To investigate the transverse standing wave modes on a stretched wire using a sonometer soundboard, verify Mersenne's laws relating fundamental frequency to vibrating length, tension, and linear density, observe sympathetic acoustic resonance and acoustic beats, and determine the transverse wave velocity and wire linear mass density using linear regression. Explore a virtual apparatus, collect readings and analyse a graph.
Ready for a virtual observation.
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 ideally flexible, infinitely thin wire with zero bending stiffness, negligible damping, and perfectly rigid bridge knife-edges. In a physical laboratory, small wire flexural rigidity (stiffness inharmonicity) slightly elevates higher overtone frequencies, minor bridge compliance adds small effective length offsets, and auditory beat detection threshold introduces slight visual/auditory setting scatter. The sonometer experiment provides a definitive validation of Mersenne's acoustic laws for continuous 1D elastic media. By tuning the string segment between movable fret bridges to unison with calibrated tuning forks using sympathetic paper rider ejection and beat minimization, students confirm the half-wave relation L = (v/2)(1/f). Linear regression yields the transverse propagation speed v = 138.2 m s⁻¹ and wire linear density μ = 1.541 g m⁻¹ with exceptional precision, demonstrating the physical foundation of stringed musical instrument acoustics.
A rigid hollow acoustic resonator box (length 110 cm, width 14 cm, depth 10 cm) constructed of seasoned Sitka spruce soundboard with internal spruce bracing and twin acoustic sound holes, designed to provide high mechanical-acoustic impedance matching that efficiently amplifies string vibrations into audible sound waves.
A precision-drawn metallurgical music spring wire (diameter d = 0.500 mm ± 0.005 mm, density ρ = 7850 kg/m³, linear density μ = 1.541 g/m) anchored at one end to a heavy cast brass hitch pin and passing over a low-friction ball-bearing pulley at the opposite end.
A nickel-plated steel weight hanger suspended vertically from the wire over the pulley, loaded with calibrated brass weights totaling M = 3.000 kg (yielding string mechanical tension T = M g = 29.420 N), establishing uniform, creep-free axial tension.
A pair of prism-shaped hardwood bridges capped with narrow nickel-silver fret wire knife-edges that slide smoothly along the soundboard beneath the wire, defining sharp, unequivocal displacement nodes that delineate the active vibrating segment length L.
A matched laboratory set of steel tuning forks milled to precise frequencies spanning the musical scale (C3 128.0 Hz, E3 160.0 Hz, G3 192.0 Hz, Middle C 256.0 Hz, E4 320.0 Hz, and G4 384.0 Hz), used to transfer pure sinusoidal reference vibrations to the soundboard.
Ultra-lightweight V-shaped folded slips of tissue paper (mass ~10 mg) straddling the wire midpoint to provide visual detection of sympathetic resonance agitation and violently jump off the wire at resonance, alongside an engraved 100 cm metric scale reading to 0.5 mm.
To investigate the transverse standing wave modes on a stretched wire using a sonometer soundboard, verify Mersenne's laws relating fundamental frequency to vibrating length, tension, and linear density, observe sympathetic acoustic resonance and acoustic beats, and determine the transverse wave velocity and wire linear mass density using linear regression. Suggested independent settings: 128.0, 160.0, 192.0, 256.0, 320.0, 384.0 Hz.
Inspect the steel sonometer wire to ensure it is completely straight, free from kinks, rust, or surface bends. Load the weight hanger with slotted masses until the total suspended mass equals M = 3.000 kg (tension T = 29.420 N). Verify that the wire runs smoothly over the center of the frictionless ball-bearing pulley and hangs vertically without rubbing against the soundboard frame.
Eliminating pulley friction and ensuring free vertical suspension guarantees that the mechanical tension T across the active segment equals the gravitational weight M g without frictional reduction.
Place the fixed zero-bridge firmly at the left end of the soundboard (aligned with the 0.0 cm graduation mark). Position the sliding bridge approximately 60 cm to the right. Verify that the wire rests snugly in the shallow guide notches atop the metal frets, ensuring both bridges establish rigid, unambiguous displacement nodes with zero lateral rocking.
Rigid bridge contact prevents rocking compliance that would effectively lengthen the vibrating segment beyond the visually measured distance between fret edges.
Cut a small V-shaped rider from lightweight paper (approximately 15 mm long by 2 mm wide) and fold it into an inverted 'V'. Carefully balance the rider upside down across the wire at the exact geometric midpoint between the two bridges (x = L / 2).
The midpoint of a string vibrating in its fundamental mode is the spatial displacement antinode; placing the rider here maximizes sympathetic agitation and ensures instantaneous ejection at true resonance.
Strike the 256.0 Hz tuning fork firmly with the soft rubber mallet. Immediately press the solid stem base of the vibrating fork firmly against the spruce soundboard near the fixed bridge. While the fork vibrates, slowly slide the movable bridge toward the fixed bridge. Observe the paper rider: it will remain motionless until the segment length approaches resonance, begin to flutter vigorously, and finally be thrown completely off the wire when the segment achieves perfect sympathetic resonance. Record this length as L.
Vibrational energy from the fork conducts through the soundboard into the wire; when the wire's natural fundamental frequency matches the fork frequency (unison), resonant energy transfer builds up standing wave amplitude until the rider is violently ejected.
Replace the paper rider. Pluck the wire gently near one bridge while simultaneously holding the sounding tuning fork against the soundboard. Listen intently for acoustic beats (pulsations in loudness caused by interference between the fork and string). Finely adjust the bridge position in sub-millimetre increments until the beats become slower and slower, finally dying out into a pure, unified, beatless unison note. Record the verified resonant length L to 0.5 mm.
The acoustic beat frequency equals the difference Δf = |f_fork − f_string|. Eliminating beats guarantees frequency matching to within a fraction of a Hertz, providing exceptional metrological accuracy.
Repeat the sympathetic rider resonance and beat-elimination procedure for each of the remaining tuning forks: 128.0 Hz, 160.0 Hz, 192.0 Hz, 320.0 Hz, and 384.0 Hz. For each frequency, execute three independent settings approaching from both shorter and longer directions, and record all observations in the formal laboratory notebook.
Collecting data across a wide range of frequencies spanning 128 Hz to 384 Hz provides an extended lever arm for linear regression of L against 1/f, isolating the transverse wave speed v and linear density μ.
Points are your observations. The fit uses SI x and y values before display scaling.
| # | Tuning fork frequency f (Hz) | Resonant string segment L (cm) | Inverse frequency (1 / f) | Resonant segment length L | Remove |
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