Tube effect

This activity allows students to measure the speed of sound by analyzing the resonant frequencies of a tube. It illustrates how resonance phenomena can be used to determine physical constants.

Every wind instrument, from a piccolo to a pipe organ, produces sound through the resonance of air columns in tubes. When air is disturbed inside a tube, only certain frequencies are amplified through constructive interference of standing waves, while all others are suppressed. These resonant frequencies follow precise mathematical relationships that depend on the tube's length and diameter. For a tube open at both ends, the resonant frequencies are integer multiples of a fundamental frequency determined by the tube length and the speed of sound. By exciting all frequencies simultaneously using white noise and observing which ones are amplified, students can identify these resonant modes and use them to calculate the speed of sound. This experiment demonstrates the same physics that allows an organ builder to predict the pitch of a pipe from its dimensions, and connects acoustic resonance to one of the most fundamental constants in physics.

Learning objectives:

The student generates white noise with a smartphone placed at the entrance of a cylindrical tube then uses a second smartphone with FizziQ to analyze the spectrum of frequencies amplified by resonance. By identifying amplitude peaks in the spectrum and applying the resonance formula for an open tube the student can calculate the speed of sound and compare its value to standard references.

Level:

High school

FizziQ

Author:

Duration (minutes) :

30

What students will do :

- Excite the resonant modes of a cylindrical tube using white noise
- Identify the resonant frequency peaks in the spectrum analyzer output
- Verify that the resonant frequencies follow the pattern f_n = n × c / (2L_eff)
- Calculate the speed of sound from the measured resonant frequencies and tube dimensions
- Understand the physics of standing waves in open tubes and the role of end corrections

Scientific concepts:

- Acoustic resonance
- Natural frequencies
- Spectral analysis
- Standing waves
- Sound propagation

Sensors:

- Microphone (spectrum analyzer / FFT)
- FizziQ synthesizer (white noise generator)

What is required:

- Two smartphones with the FizziQ application
- A cylindrical tube (roll of toilet paper cardboard or other)
- A tape measure to measure the dimensions of the tube
- FizziQ experience notebook

Experimental procedure:

Measure the length L and internal diameter D of your cylindrical tube using a ruler or tape measure.
On the first smartphone, open FizziQ and use the Synthesizer to generate white noise at moderate volume.
Place this smartphone at one end of the tube so the speaker emits directly into the tube opening.
On the second smartphone, open FizziQ and select the Spectrum Analyzer (FFT). Place the microphone near the other end of the tube.
Start the spectrum analysis. You should see several prominent peaks rising above the general noise level. These are the resonant frequencies.
Record the frequencies of the first 4-6 resonant peaks: f₁, f₂, f₃, f₄, etc.
Verify that the peaks are approximately equally spaced in frequency. For an open-open tube, f_n = n × f₁.
Calculate the ratio between consecutive peaks: f₂/f₁, f₃/f₁, f₄/f₁. These should be approximately 2, 3, 4...
Calculate the speed of sound using the formula: c = f₁ × (2L + 1.24D), where the term 1.24D is the combined end correction for both open ends.
Average the speed calculated from each resonant frequency: c = f_n × (2L + 1.24D) / n for each n.
Compare your average with the accepted value of approximately 343 m/s at 20°C.
Try tubes of different lengths and verify that longer tubes have lower fundamental frequencies.

Expected results:

For a cardboard tube 30 cm long with a 4 cm diameter, the fundamental resonant frequency should be approximately 543 Hz (using c = 343 m/s and the end correction). Higher resonances appear at approximately 1086, 1629, 2172 Hz, etc. The calculated speed of sound should fall within 320-370 m/s for careful measurements. The end correction (1.24D) is important: without it, the calculated speed of sound will be systematically too high by about 8-15% for typical tube dimensions. Students should observe that all resonant frequencies are integer multiples of the fundamental, confirming the standing wave pattern for an open-open tube.

Scientific questions:

- Why are only certain frequencies amplified by the tube while others are not?
- What is a standing wave, and how does it form inside the tube?
- Why does the end correction depend on the tube diameter?
- How would the resonant frequencies change for a tube closed at one end?
- What is the relationship between the tube length and the wavelength of the fundamental resonance?
- How do organ builders use these principles to design pipes of specific pitches?

Scientific explanations:

When white noise (containing all frequencies at equal intensity) is emitted at the input of a tube, the frequencies which correspond to the natural resonance modes of the tube are amplified by resonance. For a tube open at both ends, the resonant frequencies are given by the formula: f = n×c/(2L+1.24D), where n is an integer (1,2,3,...) representing the mode, c is the speed of sound, L is the length of the tube, and D its diameter.

The corrective term 1.24D takes into account the edge effect at the end of the tube (the pressure belly does not form exactly at the opening but slightly beyond). FizziQ's frequency spectrum uses a fast Fourier transform (FFT) to decompose the sound signal picked up by the microphone and display its frequency content.

The observed peaks correspond to the harmonics of the fundamental frequency. By precisely measuring these frequencies, the length and the diameter of the tube, we can calculate the speed of sound: c = (2L+1.24D)×f₁, where f₁ is the fundamental frequency (first peak).

The theoretical speed of sound in air at 20°C is approximately 343 m/s, but varies with temperature according to: c = 331.3 + 0.606×T (T in °C). This experiment makes it possible to obtain a measurement with a precision of around ±2%.

Extension activities:

Frequently asked questions:

Q: I cannot see clear resonant peaks in the spectrum. What should I do?
R: Increase the white noise volume and ensure it is directed into the tube opening. Reduce background noise. Try a tube with a smaller diameter-to-length ratio for sharper resonances.

Q: The peaks are not exactly equally spaced. Is this normal?
R: Small deviations (±2-5%) are normal due to the end correction varying slightly with frequency. The overall pattern of nearly equal spacing should be clear.

Q: Why is the end correction needed?
R: The pressure antinode at an open end does not form exactly at the physical opening but slightly beyond it. This effectively lengthens the tube by approximately 0.62D at each open end (total 1.24D for both ends).

Q: Can I use a toilet paper roll as the tube?
R: Yes, a toilet paper roll is an excellent tube for this experiment. Its typical dimensions (10-11 cm long, 4 cm diameter) give a fundamental frequency around 1400-1600 Hz.