Topic 9.7.213 min read
Modes of Vibration in a Closed Pipe
A closed pipe supports stationary waves with a displacement node at its closed end and an antinode at its open end. Its allowed wavelengths are λₙ=4L/(2n−1), and its allowed frequencies are fₙ=(2n−1)v4L=(2n−1)f₁ for n=1,2,3,…; therefore, only odd harmonics occur.
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How the ends control vibration
Air cannot move at the closed end, so that end is a displacement node. Air can move freely at the open end, so the open end is a displacement antinode. These unlike end conditions allow one, three, five and higher odd numbers of quarter-wavelengths.
First three allowed modes
- First mode: one quarter-wavelength; frequency f₁.
- Second mode: three quarter-wavelengths; frequency 3f₁.
- Third mode: five quarter-wavelengths; frequency 5f₁.
- Only odd harmonics occur.
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Closed-pipe boundary conditions
Consider a pipe of length L that is closed at one end and open at the other. The rigid closed end prevents longitudinal motion of air and is a displacement node. Air at the open end can move freely and forms a displacement antinode.
Fundamental frequency or first harmonic
The lowest-frequency pattern extends from a displacement node to the nearest displacement antinode. This distance is one quarter of a wavelength.
Second vibration mode: third harmonic
The next permitted pattern contains one half-wavelength plus one quarter-wavelength, giving a total of three quarter-wavelengths. This is the second vibration mode, but its frequency is three times the fundamental, so it is the third harmonic and first overtone.
Third vibration mode: fifth harmonic
The next permitted pattern contains two half-wavelengths plus one quarter-wavelength, giving five quarter-wavelengths. It is the third vibration mode, the fifth harmonic and the second overtone.
General nth mode
The allowed patterns contain 1, 3, 5, … quarter-wavelengths. For mode number n, the number of quarter-wavelengths is 2n−1.
Mode and harmonic comparison
| Vibration mode | Quarter-wavelengths | Wavelength | Frequency | Harmonic |
|---|---|---|---|---|
| First | 1 | λ1=4L | f1=v4L | First |
| Second | 3 | λ2=4L3 | f2=3f1 | Third |
| Third | 5 | λ3=4L5 | f3=5f1 | Fifth |
| nth | 2n−1 | λn=4L2n-1 | fn=(2n-1)f1 | (2n−1)th |
Why even harmonics are absent
An even number of quarter-wavelengths cannot place a displacement node at one end and an antinode at the other. Therefore frequencies 2f₁, 4f₁, 6f₁, … do not satisfy the boundary conditions of an ideal closed pipe.
First three allowed frequencies
A closed pipe is 0.50 m long. Find its first three allowed frequencies when the speed of sound is 340 m s⁻¹.
- L=0.50m
- v=340m s-1
f1=v4L=3404(0.50)=170Hz
f2=3f1=3(170)=510Hz
f3=5f1=5(170)=850Hz
Final answer: The first three allowed frequencies are 170 Hz, 510 Hz and 850 Hz.
Common mistakes
Check your understanding
Q. Why does a closed pipe contain only odd harmonics?
A. Only odd numbers of quarter-wavelengths can connect a displacement node at the closed end to an antinode at the open end.
Q. What is the first overtone of a closed pipe?
A. It is the second vibration mode with frequency 3f₁, so it is the third harmonic.
Q. What is the nth allowed frequency?
A. fₙ=(2n−1)v4L=(2n−1)f₁.
Quick revision
- The closed end is a displacement node and the open end is a displacement antinode.
- In the fundamental mode, one quarter-wavelength fits in the pipe: L=λ₁/4.
- The fundamental wavelength is λ₁=4L and the fundamental frequency is f₁=v4L.
- The second vibration mode contains three quarter-wavelengths: λ₂=4L/3 and f₂=3f₁; it is the third harmonic.
- The third vibration mode contains five quarter-wavelengths: λ₃=4L/5 and f₃=5f₁; it is the fifth harmonic.
- For mode number n, the pipe contains (2n−1) quarter-wavelengths.
- The general relations are λₙ=4L/(2n−1) and fₙ=(2n−1)v4L=(2n−1)f₁.
- Only odd harmonics—first, third, fifth and higher odd multiples—are present in a closed pipe.
Exam Focus
Exam Focus
Important points
- Closed end: displacement node; open end: displacement antinode.
- L=λ₁/4 and f₁=v4L.
- λₙ=4L/(2n−1).
- fₙ=(2n−1)v4L=(2n−1)f₁.
- Only odd harmonics are present.
Important formulas
- L=(2n-1)λn4
- λn=4L2n-1
- fn=(2n-1)v4L=(2n-1)f1
Common exam traps
- Mode number and harmonic number differ after the fundamental.
- Even harmonics are absent.
Long question structure
Unlike end conditions → first mode → second mode/third harmonic → third mode/fifth harmonic → nth mode → odd harmonics
Common Mistakes
Common Mistakes
- Mistake
Calling the second vibration mode the second harmonic.
- Mistake
Including even harmonics.
- Mistake
Using the same displacement condition at both ends.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
The closed end is a displacement node and the open end is a displacement antinode.
In the fundamental mode, one quarter-wavelength fits in the pipe: L=λ₁/4.
The fundamental wavelength is λ₁=4L and the fundamental frequency is f₁=v4L.
The second vibration mode contains three quarter-wavelengths: λ₂=4L/3 and f₂=3f₁; it is the third harmonic.
A closed pipe is closed at one end and open at the other. The closed end is a displacement node and the open end is a displacement antinode, so only odd harmonics can satisfy its boundary conditions.
- Definition
- A closed pipe is closed at one end and open at the other. The closed end is a displacement node and the open end is a displacement antinode, so only odd harmonics can satisfy its boundary conditions.
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Exam Practice
Important Question
Q1. Derive the fundamental frequency of a closed pipe.
Important Question
Q2. Derive the frequency of the second vibration mode in a closed pipe.
Important Question
Q3. Derive the frequency of the third vibration mode in a closed pipe.
Important Question
Q4. Obtain the general expressions for wavelength and frequency in a closed pipe.
Numerical
Q5. A closed pipe is 0.50 m long. Find its first three frequencies if v=340 m s⁻¹.