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The air in an open pipe of length 36 cm long is vibrating...

The air in an open pipe of length $36 \mathrm{~cm}$ long is vibrating with 2 nodes and 2 antinodes. The temperature of the air inside the pipe is $51^{\circ} \mathrm{C}$. What is the wavelength of waves produced in air outside the tube where the temperature of air is $16^{\circ} \mathrm{C}$ ? (A) $32.1 \mathrm{~cm}$ (B) $68 \mathrm{~cm}$ (C) $34 \mathrm{~cm}$ (D) $10.2 \mathrm{~cm}$

The air in an open pipe of length $36 \mathrm{~cm}$ long is vibrating with 2 nodes and 2 antinodes. The temperature of the air inside the pipe is $51^{\circ} \mathrm{C}$. What is the wavelength of waves produced in air outside the tube where the temperature of air is $16^{\circ} \mathrm{C}$ ?
(A) $32.1 \mathrm{~cm}$
(B) $68 \mathrm{~cm}$
(C) $34 \mathrm{~cm}$
(D) $10.2 \mathrm{~cm}$

Key Concepts

Frequency Invariance Across Media

When a vibrating system produces a sound, the frequency of the sound remains constant as it radiates into the surrounding medium even if that medium has different properties (such as temperature). As a result, a change in the speed of sound in the new medium leads to a corresponding change in the wavelength, since frequency and wavelength are related through the speed of sound by the formula ? = v/f.

Temperature Dependence of the Speed of Sound

The speed of sound in air is affected by the temperature of the medium; as temperature increases, the speed of sound increases. This relationship is often given by an approximate formula (such as v = 331 m/s + 0.6T, with T in degrees Celsius), meaning that sound travels faster in warmer air. This dependence is crucial when comparing sound wave behavior in regions of differing temperatures.

Resonance in Air Columns

Resonance in air columns occurs when the length of the column supports a standing wave pattern that satisfies the boundary conditions (such as free or fixed ends). In pipes open at one or both ends, certain harmonics (or modes) are excited. The number and location of nodes and antinodes dictate the relationship between the pipe’s length and the wavelength of the resonating sound wave.

Standing Waves

Standing waves are produced when two waves of the same frequency and amplitude travel in opposite directions and interfere. They are characterized by nodes (points of no displacement) and antinodes (points of maximum displacement). This concept is central to understanding resonant phenomena in various physical systems, including air columns in pipes.

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