corrected for in a class-11 practical; just note it.

• Non-uniform tube cross-section. A slightly tapered or non-circular tube gives two different end corrections, and the subtraction trick fails at the 0.1\% level. Fix: use a tube whose mouth and body have the same radius.

• Temperature gradient in the tube. On a hot day, the air at the bottom of the tube (just above the water) is typically 1–2 °C cooler than at the top. Since v depends on T, this smears the effective wave speed slightly. Fractional error: about 0.1–0.2\%.

Adding in quadrature: a careful experiment with a calibrated fork and no humidity correction reaches about \pm 0.6\% precision — better than \pm 2 m/s out of \sim 345 m/s. That is impressive for a sub-₹500 apparatus.

Common confusions

• "The air column length is the distance from the bottom of the glass tube to the top." No. The resonating column is the column of air above the water surface, from the water level up to the mouth. The glass below the water plays no acoustic role.

• "The end correction is negligible, so just use L_1 = \lambda/4." You can do this for a first estimate, but it gives a systematic over-estimate of \lambda (and hence of v) of about 3\% for a typical tube. The two-resonance method eliminates e and is almost free; use it.

• "Higher-frequency forks give better precision." Partially true. A higher f gives a shorter \lambda/2, which means a smaller absolute spacing L_2 - L_1. If your scale error is fixed at 1 mm, the fractional error 1\,\text{mm}/(L_2 - L_1) is larger for smaller spacings. Optimum: pick f such that L_2 - L_1 is 25–40 cm (so \lambda/2 fits in the tube with a few cm to spare on each side). For a 1 m tube, 384 Hz or 512 Hz are ideal; 256 Hz is too low (L_2 falls off the top of the tube).

• "Resonance means the fork gets louder, so strike it harder." Striking harder creates overtones, which muddy the resonance. A gentle strike with a rubber mallet gives the cleanest fundamental and the sharpest resonance.

• "The experiment only works if the tube is exactly vertical." Small tilts are fine acoustically — the wave doesn't care about gravity. But tilts cause parallax errors in reading L off the scale and shift the water level relative to the scale. So keep the tube vertical for metrological reasons, not acoustic ones.

• "This gives the speed of sound in water." No. The sound is in the air column above the water. The water's job is to be a closed-end reflector. The apparatus never measures wave speeds inside the liquid; for that, you would need a different setup (an interferometric method, or a direct time-of-flight measurement).

Where this leads next

• Sound Waves — Nature and Propagation — derives v = \sqrt{B/\rho} and Laplace's formula v = \sqrt{\gamma P/\rho}, the theoretical side that this experiment tests.
• Vibrations of Strings and Pipes — the full derivation of the closed-pipe harmonic series f_n = nv/(4L) used here, plus the open-pipe and stringed-instrument companions.
• Standing Waves and Normal Modes — the superposition argument that creates the resonance conditions.
• The Simple Pendulum Experiment — the same "measure a slope, extract a fundamental constant" philosophy, applied to g instead of v.
• Errors in Measurement — the formal tools for propagating the uncertainties discussed in the error-analysis section above.