An open pipe is in resonance in 2^nd harmonic with frequency f1. Now one end of the tube is clo

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An open pipe is in resonance in 2^nd harmonic with...

An open pipe is in resonance in $2^{\text {nd }}$ harmonic with frequency $\mathrm{f}_{1}$. Now one end of the tube is closed and frequency is increased to $\mathrm{f}_{2}$, such that the resonance again occurs in the nth harmonic. Choose the correct option. (A) $n=3, f_{3}=(3 / 4) f$ (B) $n=3, f_{2}=(5 / 4) f_{1}$ (C) $\mathrm{n}=5, \mathrm{f}_{2}=(5 / 4) \mathrm{f}$ (D) $\mathrm{n}=5, \mathrm{f}_{2}=(3 / 4) \mathrm{f}_{1}$

An open pipe is in resonance in $2^{\text {nd }}$ harmonic with frequency
$\mathrm{f}_{1}$. Now one end of the tube is closed and frequency is increased to $\mathrm{f}_{2}$, such that the resonance again occurs in the nth harmonic. Choose the correct option.
(A) $n=3, f_{3}=(3 / 4) f$
(B) $n=3, f_{2}=(5 / 4) f_{1}$
(C) $\mathrm{n}=5, \mathrm{f}_{2}=(5 / 4) \mathrm{f}$
(D) $\mathrm{n}=5, \mathrm{f}_{2}=(3 / 4) \mathrm{f}_{1}$

Key Concepts

Standing Waves in Pipes

Standing waves are formed when waves reflect back and forth in a confined medium, such as an air column in a pipe, creating a pattern of nodes and antinodes. The formation of standing waves is crucial for resonance, where the frequency of the source matches one of the natural frequencies of the system, leading to large amplitude oscillations.

Resonance and Harmonics

Resonance in air columns occurs when the frequency of a driving source matches one of the natural frequencies of the pipe. These natural frequencies are called harmonics. The harmonic frequencies depend on the boundary conditions at the ends of the pipe and are integral multiples of a fundamental frequency in an open pipe, whereas in a pipe closed at one end only odd harmonics are present.

Open Pipe Harmonics

For an open pipe, which is open at both ends, the boundary conditions require antinodes at both ends. This leads to the harmonic frequencies being given by f? = n·v/(2L), where n is an integer (1, 2, 3, ...), v is the speed of sound, and L is the length of the pipe. All harmonics can be present in open pipes.

Closed Pipe Harmonics

In a pipe that is closed at one end and open at the other, the boundary conditions impose a node at the closed end and an antinode at the open end. This configuration only supports odd harmonics, and the harmonic frequencies are given by f? = n·v/(4L), where n takes on odd integer values (1, 3, 5, ...). This results in a different spacing of resonant frequencies compared to an open pipe.

Frequency Relationship upon Changing Boundary Conditions

When the boundary conditions of a resonating pipe change (for instance, when one end is closed), the allowed wavelengths and thus the frequencies of resonance change. While an open pipe supports all harmonics with given frequency spacing, a closed pipe supports only odd harmonics, and its resonant frequencies are typically lower for the same pipe length. Comparing the resonant frequency of a particular harmonic between the two configurations is essential to understanding frequency shifts that occur due to changes in the end conditions.

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