A cylindrical tube open at both ends has a fundamental frequency f in air. The tube is dipped ve

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A cylindrical tube open at both ends has a fundamental...

A cylindrical tube open at both ends has a fundamental frequency $\mathrm{f}$ in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now........ (A) $\mathrm{f} / 2$ (B) $\mathrm{f}$ (C) $3 \mathrm{f} / 4$ (D) $2 \mathrm{f}$

A cylindrical tube open at both ends has a fundamental frequency $\mathrm{f}$ in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now........
(A) $\mathrm{f} / 2$
(B) $\mathrm{f}$
(C) $3 \mathrm{f} / 4$
(D) $2 \mathrm{f}$

Key Concepts

Effect of Changing Boundary Conditions

Altering the boundary conditions, such as by submerging one end in water, effectively changes the nature of the end from open to closed. This modification results in a different set of resonant modes in the tube, thus shifting the fundamental frequency. Understanding this effect is vital for analyzing how environmental changes impact sound wave behavior in tubes.

Boundary Conditions in Tubes

The behavior of the air column in a tube is largely determined by its boundary conditions. For an open end, the air pressure remains close to atmospheric, creating an antinode, while a closed end forces a node. These conditions directly influence the possible standing wave patterns and thus the fundamental and higher resonant frequencies.

Standing Waves in Air Columns

This concept explains how sound waves reflect within a tube to form standing waves, with nodes and antinodes arranged according to the geometry of the tube. The specific pattern of standing waves determines the resonant frequencies that can be sustained, which is central to understanding acoustic resonance in tubes.

Fundamental Frequency Determination

The fundamental frequency is the lowest resonant frequency of the air column and depends on the effective length of the column and the speed of sound. For example, a tube with both ends open supports a half-wavelength resonance, whereas a tube with one end closed supports a quarter-wavelength resonance. Changes in the boundary conditions alter the effective length and, consequently, the fundamental frequency.

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