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. Explore a virtual apparatus, collect readings and analyse a graph.
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Adjust the apparatus, then take a reading.
Timing is generated from the model; a short on-screen trial represents a longer physical interval. Animation is illustrative.
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.
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.
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.
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.
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.
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.
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.
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. Suggested independent settings: 256.0, 288.0, 320.0, 384.0, 480.0, 512.0 Hz.
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.
Determining internal radius r with high accuracy is essential for calculating Rayleigh's theoretical dual end correction 2c = 1.2266 r, while zeroing the telescoping scale eliminates zero-offset errors in subsequent physical length readings.
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.
Maintaining an open gap preserves the natural radiation impedance of the unflanged pipe rim, preventing the driver horn from behaving as an acoustic acoustic mass or partial closed termination.
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.
At fundamental resonance, the acoustic impedance looking into the open pipe reaches an acoustic minimum, creating maximum volume velocity and acoustic sound radiation into the laboratory.
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.
Varying frequency across a full acoustic octave provides an extended lever arm for linear least-squares regression of L against 1/f, minimizing slope variance and isolating the propagation velocity v.
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.
Demonstrating that open pipes support both even and odd harmonics validates the symmetric open-open boundary condition, where the wave must satisfy pressure nodes at both terminations.
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).
Directly contrasting open vs closed configurations using the exact same physical tube provides unambiguous experimental proof that boundary conditions govern the allowable harmonic eigenvalues and acoustic timbre.
Points are your observations. The fit uses SI x and y values before display scaling.
| # | Excitation frequency f (Hz) | Open pipe fundamental length L (cm) | Inverse frequency (1 / f) | Open pipe resonant length L | Remove |
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