Kundt's Tube Standing Waves lab report

1. Title and aim

Interference of Waves: Kundt's Tube, Standing Waves in Air, and Sound Velocity in Solids and Gases. To investigate acoustic wave interference and stationary wave formation in an air column using Kundt's tube, observe powder striations at displacement nodes and antinodes, determine the velocity of longitudinal sound waves in a solid metal rod and dry air, and calculate the dynamic Young's modulus of the rod using linear regression.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Sound velocity in brass rod of 3496.9 m s⁻¹, differing by 0.087% 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

• Kundt's Acoustic Resonance Tube: Heavy-wall precision borosilicate glass tube (1.5 m length, 32 mm internal diameter) equipped with metric scale graduations, gas inlet/outlet stopcocks, and vibration-isolated cradle mounts.
• Adjustable Acoustic Reflector Piston: Tightly fitting rubber-edged reflector piston mounted on an extended steel control rod, allowing continuous acoustic air column length tuning to exact standing wave resonance.
• Standard Brass Sounding Rod: Precision-turned solid brass rod (12.7 mm diameter, selectable lengths from 80 cm to 130 cm) clamped at its geometric center, fitted with a lightweight cork coupling disc at one end.
• Center-Clamping Vise Stand: Massive cast-iron bench vise with vulcanized leather clamp jaws designed to firmly grip the rod at its exact nodal center without damping longitudinal vibrations.
• Lycopodium Powder / Fine Cork Dust: Dehydrated, uniform microscopic spores (clubmoss) providing high fluidic mobility to form sharp, well-defined nodal heap striations under acoustic excitation.
• Excitation Cloth and Rosin: Coarse woolen or leather cloth impregnated with powdered rosin to stroke the free half of the metal rod and excite strong longitudinal vibrations.
• Ambient room temperature: T = 20.0 °C (Speed of sound in dry air: v_air = 343.20 m s⁻¹).
• Reference longitudinal wave velocity in brass rod: v_rod = 3500.0 m s⁻¹ (Dynamic Young's modulus E ≈ 104.1 GPa).
• Sounding rod material: High-tensile solid brass (diameter d_rod = 12.7 mm, density ρ = 8500 kg m⁻³).
• Acoustic resonance tube: Borosilicate glass (length = 1.50 m, internal bore = 32.0 mm).
• Coupling disc clearance: Radial air gap = 1.5 mm (frictionless acoustic transmission).
• Acoustic indicator powder: Dehydrated lycopodium spore powder (particle diameter ≈ 30 μm).

4. Theory and graph choice

When two coherent longitudinal acoustic waves of identical frequency and amplitude propagate in opposite directions along a bounded gaseous column, their linear superposition establishes an acoustic standing (stationary) wave. In Kundt's tube apparatus, invented in 1866, longitudinal sound waves are excited inside a long glass tube by the periodic vibrations of a solid metal rod clamped firmly at its center. One free end of the rod is fitted with a light cork disc that projects into the mouth of the tube, while the far end of the tube is terminated by an adjustable acoustic reflector piston. Stroking the protruding half of the rod with a resined cloth generates intense longitudinal standing waves in the metal bar, producing a displacement node at the rigid center clamp and displacement antinodes at both free ends. Consequently, the fundamental wavelength in the rod is twice its length: λ_rod = 2 L_rod, vibrating at natural frequency f = v_rod / (2 L_rod). As the cork disc oscillates back and forth at this exact frequency, it launches longitudinal compression waves down the air column. When the reflector piston is adjusted so that the tube's effective acoustic length is an integral multiple of half-wavelengths, multiple reflections constructively interfere to form a stationary wave pattern in the air. Lightweight lycopodium powder or fine cork dust distributed along the bottom of the tube acts as an aerodynamic indicator: at displacement antinodes, acoustic particle velocity reaches violent maxima, blowing the particles into agitated transverse striations; at displacement nodes, the acoustic particle velocity is zero, allowing dust particles to come to rest and accumulate into sharp, quiet nodal heaps. The distance d between consecutive dust heaps represents exactly one-half of the acoustic wavelength in air: d = λ_air / 2. Because the air column is driven synchronously at the rod's frequency (f_air = f_rod), equating frequencies yields the fundamental Kundt velocity ratio: v_rod / v_air = λ_rod / λ_air = (2 L_rod) / (2 d) = L_rod / d, or d = (v_air / v_rod) L_rod. By measuring the dust node spacing d across various rod lengths L_rod, ordinary least-squares regression yields the velocity ratio v_air / v_rod as the slope m. With the speed of sound in air known from Laplace's thermodynamic formula v_air = √(γ R T / M) (343.20 m/s at 20.0 °C), the longitudinal wave propagation speed in the solid metal rod is determined as v_rod = v_air / m. Furthermore, knowing the material density ρ_rod allows direct calculation of the dynamic Young's modulus E = ρ_rod v_rod² without relying on static mechanical deformation.

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 m⁻¹.

5. Method

1. Thoroughly dry the interior of the Kundt's glass tube using a dry flannel swab. Distribute a light, uniform track of dry lycopodium powder or fine cork dust along the entire bottom length of the horizontal tube by gently rocking and tapping the tube cradle.
2. Measure the total length of the brass rod L_rod with a steel meter rule. Mark the exact geometric center (L_rod / 2) with a fine pencil line, and clamp the rod firmly in the heavy vise jaws at this exact midpoint, ensuring the cork disc enters the tube mouth without touching the glass walls.
3. Insert the adjustable reflector piston into the far end of the glass tube. Position the tube so that the vibrating cork disc on the rod penetrates approximately 1 to 2 cm into the near end, ensuring an airtight but frictionless acoustic coupling.
4. Grip the free protruding end of the brass rod with the rosin-dusted leather cloth. Pull firmly along the axis of the rod toward the free tip with a steady, decisive stroke to emit a piercing, clear musical note. Simultaneously adjust the position of the far reflector piston until the powder inside the tube violently agitates and settles into sharp, well-defined transverse striations.
5. Using the tube's graduated metric scale, record the exact positions of five consecutive well-defined dust heaps (displacement nodes). Calculate the distance between successive heaps and compute the mean nodal spacing d = λ_air / 2.
6. Repeat the excitation, piston tuning, and nodal distance recording for six distinct calibrated rod lengths (80.0, 90.0, 100.0, 110.0, 120.0, and 130.0 cm). Tabulate rod length L_rod against dust node spacing d for subsequent linear regression.

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 = 0.0873%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 4.7619e-09; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 8.2479e-05. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 8.7741e-05.

Propagating only the relevant fit coefficient gives a fit-only standard error of 2.93881 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.

Dust heap apex locating visual uncertainty (graduated glass scale parallax ±0.5 mm), acoustic air column standing wave resonance width (±1 to 2 mm in piston tuning), room air temperature gradient and drafts along the 1.5 m glass tube (±0.3 °C producing ±0.18 m/s sound speed uncertainty), relative humidity and air density fluctuations inside the closed tube, clamp jaw compliance and finite clamp width at the rod center (effective node width ~1 mm), radial Poisson contraction dispersion in thick rods (Pochhammer-Chree correction ~0.15%), and Helmholtz-Kirchhoff acoustic boundary-layer viscous drag along the tube walls reducing wave speed by ~0.3% contribute experimental uncertainty. Ordinary least-squares regression of dust node spacing d against rod length L_rod isolates the pure sound velocity ratio v_air / v_rod independent of constant end-gap 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

• Ensure the vibrating cork coupling disc never touches the inner wall of the glass tube during excitation to avoid damping the vibrations or cracking the glass.
• Stroke the metal rod strictly in the longitudinal axial direction; avoid lateral or twisting forces that excite undesirable flexural or torsional vibration modes.
• Keep the interior of the glass tube completely dry and free of condensation; even slight humidity causes lycopodium powder to aggregate into immobile clusters.
• Wear acoustic ear defenders during longitudinal rod excitation, as high-frequency acoustic standing waves can exceed 95 dB SPL near the apparatus.

The analytical model treats the sounding rod as an ideal, infinitely thin elastic bar with strictly one-dimensional longitudinal stress (Pochhammer-Chree radial inertia dispersion neglected), assumes perfectly planar acoustic wavefronts inside the glass tube with zero viscous tube boundary-layer drag (Helmholtz-Kirchhoff wall attenuation), and models the cork disc as a massless acoustic source. In a physical laboratory, small radial Poisson expansion in thick rods slightly alters overtone frequencies, narrow glass bore introduces small acoustic velocity depression (~0.2% to 0.4% from wall friction), and room temperature gradients cause minor sound velocity shifts in air.

12. Conclusion

The worked example illustrates how to obtain Sound velocity in brass rod from measurements, graph variables and a fitted coefficient. The estimate is 3496.9 m s⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.