An air column in a pipe which is closed at one end will be...

An air column in a pipe which is closed at one end will be in resonance with a vibrating tuning fork of frequency $264 \mathrm{~Hz}$ if the length of the air column in $\mathrm{cm}$ is: (Speed of sound in air $=340 \mathrm{~m} / \mathrm{s}$ ) (a) $32.19 \mathrm{~cm}$ (b) $64.39 \mathrm{~cm}$ (c) $100 \mathrm{~cm}$ (d) $140 \mathrm{~cm}$

An air column in a pipe which is closed at one end will be in resonance with a vibrating tuning fork of frequency $264 \mathrm{~Hz}$ if the length of the air column in $\mathrm{cm}$ is:
(Speed of sound in air $=340 \mathrm{~m} / \mathrm{s}$ )
(a) $32.19 \mathrm{~cm}$
(b) $64.39 \mathrm{~cm}$
(c) $100 \mathrm{~cm}$
(d) $140 \mathrm{~cm}$

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Key Concepts

Standing Waves

Standing waves are patterns of vibration that occur when waves reflect and interfere with each other, creating points of no motion (nodes) and maximum motion (antinodes). In air columns, standing wave patterns form under specific conditions based on the boundaries of the medium.

Resonance in Closed Pipes

In pipes closed at one end, resonance occurs only at specific frequencies where the standing wave pattern fits the constraint of a node at the closed end and an antinode at the open end. This results in the formation of only odd harmonics, meaning the fundamental frequency corresponds to a quarter of a wavelength fitting into the length of the pipe.

Wave Speed, Frequency, and Wavelength Relationship

The relationship between wave speed, frequency, and wavelength is given by the equation v = f × ?. This principle allows the determination of any one of these quantities if the other two are known, which is crucial in solving resonance problems in acoustic systems.

Fundamental Mode in Closed Pipes

The fundamental mode is the lowest resonant frequency of an air column in a closed pipe. For this mode, the length of the column is equal to one-fourth of the wavelength. This concept is essential for calculating the resonance conditions in instruments and physical systems involving sound waves.

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