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 $264 \mathrm{~Hz}$, What is the length of the column if it is in $\mathrm{cm} ?$ (speed of sound in $\operatorname{air}=330 \mathrm{~m} / \mathrm{s}$ ) (A) $62.50$ (B) $15.62$ (C) 125 (D) $93.75$

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Key Concepts

Standing Waves

Standing waves are formed by the interference of two waves of the same frequency traveling in opposite directions. In systems like pipes or strings, they result in fixed points called nodes (where the amplitude is always zero) and antinodes (where the amplitude is maximum), and only specific wavelengths that meet the boundary conditions can be sustained.

Resonance in Closed Pipes

In a pipe that is closed at one end, resonance occurs only for those wavelengths that form a standing wave with a node at the closed end and an antinode at the open end. The allowed lengths of the air column are given by L = (2n - 1)(?/4), where n is a positive integer, with the fundamental mode corresponding to n = 1 and higher resonances (overtones) corresponding to odd multiples.

Wavelength-Frequency Relationship

The wavelength, frequency, and speed of sound are related by the equation v = f?, where v is the speed of sound, f is the frequency, and ? is the wavelength. This relationship allows one to calculate any of the three parameters if the other two are known, which is essential for analyzing wave phenomena and resonance conditions in air columns.

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