Modes of Vibration of an Air Column lab report

1. Title and aim

Modes of Vibration of an Air Column: Open and Closed Organ Pipes, Harmonics, and Dual End Corrections. To investigate the stationary longitudinal standing wave modes in open and closed cylindrical air columns, determine the fundamental resonance lengths across variable acoustic frequencies, measure the dual-aperture acoustic end correction 2c, compare full-integer versus odd harmonic spectra, and calculate the speed of sound in air at laboratory room temperature.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Speed of sound in air v of 343.22 m s⁻¹, differing by 0.004% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Precision Telescoping Open Resonance Tube Assembly: A pair of concentric, close-tolerance seamless brass and acrylic cylindrical tubes (internal diameter 2r = 4.00 cm, radius r = 2.00 cm) equipped with Teflon sliding low-friction acoustic O-ring seals, permitting smooth, continuous variation of the open-open air column length from 25.0 cm to 80.0 cm without acoustic leakage along the seam.
• Variable-Frequency Acoustic Siren Disk & Driver System: A motorized Cagniard de la Tour acoustic siren disk with concentric circle hole rings driven by a variable-speed rotator and optical tachometer counter, complemented by a lab audio frequency generator and horn compression driver delivering pure sinusoidal longitudinal acoustic pressure waves from 200 Hz to 1200 Hz.
• Removable Rigid End Caps (Closed-Pipe Conversion Kit): A set of dense, acoustically reflective Delrin and brass end stoppers fitted with silicone gaskets that clamp hermetically over one open end of the telescoping tube, instantly converting the open-open acoustic waveguide into a closed-open organ pipe for direct comparative mode analysis.
• Precision Metric Scale and Linear Vernier Slider: A photo-etched high-contrast metric scale graduated in millimetres bonded to the outer brass tube with an optical vernier index magnifier reading to 0.1 mm to measure the exact physical tube length L between the metal lip rims.
• Internal Miniature Acoustic Pressure Probe (Microphone Array): A slender 2.0 mm diameter probe electret microphone mounted on a rigid sliding brass wand that can be traversed longitudinally along the central axis inside the pipe to map spatial acoustic pressure distributions, locating displacement nodes and antinodes directly.
• Digital Oscilloscope and Fast Fourier Transform (FFT) Analyzer: A dual-channel digital storage oscilloscope with real-time FFT spectrum analyzer displaying the frequency spectrum (harmonics 1f, 2f, 3f, 4f, 5f) and calculating the Total Harmonic Distortion (THD) and acoustic quality factor Q of each resonance peak.
• Pipe configuration: precision open-ended brass/acrylic telescoping cylinder (bore diameter 2r = 4.00 cm, radius r = 2.00 cm)
• Acoustic boundary conditions: open-open (displacement antinodes at both apertures, effective length L_eff = L + 2c)
• Excitation source: calibrated variable acoustic driver / precision tuning fork set (256.0 Hz to 512.0 Hz)
• Harmonic mode index: fundamental mode n = 1 (half-wavelength λ₁ / 2 = L + 2c)
• Dual end correction: theoretical Rayleigh offset 2c = 2(0.6133 r) = 2.45 cm (1.23 cm per aperture)
• Ambient laboratory conditions: dry atmospheric air at t = 20.0 °C (293.15 K), reference v = 343.20 m/s

4. Theory and graph choice

The acoustic modes of vibration in bounded air columns govern the operational physics of musical wind instruments, pipe organs, architectural ducts, and acoustic resonators. When longitudinal sound waves travel through a cylindrical pipe, their behavior at the pipe terminations is dictated by the acoustic radiation impedance mismatch with the surrounding atmosphere. In an open pipe—where both ends communicate freely with the ambient air—the acoustic pressure must drop to ambient atmospheric pressure at both terminations, establishing an acoustic pressure node (displacement antinode) at each open aperture. Consequently, the fundamental resonant standing wave mode in an open pipe contains a displacement node at its spatial center and displacement antinodes at both extremities, accommodating exactly one half-wavelength between its effective acoustic boundaries: L_eff = λ₁ / 2. Because acoustic pressure does not equalize instantaneously at the physical rim of an aperture, each open end exhibits an acoustic end correction c ≈ 0.6133 r (for an unflanged pipe of radius r), so that the effective acoustic length of an open pipe is L_eff = L + 2c = L + 1.2266 r. The allowable normal modes for an open column form a complete harmonic series containing all positive integers: L + 2c = n (λ_n / 2) = n v / (2 f_n), where n = 1, 2, 3, 4, ... By contrast, closing one end of the pipe with a rigid stopper enforces a displacement node (pressure antinode) at that boundary, restricting the standing wave to odd quarter-wavelengths: L + c = (2m − 1) λ_m / 4, which permits only odd harmonics (1f, 3f, 5f, ...). In this classic university acoustic waveguide laboratory investigation, students utilize variable-length telescoping pipes and calibrated acoustic frequency drivers to isolate fundamental resonance lengths across the musical scale. Rearranging the fundamental open-pipe condition yields the linear regression relationship L = (v / 2)(1 / f₁) − 2c. By plotting the physical resonant length L against the reciprocal frequency 1/f₁, students obtain the speed of sound v directly from twice the regression slope (v = 2m) and the dual-end aperture correction from the negative vertical intercept (2c = −b), providing definitive experimental verification of acoustic radiation impedance and normal-mode waveguide theory.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units m s⁻¹.

5. Method

1. Measure the internal diameter 2r of the telescoping tube at both open apertures using precision internal vernier calipers across three orthogonal diameters to establish the mean bore radius r = 2.00 cm. Fully collapse the telescoping inner sleeve until the end rims are flush, verifying that the sliding metric index registers zero on the engraved graduation scale.
2. Mount the telescoping tube horizontally on vibration-isolation laboratory V-blocks. Position the acoustic driver horn coaxially with the left open aperture, maintaining an axial air gap of approximately 10 to 15 mm to ensure the aperture remains unbaffled and free to radiate sound into ambient space without acoustic boundary clamping.
3. Set the acoustic driver to the standard Middle C frequency f₁ = 256.0 Hz with pure sinusoidal excitation. Slowly extend the sliding inner tube while monitoring both the auditory loudness and the external sound level meter. Locate the position of maximum acoustic amplification where the tube roars into fundamental resonance. Bracket the setting by moving the sleeve back and forth across a 1 cm span, locking the brass clamp screw at the peak. Record this physical length as L.
4. Repeat the resonance length optimization for each of the remaining calibrated test frequencies: 288.0 Hz, 320.0 Hz, 384.0 Hz, 480.0 Hz, and 512.0 Hz. For each frequency, execute three independent settings from opposite directions, averaging the results to obtain the representative fundamental resonance length L in centimetres.
5. Fix the open tube length at L = 64.6 cm (the 256 Hz fundamental). Slowly increase the driver frequency from 250 Hz up to 1200 Hz, noting each frequency where a sharp acoustic resonance peak occurs. Record the resonance frequencies f₁, f₂, f₃, f₄ and verify that they form an integer harmonic progression f_n ≈ n f₁ (ratio 1 : 2 : 3 : 4). Insert the miniature microphone probe along the tube axis to confirm that the n = 2 mode exhibits two displacement nodes inside the bore.
6. Clamp the Delrin end cap tightly over the distal end of the tube, converting it to a closed-open pipe of identical physical length L = 64.6 cm. Sweep the driver frequency again from 100 Hz to 1200 Hz. Identify the new fundamental frequency f₁' and the subsequent overtones. Show that the fundamental frequency is halved (f₁' ≈ f₁ / 2 ≈ 130 Hz) and that even harmonics are completely absent, yielding only odd harmonics 1f₁', 3f₁', 5f₁' (ratio 1 : 3 : 5).

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0044%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 3.7895e-09; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.01768. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 5.2058e-05.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.03536 m s⁻¹. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Acoustic resonance peak auditory bandwidth (human loudness discrimination threshold ±3 to 4 mm along the sliding telescoping tube sleeve), acoustic end vortex turbulence and aperture reflection loss at high acoustic driving power, non-planar wavefront curvature near the open lips introducing higher-order radial evanescent modes, ambient room temperature drift and convective drafts across the horizontal bench (±0.3 °C producing ±0.18 m/s sound speed variation), sleeve mechanical seal compliance and friction, and stamped driver frequency calibration tolerance (±0.1 Hz) govern the experimental error budget. Ordinary least-squares regression across multiple frequencies separates the pure propagation velocity from constant dual end-correction offsets.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Maintain an axial air gap of 10 to 15 mm between the acoustic driver horn and the pipe aperture to prevent acoustic coupling mass loading or partial closure.
• Ensure the sliding telescoping joint is well-lubricated with a trace of light silicone grease to guarantee a hermetic acoustic seal without mechanical binding or distortion.
• Keep hands off the cylindrical tube walls during resonance measurement to avoid creating internal thermal gradients and localized convective eddies.
• Approach the resonance peak slowly from both shortening and lengthening directions to eliminate sliding friction hysteresis and ear perception lag.

The teaching model assumes an idealized planar 1D acoustic wave without transverse radial modes and treats the tube wall as perfectly rigid and isothermal. In a physical laboratory, small viscous and thermal boundary-layer damping at the inner wall produces slight Kirchhoff-Helmholtz phase velocity dispersion, minor acoustic radiation leaks through sliding joints can slightly broaden resonance peaks, and ambient room temperature gradients introduce small velocity uncertainties. The open air column experiment demonstrates the harmonic mode structure of acoustic waveguides with symmetric open-open boundary conditions. Because acoustic pressure drops to ambient atmospheric levels outside both apertures, displacement antinodes form at both ends, requiring a total dual end correction of 2c ≈ 1.2266 r. Ordinary least-squares regression of fundamental tube length L against reciprocal frequency 1/f isolates the propagation velocity v = 2m = 343.2 m s⁻¹ and the dual-end aperture offset 2c = 2.45 cm, confirming the half-wave resonance eigenvalue L + 2c = λ/2.

12. Conclusion

The worked example illustrates how to obtain Speed of sound in air v from measurements, graph variables and a fitted coefficient. The estimate is 343.22 m s⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.