A narrow column of 20^∘ C air is found to have standing...

A narrow column of $20^{\circ} \mathrm{C}$ air is found to have standing waves at frequencies of $390 \mathrm{Hz}, 520 \mathrm{Hz},$ and $650 \mathrm{Hz}$ and at no frequencies in between these. The behavior of the tube at frequencies less than 390 Hz or greater than $650 \mathrm{Hz}$ is not known. Is this an open-open tube or an open-closed tube? Explain. b. How long is the tube? c. Draw a displacement graph of the 520 Hz standing wave in the tube. d. The air in the tube is replaced with carbon dioxide, which has a sound speed of $280 \mathrm{m} / \mathrm{s} .$ What are the new frequencies of these three modes?

A narrow column of $20^{\circ} \mathrm{C}$ air is found to have standing waves at frequencies of $390 \mathrm{Hz}, 520 \mathrm{Hz},$ and $650 \mathrm{Hz}$ and at no frequencies in between these. The behavior of the tube at frequencies less than 390 Hz or greater than $650 \mathrm{Hz}$ is not known.
Is this an open-open tube or an open-closed tube? Explain.
b. How long is the tube?
c. Draw a displacement graph of the 520 Hz standing wave in the tube.
d. The air in the tube is replaced with carbon dioxide, which has a sound speed of $280 \mathrm{m} / \mathrm{s} .$ What are the new frequencies of these three modes?

Key Concepts

Effect of Medium on Sound Speed and Resonant Frequencies

The speed at which sound travels in a medium influences the wavelength and, consequently, the resonant frequencies of an air column. Substituting one gas for another (for example, replacing air with carbon dioxide) changes the speed of sound, thereby shifting the frequencies of the standing waves while the mode structure remains unchanged.

Harmonic Series and Mode Patterns

The series of resonant frequencies, or harmonics, in a tube directly results from its boundary conditions. Open-open tubes support all harmonics (integer multiples of the fundamental frequency), whereas open-closed tubes only allow odd harmonics. Recognizing the pattern of observed modes helps in determining the tube type.

Relationship Between Frequency, Wavelength, and Tube Length

There is a direct mathematical relationship between the frequency of a standing wave, its wavelength, and the length of the tube, typically expressed as f = v/?. For a tube with given boundary conditions, the mode shapes determine how the wavelength relates to the tube length, which is pivotal for calculating the tube’s dimensions.

Standing Waves in Air Columns

Standing waves form when waves reflecting at the boundaries of a tube interfere constructively with incoming waves. This concept is essential for understanding the resonant modes in an air column, which are determined by the specific boundary conditions at the tube ends.

Boundary Conditions at Tube Ends

The behavior of the wave at the boundaries depends on whether the tube’s ends are open or closed. Open ends force displacement antinodes because the air is free to move, while closed ends force displacement nodes. This understanding is key when distinguishing between open-open and open-closed tubes.

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