JEE Mains Physics · Waves

Organ Pipes and the Resonance Tube

An open pipe has antinodes at both ends and every harmonic nv/2L; a closed pipe has a node at the closed end and only odd harmonics (2n − 1)v/4L.

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Why this matters

Fifteen PYQs, nine of them asking for a number, and two from 2026. Eleven compare harmonics and overtones of open and closed pipes, often of two pipes sounding the same note or a pipe being part-filled with water. Four are resonance-tube experiments, where the end correction moves every length. Nearly all of them come down to knowing which harmonics each pipe allows.

Concept 1 of 2: Harmonics and overtones of open and closed pipes

An open end lets the air move freely, so it is an antinode; a closed end stops the air, so it is a node. A pipe open at both ends behaves like a string fixed at both ends: all harmonics. A pipe closed at one end fits only odd quarter-wavelengths, so its even harmonics are missing and its overtones skip numbers.

Definition

Open pipe: $f_n = \dfrac{nv}{2L}$ , n = 1, 2, 3, …; the kth overtone is the (k + 1)th harmonic.
Closed pipe: $f = \dfrac{(2n - 1)v}{4L}$ ; only odd harmonics 1, 3, 5, …; the kth overtone is the (2k + 1)th harmonic.
Resonances in the ratio 1 : 3 : 5 mark a closed pipe; 1 : 2 : 3 marks an open pipe or a string.
Two pipes in unison: set the two frequency expressions equal; v cancels when both hold the same gas.
Different gases with the same bulk modulus: $v = \sqrt{B/\rho}$ , so $v \propto 1/\sqrt{\rho}$ .
Pouring water into a closed pipe shortens its air column, which raises every frequency.

Vibrating system	Fundamental	Harmonics present	First overtone	kth overtone
String fixed at both ends	$v/2L$ ( $\lambda = 2L$ )	All: 1, 2, 3, …	2nd harmonic, $2f_1$	(k + 1)th harmonic
Pipe open at both ends	$v/2L$ ( $\lambda = 2L$ )	All: 1, 2, 3, …	2nd harmonic, $2f_1$	(k + 1)th harmonic
Pipe closed at one end	$v/4L$ ( $\lambda = 4L$ )	Odd only: 1, 3, 5, …	3rd harmonic, $3f_1$	(2k + 1)th harmonic A closed pipe has no 2nd harmonic, so its first overtone is three times the fundamental.

A closed pipe of length L has the same fundamental as an open pipe of length 2L.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 8 April 2024 · Q27Moderate

Example 1 · Waves · Organ Pipes and the Resonance Tube

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is

\left( \frac{a - 1}{a} \right)

then the value of

a

is_____.

Overtone number is not harmonic number

The first overtone is the first frequency above the fundamental. In an open pipe that is the 2nd harmonic; in a closed pipe it is the 3rd, because the 2nd does not exist.

Water makes the closed pipe shorter

Water poured into a closed pipe takes the place of air, so the vibrating column is shorter and the note is higher. Use the new air-column length, not the full pipe.

Drill 10 more on harmonics and overtones of open and closed pipes

Concept 2 of 2: Resonance tube and end correction

A resonance tube is a closed pipe whose length you change by moving the water level. It resonates with a fork when the air column fits λ/4, then 3λ/4, then 5λ/4. The antinode sits a little outside the open end, so the effective length is the column plus an end correction e. Subtracting two resonance lengths cancels e.

Definition

Resonances: $l_1 + e = \dfrac{\lambda}{4}$ , $l_2 + e = \dfrac{3\lambda}{4}$ , $l_3 + e = \dfrac{5\lambda}{4}$ .
$l_2 - l_1 = \dfrac{\lambda}{2}$ , so $v = 2f(l_2 - l_1)$ with no end correction needed.
End correction: $e = \dfrac{l_2 - 3l_1}{2}$ , or $e = 0.3d$ for a tube of inner diameter d.
Without end correction, the shortest closed pipe for a fork of frequency f is $\dfrac{\lambda}{4} = \dfrac{v}{4f}$ .
With a fixed fork, raising the water level skips from one resonance to the next shorter one: the column shortens by $\lambda/2$ .

Resonance tube

l_n + e = \frac{(2n - 1)\lambda}{4} \qquad l_2 - l_1 = \frac{\lambda}{2} \qquad e = 0.3d

Worked example

With a 480 Hz fork, a resonance tube gives its first two resonances at 16.6 cm and 52.6 cm. Find the wavelength, the speed of sound, the end correction and the length for the third resonance.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 29 June 2022 · Q113Moderate

Example 2 · Waves · Organ Pipes and the Resonance Tube

In an experiment to determine the velocity of sound in air at room temperature using a resonance is observed when the air column has a length of 20.0

cm

for a tuning fork of frequency

400\text{ }Hz

is used. The velocity of the sound at room temperature is

336{\text{ }ms}^{- 1}

. The third resonance is observed when the air column has a length of

\_\_\_\_

cm.

The end correction is added, not subtracted, to the column

The antinode is just outside the open end, so the effective length is l + e. The measured column is therefore shorter than λ/4 by e.

Use the difference to get the speed

v = 2f(l₂ − l₁) needs no end correction. Using v = 4f l₁ instead ignores e and gives a speed that is too low.

0.3 times the diameter, not the radius

The end correction is 0.3d, which is 0.6r. Using the radius in 0.3d halves the correction.

Drill 3 more on resonance tube and end correction

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Resonance tube and end correction
Resonance tube
$l_n + e = \frac{(2n - 1)\lambda}{4} \qquad l_2 - l_1 = \frac{\lambda}{2} \qquad e = 0.3d$

Reference tables (1)

Harmonics and overtones of open and closed pipes3 rows

Vibrating system	Fundamental	Harmonics present	First overtone	kth overtone
String fixed at both ends	$v/2L$ ( $\lambda = 2L$ )	All: 1, 2, 3, …	2nd harmonic, $2f_1$	(k + 1)th harmonic
Pipe open at both ends	$v/2L$ ( $\lambda = 2L$ )	All: 1, 2, 3, …	2nd harmonic, $2f_1$	(k + 1)th harmonic
Pipe closed at one end	$v/4L$ ( $\lambda = 4L$ )	Odd only: 1, 3, 5, …	3rd harmonic, $3f_1$	(2k + 1)th harmonic A closed pipe has no 2nd harmonic, so its first overtone is three times the fundamental.

A closed pipe of length L has the same fundamental as an open pipe of length 2L.

Watch out for (5)

Overtone number is not harmonic number
→ Harmonics and overtones of open and closed pipes
Water makes the closed pipe shorter
→ Harmonics and overtones of open and closed pipes
The end correction is added, not subtracted, to the column
→ Resonance tube and end correction
Use the difference to get the speed
→ Resonance tube and end correction
0.3 times the diameter, not the radius
→ Resonance tube and end correction

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