Organ Pipe HarmonicsJEE Main

Open vs closed pipe resonance — interactive Physics simulation for IIT-JEE.

Concept

A pipe resonates only at frequencies whose standing waves fit its ends. An open pipe has antinodes at both ends and supports all harmonics; a closed pipe has a node at the closed end and supports only the odd ones — making it sound an octave lower for the same length.

Key formula

\text{open: } f_n = \frac{nv}{2L}\;(n=1,2,3\ldots), \qquad \text{closed: } f_n = \frac{(2n-1)v}{4L}

Derivation

An open–open pipe fits $L = n\lambda/2$ , giving $f_n = nv/2L$ — every integer harmonic.

A closed–open pipe needs a node at the closed end and antinode at the open end, fitting $L = (2n-1)\lambda/4$ . Only odd multiples of the fundamental survive, and that fundamental ( $v/4L$ ) is half the open-pipe value.

Scenarios to explore

Organ Pipe Harmonics — Open vs closed pipe resonance and overtones.

Real-world applications

Wind and pipe instruments (flute vs clarinet).
Resonance-tube measurement of the speed of sound.
Acoustic design of ducts and exhausts.

JEE exam tips

Closed-pipe fundamental is one octave below an open pipe of the same length.
Closed pipe harmonic ratio 1:3:5; open pipe 1:2:3.
Resonance-tube experiment: $v = 2f(\ell_2 - \ell_1)$ removes the end correction.

Common mistakes

Using $v/2L$ for a closed pipe (it is $v/4L$ ).
Expecting even harmonics from a closed pipe — they're absent.
Ignoring the end correction in precise work.

Exam traps to avoid

A closed pipe's 2nd overtone is its 5th harmonic, not its 3rd.
Frequency rises with temperature because $v \propto \sqrt{T}$ .