Interactive physics simulator
Closed Pipe
Discover how an air column closed at one end forms standing sound waves with only odd harmonics. Watch the displacement node at the closed wall, the antinode at the open mouth, and see how pipe length, sound speed, and harmonic number change the resonant frequency.
Closed Pipe Resonance Simulator
Adjust pipe length, sound speed, pipe radius, and odd harmonic number. The simulator shows the standing wave pattern, air particle motion, displacement node at the closed end, and live resonance frequency.
Live Wave Telemetry
- Harmonic (n)
- n = 1
- Frequency
- 100 Hz
- Length
- 0.85 m
- Wavelength
- 3.40 m
- Nodes / Antinodes
- D node 1
- End Correction
- 0.012 m
What is a Closed Pipe?
A closed pipe is an air column that is sealed at one end and open at the other. The closed end is usually a rigid wall, a piston, or a reed — anything that stops air from moving freely. Because the wall cannot flex, air particles pile up against it, creating a displacement node and a pressure antinode at that end.
At the open end, air pressure relaxes to atmospheric pressure, so the open end is a displacement antinode and a pressure node. Standing waves can only form when an exact number of quarter-wavelengths fit inside the pipe — and since the two ends must always have opposite characters, only odd harmonics (n = 1, 3, 5, ...) are resonant.
Key Definition
- Closed end: displacement node, pressure antinode.
- Open end: displacement antinode, pressure node.
- Only odd harmonics are allowed: n = 1, 3, 5, ...
- Fundamental has one quarter-wavelength inside.
- Shorter length or higher harmonic = higher pitch.
Closed Pipe Formula
λn = 4Leff / n (n = 1, 3, 5, ...)
fn = n v / 4Leff
With end correction: Leff = L + e and e ≈ 0.6r.
Closed vs Open Pipe
| Feature | Closed pipe | Open pipe |
|---|---|---|
| Ends | One closed, one open | Both open |
| Harmonics | Odd only (1,3,5,...) | All integers |
| Fundamental | f = v / 4L | f = v / 2L |
| Real example | Clarinet | Flute, organ |
| End corrections | One open end: +e | Two open ends: +2e |
Real-life Example
A clarinet behaves like a closed pipe. The reed-mouthpiece end creates a pressure antinode (displacement node), while the open bell radiates sound as a displacement antinode. This is why a clarinet sounds an octave lower than a flute of the same tube length and has a hollow, dark tone with only odd harmonics.
- Longer pipe: lower note (f decreases).
- Covering tone holes: longer effective air column, lower pitch.
- Opening holes: shorter effective column, higher pitch.
Solved Examples
A closed pipe has length 0.85 m. If sound speed is 340 m/s and end correction is ignored, find the fundamental frequency.
- For a closed pipe, only odd harmonics are allowed. The fundamental has a quarter wavelength inside the pipe.
- The fundamental wavelength is λ1 = 4L = 4 × 0.85 = 3.40 m.
- Frequency: f1 = v / λ1 = 340 / 3.40 = 100 Hz.
Answer: Fundamental frequency = 100 Hz
A closed pipe resonates at its third harmonic (n = 3) with a frequency of 510 Hz. If v = 340 m/s, find the pipe length.
- For a closed pipe: fn = n v / 4L, where n is odd only (1, 3, 5, ...).
- Rearranging: L = n v / 4f = 3 × 340 / (4 × 510).
- L = 1020 / 2040 = 0.50 m.
Answer: Pipe length = 0.50 m
A closed pipe of effective length 0.34 m. Find the first three resonant frequencies if v = 340 m/s.
- For a closed pipe: fn = n v / 4Leff, n = 1, 3, 5 only.
- 1st harmonic (fundamental): f1 = 1 × 340 / (4 × 0.34) = 250 Hz.
- 3rd harmonic: f3 = 3 × 250 = 750 Hz.
- 5th harmonic: f5 = 5 × 250 = 1250 Hz.
Answer: f1 = 250 Hz, f3 = 750 Hz, f5 = 1250 Hz
Common Mistakes
- Using the open-pipe formula (dividing by 2L) instead of 4L.
- Using even harmonics — only odd n are allowed in a closed pipe.
- Forgetting that the closed end is a displacement node, not an antinode.
- Adding two end corrections — a closed pipe has only one open end, so only one e is added.
- Confusing pressure nodes and displacement nodes at each end.
Practice Questions
1. What boundary conditions form at each end of a closed pipe?
The closed end forces a displacement node (air cannot move) and a pressure antinode. The open end is a displacement antinode and a pressure node.
2. A closed pipe is 0.40 m long. If v = 320 m/s, find the fundamental frequency (ignore end correction).
f1 = v / 4L = 320 / (4 × 0.40) = 200 Hz.
3. Why does a closed pipe produce only odd harmonics?
The closed end forces a displacement node and the open end forces an antinode. Only quarter-wavelength multiples that satisfy both conditions simultaneously are odd-numbered harmonics.
4. How is end correction applied to a closed pipe?
The effective length is Leff = L + e where e ≈ 0.6r per open end. Only the open end gets the correction, so just one e is added.
5. Compare the fundamental frequency of a closed pipe to an open pipe of the same length.
The closed pipe fundamental is half that of an open pipe of equal length: fclosed = v/4L versus fopen = v/2L. The closed pipe sounds an octave lower.
Quick Summary
- A closed pipe has one sealed end (displacement node) and one open end (displacement antinode).
- Only odd harmonics resonate: n = 1, 3, 5, ...
- The ideal formula is fn = n v / 4L (n odd).
- With end correction, use Leff = L + e, where e ≈ 0.6r.
- A closed pipe sounds an octave lower than an open pipe of the same length.
- The clarinet is the classic real-world closed pipe instrument.
Frequently Asked Questions
What is a closed pipe in physics?
A closed pipe is an air column that is closed at one end and open at the other. The closed end reflects sound with a displacement node, while the open end is a displacement antinode.
What is the formula for a closed pipe?
The ideal closed pipe formula is fn = n v / 4L, where n must be an odd integer (1, 3, 5, ...). With end correction: fn = n v / 4(L + e).
What is the fundamental frequency of a closed pipe?
The fundamental frequency is f1 = v / 4L for an ideal closed pipe, where L is the pipe length and v is the speed of sound.
Why do closed pipes only produce odd harmonics?
Because the closed end always produces a displacement node and the open end always produces a displacement antinode. This means only an odd number of quarter-wavelengths can fit inside the pipe.
What is a displacement node in a closed pipe?
A displacement node is a point where air particles cannot move back and forth. In a closed pipe, the closed wall forces a node because the wall is rigid and prevents air motion.
What is the end correction for a closed pipe?
The open end of a closed pipe has an end correction of approximately e = 0.6r, where r is the pipe radius. Only one end is open, so only one end correction is added: Leff = L + e.
How is a closed pipe different from an open pipe?
An open pipe supports all harmonics (n = 1, 2, 3, ...) and has antinodes at both ends. A closed pipe supports only odd harmonics (n = 1, 3, 5, ...) and has a node at the closed end.
What real instruments use closed pipe acoustics?
The clarinet behaves approximately like a closed pipe because one end is the reed (nearly closed) and the other end is open. This is why clarinets sound an octave lower than open-pipe instruments of the same length.
Why is a clarinet called a closed pipe instrument?
The mouthpiece reed of a clarinet acts as a pressure antinode (displacement node) similar to a closed pipe. This allows only odd harmonics and gives the clarinet a distinctive hollow, dark tone.
What happens to a closed pipe fundamental if the pipe is made longer?
The resonant frequency decreases because f1 = v / 4L. A longer L means a smaller f, so the sound gets lower in pitch.
How does temperature affect a closed pipe frequency?
Higher temperature increases the speed of sound v. Since f = v / 4L, a higher v gives a higher frequency. Brass instruments tune sharp in warm conditions for this reason.
Can a closed pipe produce even harmonics?
No. A strictly closed-open pipe produces only odd harmonics: 1st, 3rd, 5th, etc. Even harmonics require a pressure node at the closed end, which is physically impossible for a rigid wall.