Resonance of Air Columns lab report

1. Title and aim

Resonance of Air Columns: Speed of Sound in Air and Acoustic End Correction. To determine the speed of sound in air at laboratory ambient temperature, calculate its extrapolated value at 0 °C, and measure the acoustic end correction of a cylindrical glass resonance tube using calibrated tuning forks and an adjustable water-level reservoir.

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 343.14 m s⁻¹, differing by 0.018% 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 Calibrated Resonance Tube with Water Reservoir: A vertical heavy-walled borosilicate glass cylinder (length 120 cm, inner diameter 32.0 mm, radius r = 1.60 cm) graduated in millimetres, connected at its bottom outlet via heavy-gauge flexible rubber tubing to a counterweighted water reservoir mounted on an adjustable vertical support rod, enabling continuous and rapid hydrostatic manipulation of the internal water meniscus.
• Standard Acoustic Tuning Fork Set with Striking Mallet: A precision-matched set of metallurgical acoustic tuning forks milled from high-tensile chrome-vanadium steel with stamped nominal frequencies across the musical scale (Middle C 256.0 Hz, D 288.0 Hz, E 320.0 Hz, G 384.0 Hz, B 480.0 Hz, and C' 512.0 Hz), accompanied by a soft vulcanized rubber striking mallet to excite clean, fundamental sinusoidal transverse oscillations without non-harmonic overtone chattering.
• Graduated Millimetre Scale and Sliding Index Markers: A high-contrast photo-etched metric scale bonded along the longitudinal axis of the glass tube accompanied by twin adjustable elastic rubber locator rings that slide along the tube exterior to clamp and mark the auditory loudness maxima during bidirectional meniscus searching.
• Laboratory Precision Thermometer & Digital Barometer: A mercury-in-glass thermometer calibrated to 0.1 °C suspended adjacent to the tube mouth to measure the ambient room temperature t for adiabatic velocity calculation, alongside an aneroid barometer recording ambient atmospheric pressure P_atm.
• Vernier Caliper for Tube Bore Metrology: A stainless-steel precision vernier caliper with internal knife-edge measuring jaws reading to 0.02 mm, utilized to measure the internal diameter 2r of the glass tube at multiple azimuthal angles to compute the theoretical Rayleigh end correction c = 0.6133 r.
• Acoustic Sound Level Meter / Resonance Microphone Probe: An auxiliary digital sound pressure level (SPL) meter with decibel readout (dBA) or narrow-band microphone positioned near the tube lip to provide quantitative, objective verification of auditory resonance peaks alongside human sensory ear tuning.
• Acoustic medium: ambient laboratory air at constant room temperature t = 20.0 °C (293.15 K)
• Resonance tube: heavy-walled borosilicate glass (internal diameter 2r = 32.0 mm, radius r = 1.60 cm)
• Water leveling system: counterweighted reservoir with flexible hydraulic rubber hose coupling
• Calibrated tuning fork set: steel forks stamped at 256.0 Hz, 288.0 Hz, 320.0 Hz, 384.0 Hz, 480.0 Hz, 512.0 Hz
• Excitation technique: soft vulcanized rubber mallet strike one-third from prong tip to suppress higher overtones
• Theoretical standards: speed of sound at 0 °C v₀ = 331.30 m/s; Rayleigh end correction c = 0.6133 r = 0.98 cm

4. Theory and graph choice

Acoustic resonance in closed-end cylindrical tubes represents one of the most fundamental manifestations of longitudinal standing waves in fluid mechanics and classical continuum acoustics. When a vibrating tuning fork is held immediately above the open aperture of a vertical cylindrical tube partially filled with water, longitudinal sound waves propagate downward through the air column, undergo reflection at the dense water-air boundary with an acoustic phase inversion of displacement, and travel upward to superimpose with newly incident wavefronts. At particular discrete air column lengths, the incident and reflected acoustic pressure waves interfere constructively, establishing stationary longitudinal wave modes characterized by a displacement node (pressure antinode) at the rigid water meniscus and a displacement antinode (pressure node) at the open upper termination. Because the open aperture communicates with an infinite half-space of unconfined ambient atmosphere, the effective acoustic boundary condition does not coincide precisely with the physical lip of the tube; rather, the radiation impedance of the escaping sound waves forces the displacement antinode to form at a small distance c (the tube end correction) outside the tube mouth. In 1860, Hermann von Helmholtz developed the mathematical foundations of acoustic aperture impedance, and Lord Rayleigh subsequently demonstrated that for a cylindrical pipe of radius r, the acoustic end correction is approximately c ≈ 0.6133 r for an unflanged lip and c ≈ 0.82 r for an infinitely flanged pipe. For a tube of length L closed at one end, the fundamental resonance condition requires that the effective acoustic length L + c equals an odd quarter-wavelength: L₁ + c = λ / 4 = v / (4f), where v is the speed of sound in air and f is the vibrational frequency of the exciting source. By systematic variation of the tuning fork frequency across a calibrated acoustic set (e.g., 256 Hz to 512 Hz), students obtain a linear relationship between the physical first resonance column length L₁ and the reciprocal acoustic frequency 1/f: L₁ = (v / 4)(1 / f) − c. Applying ordinary least-squares linear regression to this acoustic dataset allows simultaneous and independent determination of the propagation speed of sound v from four times the regression slope (v = 4m) and the physical end correction c from the negative vertical intercept (c = −b). Furthermore, by recording the ambient room temperature t, students can compare their experimental velocity with Laplace's adiabatic acoustic wave equation v(T) = √(γ R T / M) = v₀ √(1 + t / 273.15), providing rigorous validation of thermodynamic gas compressibility, acoustic radiation impedance, and boundary-value continuum mechanics.

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 borosilicate glass resonance tube at three independent orthogonal orientations at the upper open aperture using precision vernier calipers reading to 0.02 mm. Record the ambient air temperature t to 0.1 °C using the calibrated laboratory thermometer mounted adjacent to the tube exterior, and verify that the laboratory is free from strong convective drafts, direct sunlight, and thermal plumes.
2. Fill the adjustable leveling reservoir with distilled water until the meniscus in the resonance tube rises to approximately 8 to 10 cm below the top rim. Check that the flexible connecting hose is free from trapped air bubbles or constrictions, and verify that raising and lowering the reservoir smoothly modulates the water height across the full 100 cm active measuring span of the tube.
3. Select the 256.0 Hz tuning fork. Strike one prong cleanly approximately one-third back from its distal end using the soft rubber mallet, initiating a pure fundamental sinusoidal vibration. Immediately mount or hold the vibrating fork directly above the center of the tube aperture, orienting the prongs so that their plane of vibration is vertical and parallel to the tube axis, maintaining a clearance of approximately 5 to 8 mm (1/4 inch) above the glass lip without permitting physical contact.
4. While striking the tuning fork continuously to maintain steady acoustic power, slowly lower the water reservoir, causing the internal water level to descend. Listen intently near the tube mouth for a sudden, dramatic crescendo in sound loudness signifying acoustic resonance. Slide a rubber marker band to the approximate position. Refine the location by running the water level back and forth across a 3 cm window around the marker, clamping the marker precisely where auditory loudness reaches its sharp global maximum. Record this position as the first resonance length L₁.
5. Repeat the precision resonance determination for each of the remaining tuning forks in the calibrated set: 288.0 Hz, 320.0 Hz, 384.0 Hz, 480.0 Hz, and 512.0 Hz. For each frequency, execute three independent bidirectional runs to establish reproducible first resonance lengths L₁, and record all observations in the formal laboratory notebook alongside ambient temperature readings.
6. For the higher frequency forks (such as 480.0 Hz and 512.0 Hz), continue lowering the water reservoir beyond the first resonance point to locate the second resonance position L₂ (corresponding to the 3λ/4 mode). Record the distance between successive resonance points ΔL = L₂ − L₁. Compute the independent wavelength λ = 2(L₂ − L₁) and the single-fork end correction c = (L₂ − 3L₁) / 2 to cross-validate the multi-frequency regression results.

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 = 0.0183%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 3.2487e-09; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.01637. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 4.8201e-05.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.06548 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 meniscus locating uncertainty (visual parallax against graduated glass scale ±0.5 mm), acoustic auditory maximum discrimination width (±2 to 3 mm of water column displacement around peak resonance loudness), room air temperature gradient and convective drafts between top of tube and water reservoir surface (±0.3 °C producing ±0.18 m/s speed uncertainty), water vapor saturation inside the air column altering effective adiabatic index γ and molar mass M_mix (up to +0.25% acoustic speed increase over bone-dry air), higher-order acoustic transverse modes and tube wall elasticity compliance, and nominal stamped fork frequency calibration tolerance (±0.1 Hz) contribute experimental uncertainty. The OLS regression standard error reflects random scatter across frequency stations, with the slope isolating the true acoustic speed independent of constant zero-offset end corrections.

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

• Strike the tuning forks exclusively with the vulcanized soft rubber mallet; never strike the forks against metallic stands, glass tube rims, or bench tops to prevent permanent detuning or scratching.
• Hold the vibrating tuning fork strictly in the vertical plane centered 5 to 8 mm above the tube lip, ensuring the vibrating prongs never touch the glass lip during acoustic resonance searching.
• Avoid breathing directly into the tube mouth or placing hot hands on the borosilicate glass cylinder, which would establish internal thermal gradients and convection currents.
• Execute bidirectional water level searching (approaching resonance loudness from both above and below) to eliminate auditory loudness perception lag and glass wetting meniscus hysteresis.

The teaching model assumes an idealized inviscid acoustic gas column with perfect cylindrical planar wavefront propagation and a sharp Dirichlet boundary condition at the water meniscus. In a physical laboratory, small thermal boundary-layer conduction and viscous drag at the inner glass wall produce slight Kirchhoff-Helmholtz acoustic dispersion, water evaporation introduces a finite humidity gradient altering gas density, and finite human auditory discrimination width creates small random setting uncertainties. The resonance tube method combines boundary-value wave mechanics with acoustic impedance matching at an open aperture. By measuring the first resonance length L₁ across six calibrated tuning fork frequencies, students linearize the quarter-wave condition L₁ = (v/4)(1/f) − c. Ordinary least-squares regression isolates both the propagation velocity v = 4m = 343.2 m s⁻¹ and the boundary antinode offset c = 0.98 cm with sub-percent fidelity, confirming Helmholtz's end-correction theory.

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 343.14 m s⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.