Resonance Tubes: Finding Speed of Sound for H2 Physics Practicals

TL;DR
Turn a PVC pipe and tuning fork into a precision sound speed measurement device. This guide shows how to find resonance positions to ±1mm, apply end corrections properly, account for temperature variations, and achieve results within 1% of the theoretical 343 m/s. Plus smartphone alternatives that work just as well.

The Physics of Pipe Resonance

When sound waves bounce inside a tube, magic happens at specific lengths - the reflected waves reinforce the original, creating resonance. This experiment elegantly connects:

• Wave properties and standing waves
• Boundary conditions (closed vs open ends)
• Temperature dependence of sound speed
• Practical applications (organ pipes, wind instruments)

Your measurement of sound speed will be as accurate as professional equipment, using basic materials.

Core Theory and Equations

Standing Waves in Tubes

For a tube closed at one end:

• Closed end: Displacement node (pressure antinode)
• Open end: Displacement antinode (pressure node)

Resonance condition: \[L = \frac{(2n-1)\lambda}{4}\]

Where:

• \(L\) = Effective length of air column
• \(n\) = 1, 2, 3... (resonance number)
• \(\lambda\) = Wavelength

Speed of Sound Calculation

Since \(v = f\lambda\):

\[v = f \times 4(L_2 - L_1)\]

Where \(L_2 - L_1\) is the distance between successive resonances.

Temperature Dependence

\[v = 331.3\sqrt{1 + \frac{T}{273.15}} \text{ m/s}\]

Or approximately: \(v ≈ 331.3 + 0.606T\) for \(T\) in °C

Equipment Setup

Method 1: Traditional Water Column

Materials:

• Clear tube (1-1.5m long, 3-5cm diameter)
• Water reservoir with flexible tube
• Tuning forks (256Hz, 512Hz typical)
• Meter ruler
• Thermometer
• Striking pad (rubber/cork)

Assembly:

1. Mount tube vertically with ruler alongside
2. Connect water reservoir to bottom
3. Fill system (no air bubbles!)
4. Practice raising/lowering water level

Method 2: Modern PVC Pipe

Simpler alternative:

• PVC pipe (1-2m, smooth interior)
• Moveable piston (tight-fitting disc)
• Marked rod attached to piston
• No water needed!

Method 3: Smartphone Speaker

21st century approach:

• Frequency generator app
• Bluetooth speaker
• Any tube
• Sound meter app for detection

Experimental Procedure

Finding First Resonance

1. Strike tuning fork firmly (consistent amplitude)
2. Hold vibrating fork above tube (~1cm gap)
3. Lower water level slowly (or move piston out)
4. Listen for sudden amplification
5. Fine-tune position for maximum sound
6. Record position \(L_1\) carefully

Finding Subsequent Resonances

Continue lowering water/extending tube:

• Second resonance at \(L_2 ≈ 3L_1\)
• Third at \(L_3 ≈ 5L_1\)
• Pattern: Odd multiples

Critical: Use same tuning fork throughout!

Measurements to Record

For each resonance:

• Position (±1mm)
• Which resonance (1st, 2nd, etc.)
• Room temperature
• Tuning fork frequency
• Background noise level

The End Correction Mystery

Why It Matters

The effective length extends beyond the tube opening:

\[L_\text{effective} = L_\text{measured} + e\]

Where end correction \(e ≈ 0.3d\) to \(0.6d\) (\(d\) = tube diameter)

Experimental Determination

Since \(L_2 - L_1 = \frac{\lambda}{2}\):

1. Measure multiple \((L_{n+1} - L_n)\) intervals
2. These equal \(\frac{\lambda}{2}\) regardless of \(e\)
3. Calculate \(v = f \times 2(L_{n+1} - L_n)\)
4. End correction cancels out!

Verifying End Correction

Plot \(L_n\) vs \(n\):

• Gradient = \(\frac{\lambda}{2}\)
• Y-intercept = \(-e\)

Typical result: \(e ≈ 0.4d\)

Data Collection Best Practices

Maximizing Precision

1. Approach resonance from both directions
   • Note positions approaching from above/below
   • Average for best estimate
2. Multiple frequency method
   • Use 2-3 different tuning forks
   • Plot \(v\) vs \(f\) (should be horizontal)
   • Confirms wave equation
3. Temperature monitoring
   • Record at start and end
   • Calculate expected change in \(v\)
   • Apply correction if needed

Common Measurement Pitfalls

"Can't hear clear resonance"

• Fork not vibrating strongly enough
• Holding fork too far from tube
• Background noise interference
• Tube diameter too small/large

"Multiple resonance positions"

• Temperature gradients in water
• Tube not perfectly vertical
• Air bubbles in water column

Alternative Methods

Method 1: Two-Microphone Phase

Using smartphones/computers:

1. Place mics at known separation
2. Generate pure tone
3. Measure phase difference
4. Calculate wavelength

Method 2: Kundt's Tube (Visual)

Classic demonstration:

1. Horizontal tube with speaker at one end
2. Sprinkle cork dust/lycopodium powder
3. Dust forms patterns at nodes
4. Measure node spacing = \(\frac{\lambda}{2}\)

Method 3: Echo Timing

Direct time-of-flight:

1. Sharp sound pulse (clap/click)
2. Measure echo time from distant wall
3. \(v = \frac{2d}{t}\)
4. Need \(d > 50m\) for accuracy

Smartphone App Experiments

Frequency Generator + Spectrum Analyzer

Modern approach benefits:

• Any frequency (not limited to tuning forks)
• See resonance peak on screen
• Precise frequency control
• Record data automatically

Recommended apps:

• Frequency Generator: "Signal Generator"
• Analysis: "Spectroid" (Android) or "SpectrumView" (iOS)

Procedure:

1. Set generator to pure tone (start ~500Hz)
2. Place phone speaker at tube opening
3. Position analyzer mic inside tube
4. Vary tube length watching amplitude
5. Screenshot at resonances

Automated Sweep Method

Some apps can sweep frequency:

1. Fix tube length
2. Sweep 100-1000Hz slowly
3. Note frequencies of amplitude peaks
4. Calculate wavelengths and average \(v\)

Advanced Analysis

Uncertainty Calculations

Main contributions:

• Position measurement: \(\delta L = ±1\) mm
• Frequency uncertainty: \(\delta f = ±0.5\) Hz (tuning fork)
• Temperature: \(\delta T = ±0.5\) °C

For speed of sound: \[\frac{\delta v}{v} = \sqrt{\left(\frac{\delta f}{f}\right)^2 + \left(\frac{2\delta L}{L_2-L_1}\right)^2}\]

Typical achievement: \(\delta v ≈ ±2\) m/s (0.6%)

Temperature Correction Graph

Plot your results:

1. Measure \(v\) at different room temperatures
2. Plot \(v\) vs \(T\)
3. Compare gradient to theory (0.606 m/s/°C)
4. Extrapolate to find \(v_0\) at 0°C

Dispersion Investigation

Does \(v\) depend on frequency?

1. Use multiple tuning forks (256, 512, 1024 Hz)
2. Measure \(v\) for each
3. Plot \(v\) vs \(f\)
4. Should be constant (non-dispersive medium)

Common Exam Questions

Q1: "Why must the tube be open at one end?"

Key points:

• Open end allows pressure variations
• Creates pressure node (displacement antinode)
• Closed end would give different resonance pattern
• Explains why \(L = \frac{(2n-1)\lambda}{4}\) not \(\frac{n\lambda}{2}\)

Q2: "Explain the end correction"

Model answer:

• Sound wave extends beyond tube opening
• Effective length > physical length
• Depends on tube diameter
• Cancels when using \((L_2-L_1)\) method

Q3: "How would results change in helium?"

Consider:

• \(v_\text{He} ≈ 3 \times v_\text{air}\) (lower density)
• Same frequency → 3x wavelength
• Resonances at 3x distances
• Pitch changes in voice explained

Practical Tips for Success

Getting Clear Resonances

1. Strike tuning fork correctly
   • Hit prong end on rubber pad
   • Not too hard (distorts frequency)
   • Hold by stem only
2. Minimize damping
   • Don't touch vibrating prongs
   • Keep fork close to tube
   • Work quickly before amplitude decays
3. Reduce interference
   • Work away from noisy equipment
   • Close windows/doors
   • Use sound-absorbing materials around setup

Data Recording Template

• Tuning-fork frequency \(f = 512 \space \text{Hz}\)
• Ambient temperature \(T = 23.5 \space ^{\circ}\text{C}\)

\[ \overline{v} = 343.6 \pm 2 ;\text{m/s} \]

\[ v_{\text{theory}}(23.5^{\circ}\text{C}) = 345.5 ;\text{m/s} \]

\[ \text{Percentage difference} = \frac{\lvert \overline{v} - v_{\text{theory}} \rvert}{v_{\text{theory}}}\times 100% = 0.5% \]

Links to A-Level Topics

Waves and Superposition

• Standing wave formation demonstrated
• Nodes and antinodes visible through resonance
• Constructive interference at specific lengths

Boundary Conditions

• Open end: Pressure node
• Closed end: Pressure antinode
• Determines allowed wavelengths

Sound and Music

• Why instruments have specific lengths
• How temperature affects tuning
• Harmonic series in pipes

Extensions and Investigations

1. Humidity Effects

Add humidity sensor:

• Measure on dry/humid days
• Plot \(v\) vs relative humidity
• Small but measurable effect

2. Open-Open Tube

Both ends open:

• Different resonance condition
• \(L = \frac{n\lambda}{2}\)
• Compare with closed tube

3. Real Wind Instruments

Analyze recorder/flute:

• Measure hole positions
• Predict notes from lengths
• Verify with frequency app

4. Acoustic Impedance

Advanced topic:

• Why brass instruments have flared bells
• Impedance matching
• Efficiency of sound radiation

Your Lab Success Checklist

✓ Check tuning fork frequency (often stamped on it)
✓ Measure tube diameter for end correction
✓ Record temperature at start and end
✓ Approach resonance slowly from both directions
✓ Measure multiple resonances (at least 3)
✓ Use \((L_2-L_1)\) method to eliminate \(e\)
✓ Calculate uncertainty propagation
✓ Compare with theory at measured temperature

Master this experiment and you'll understand how pipe organs work, why your voice sounds different in helium, and how submarines use sonar. You're measuring the same property that lets you hear - the speed at which pressure waves travel through air.