Closed Organ Pipe: Frequency and Odd Harmonics Only
Physics · Waves · NEET
A closed organ pipe is closed at one end and open at the other. Its fundamental frequency is f1 = v/4L, and it produces ONLY odd harmonics: f1, 3f1, 5f1, 7f1... (never 2f1 or 4f1). Memory hook: "Closed = C for odd Count" — the closed end must be a node, the open end an antinode, so only odd multiples fit.
Closed pipe standing waves: a node (N) sits at the closed end and an antinode (A) at the open end. Only odd numbers of quarter-wavelengths fit, giving frequencies f1 = v/4L, 3f1, 5f1... Even harmonics cannot form.
Your doubts, answered
Why does a closed organ pipe produce only odd harmonics?
The closed end is fixed air (a displacement node) and the open end is free air (a displacement antinode). The shortest pattern that fits a node at one end and an antinode at the other is a quarter wave (L = lambda/4). Only patterns with an ODD number of quarter-waves keep node-at-closed and antinode-at-open: L = lambda/4, 3lambda/4, 5lambda/4... These give f = v/4L, 3v/4L, 5v/4L. Even multiples would force an antinode at the closed end, which is impossible, so 2f1, 4f1... never appear.
What is the formula for the fundamental frequency of a closed pipe?
For a pipe of length L (one end closed): L = lambda/4, so lambda = 4L. Fundamental frequency f1 = v/lambda = v/(4L), where v is the speed of sound in air. All allowed frequencies are fn = n*v/(4L) with n = 1, 3, 5, 7... (odd only).
Why is the closed end a node and the open end an antinode?
At the closed (rigid) end, air cannot move back and forth, so displacement is always zero — that is a node. At the open end, air is free to oscillate with maximum displacement — that is an antinode. Sound is a longitudinal (displacement) wave, so we track where air particles move most and least.
In a closed pipe, what is the first overtone?
An overtone is any allowed frequency above the fundamental. Since only odd harmonics exist, the first overtone is the 3rd harmonic (3f1), the second overtone is the 5th harmonic (5f1), and so on. Trap: overtone number and harmonic number are NOT the same. n-th overtone = (2n+1)-th harmonic for a closed pipe.
How do I get the fundamental from two successive harmonics of a closed pipe?
Successive odd harmonics differ by 2f1 (e.g. 3f1 and 5f1 differ by 2f1). So the difference between two nearest resonant frequencies equals 2 * fundamental. Example: 260 - 220 = 40 Hz, so 2f1 = 40 and f1 = 20 Hz. This is a direct NEET shortcut.
⚠️ The NEET trap ✗ Students take the 'first overtone' of a closed pipe as 2f1 = 2v/4L, copying the open-pipe pattern. ✓ A closed pipe has NO even harmonics. Its first overtone is the 3rd harmonic = 3v/4L, second overtone = 5th harmonic = 5v/4L. Always map n-th overtone to (2n+1)-th harmonic. 🧠 Closed pipe overtones climb 1, 3, 5, 7 — never touch even numbers.
Real NEET questions
NEET 2017
Two nearest (successive) harmonics of a tube closed at one end and open at the other are 220 Hz and 260 Hz. The fundamental frequency of the system is:
A · 10 Hz
B · 20 Hz ✓
C · 30 Hz
D · 40 Hz
Solution: A closed pipe supports only odd harmonics, so successive resonances are n*f1 and (n+2)*f1. Their difference is 2*f1. Step 1: 260 - 220 = 40 Hz = 2*f1. Step 2: f1 = 40/2 = 20 Hz. (Check: 220 = 11*20 and 260 = 13*20, both odd multiples.) Answer: 20 Hz.
NEET 2016
An air column closed at one end and open at the other resonates with a tuning fork when the smallest length of the column is 50 cm. The next larger length of the column that resonates with the same tuning fork is:
A · 66.7 cm
B · 100 cm
C · 150 cm ✓
D · 200 cm
Solution: A closed pipe resonates at odd multiples of lambda/4. Step 1: smallest length = fundamental, lambda/4 = 50 cm, so lambda = 200 cm. Step 2: next resonance is the 3rd harmonic at 3*lambda/4 = 3 * 50 = 150 cm. Answer: 150 cm.
NEET 2023
The ratio of the fundamental frequency of an open pipe to that of a closed pipe of the same length is:
A · 1 : 2
B · 2 : 1 ✓
C · 1 : 3
D · 3 : 1
Solution: Open pipe fundamental = v/(2L). Closed pipe fundamental = v/(4L). Ratio = [v/(2L)] / [v/(4L)] = 4L/2L = 2. So open : closed = 2 : 1. For the same length, an open pipe sounds one octave higher. Answer: 2 : 1.
Solved Waves NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
