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Superposition overview

Topic 4 of 9

Standing sound and air columns

A resonating air column has displacement and pressure patterns with different node positions. Identify which quantity is shown before using a boundary condition or a resonant length.

Separate particle displacement from pressure variation

Sound is longitudinal: an air particle's displacement is along the tube. Pressure variation, Δp, is the change from ambient pressure, measured in Pa. A graph of either quantity uses vertical height to show its value; it does not show particles moving transversely across the tube.

  • At an approximately closed end, air cannot move through the boundary: it is a displacement node and a pressure-variation antinode.
  • At an approximately open end, pressure stays close to ambient: it is a pressure-variation node and a displacement antinode.

A pressure node means nearly zero pressure variation, not zero absolute pressure or a vacuum. A pressure antinode has the greatest amplitude of pressure variation; it is not permanently at the highest pressure.

The displacement and pressure nodes alternate in space. A displacement node and its nearest pressure node are λ/4 apart. In a tube open at both ends, both ends have the open-end condition: displacement antinodes and pressure nodes. These are spatial relationships; a pair of amplitude envelopes does not establish a time delay between pressure and displacement.

Find resonances with a water boundary

Hold a vibrating tuning fork near an open tube mouth and change the water level to vary the air length. The water surface acts approximately as a closed end. At suitable lengths, the sound grows loud because a standing-wave mode is resonantly driven.

Measure the air length from the water surface to the fixed tube mouth. Do not use the tube's full length or the depth of water. Keep the fork frequency and tube geometry unchanged, and repeat the judgement of the loudness peak.

Separate the apparatus from the air-column model

The tube mouth stays fixed while the water level changes

A tuning fork drives two alternative resonant air lengths in one fixed tubeA tuning fork is held near the fixed open mouth of a vertical tube. The tube bottom contains water, whose surface approximately closes the air column. The lower solid water level gives physical air length 0.510 metres from the mouth. A higher dashed reference marks the alternative 0.160-metre air length; the two surfaces are not simultaneous. Both lengths use the same scale, six hundred drawing units per metre, and are measured from the same fixed mouth. Water is added or withdrawn between settings. The fork and tube geometry are unchanged. The later model graphs align closed ends for comparison, unlike this actual fixed-mouth arrangement.480 Hz fork0.160 m0.510 mFixedmouthFirst levelSecond levelWaterTwo settings, not two water surfaces

Air-length model 1: physical length 0.160 m

These model comparisons align the closed end at x = 0. They do not show the apparatus mouth moving. Grey dashed: physical mouth. Brown dotted: effective open end, 0.015 m farther out.

Separate displacement and pressure amplitudes for effective length 0.175 metresBoth this model and the other air-length model use the same horizontal position scale from the closed end. The physical mouth is at 0.160 metres; the effective open end is at 0.175 metres. Two separate nonnegative amplitude graphs show displacement relative to its own maximum above, and pressure variation relative to its own maximum below. They do not share a physical amplitude unit. Displacement has nodes at zero and, where within the column, 0.350 metres; pressure variation has nodes at 0.175 and 0.525 metres where present. At the effective open end, displacement is an antinode and pressure variation is a node. At the closed end the opposite holds. The shapes are amplitude envelopes, not transverse particle paths or simultaneous signed pressure and displacement traces.Displacement amplitude ratioPressure amplitude ratio01010.0000.1750.3500.525Mouth: 0.160 mEffective end: 0.175 mDistance from closed end / m

A pressure node means no oscillating pressure variation there, not a vacuum. The two physical lengths differ by 0.350 m = λ/2; their common end correction cancels.

Air-length model 2: physical length 0.510 m

These model comparisons align the closed end at x = 0. They do not show the apparatus mouth moving. Grey dashed: physical mouth. Brown dotted: effective open end, 0.015 m farther out.

Separate displacement and pressure amplitudes for effective length 0.525 metresBoth this model and the other air-length model use the same horizontal position scale from the closed end. The physical mouth is at 0.510 metres; the effective open end is at 0.525 metres. Two separate nonnegative amplitude graphs show displacement relative to its own maximum above, and pressure variation relative to its own maximum below. They do not share a physical amplitude unit. Displacement has nodes at zero and, where within the column, 0.350 metres; pressure variation has nodes at 0.175 and 0.525 metres where present. At the effective open end, displacement is an antinode and pressure variation is a node. At the closed end the opposite holds. The shapes are amplitude envelopes, not transverse particle paths or simultaneous signed pressure and displacement traces.Displacement amplitude ratioPressure amplitude ratio01010.0000.1750.3500.525Mouth: 0.510 mEffective end: 0.525 mDistance from closed end / m

A pressure node means no oscillating pressure variation there, not a vacuum. The two physical lengths differ by 0.350 m = λ/2; their common end correction cancels.

The apparatus mouth stays fixed while the water level changes. The separate mode comparisons align their closed ends and distinguish each physical mouth from its effective open end. Displacement and pressure-variation envelopes are separately normalised; their vertical scales do not share a physical unit.

In the closed/open model, the first two effective resonant lengths are λ/4 and 3λ/4. The second pattern contains an additional internal displacement node and antinode. Increasing the effective length by λ/2 gives the next resonance.

Estimate the wavelength and speed

A successive resonant-length difference of roughly 0.35 m suggests λ about 0.7 m. With a fork frequency of order 500 Hz, v = fλ is of order 350 m/s. This assumes the same tube and successive resonances; it is a scale estimate, not an independent sound-speed measurement.

Worked resonant lengths

Use the difference to remove a common end correction

At a supplied frequency of 480 Hz, successive physical air lengths are L1 = 0.160 m and L2 = 0.510 m. Their effective open ends lie slightly beyond the tube mouth. Model this with the same correction e at both resonances.

(L2 + e) - (L1 + e) = λ/2
λ = 2(0.510 - 0.160) = 0.700 m
v = fλ = 480 × 0.700 = 336 m/s

The common end correction cancels in the difference. If these are the first two resonances, their effective lengths are:

λ/4 = 0.175 m
3λ/4 = 0.525 m
e = 0.175 - 0.160 = 0.015 m

The second length gives the same e = 0.525 - 0.510 = 0.015 m. Using λ = 4 × 0.160 m would treat the physical mouth as the exact effective endpoint and underestimate the wavelength.

Optional check A 480 Hz fork gives successive resonances at physical air lengths 0.160 m and 0.510 m in the same closed/open tube. Assume a common end correction. What are the wavelength and wave speed?
A 480 Hz fork gives successive resonances at physical air lengths 0.160 m and 0.510 m in the same closed/open tube. Assume a common end correction. What are the wavelength and wave speed?

Interpret the measurement limitations

A broad loudness maximum limits how precisely the resonant length can be located. Approach it from both sides and repeat the reading, while keeping the fork and tube arrangement consistent. Read the position scale with the line of sight perpendicular to it to reduce parallax.

The difference method assumes genuinely successive resonances and a common effective-end correction. Skipping a resonance changes the number of half-wavelength intervals. A changed tube opening can change the correction, so repeating readings alone would not repair that change.

The supplied lengths illustrate the inference. In an experiment, retain the original position readings, their units and the spread of repeated peak locations before reporting a wavelength.