The frequency of the fundamental of a certain tube in air is 200 Hz. The frequency of the funda

step 1 We would like to find what is the speed of sound of this gas. We know that our fundamental frequ

The frequency of the fundamental of a certain tube in air is...

The frequency of the fundamental of a certain tube in air is $200 \mathrm{~Hz}$. The frequency of the fundamental is $250 \mathrm{~Hz}$ when the tube is placed in an unknown gas.
a. What is the speed of sound in this gas?
b. What type of tube was used (open or closed)?

Key Concepts

Speed of Sound

The speed of sound in a medium determines how fast sound waves travel and is influenced by the medium’s physical properties (such as its temperature, molecular composition, and density). It is key to relating the frequency and wavelength of waves in that medium by the equation v = f×?. In problems comparing two media, changes in observed frequencies for the same geometric setup usually reflect differences in the speed of sound.

Fundamental Frequency and Wavelength Relationship

In acoustics, the fundamental frequency (or first harmonic) of a resonant system is related to the dimensions of the system and the speed of sound. For a tube, the relationship between the frequency, the speed of sound, and the length of the tube is used to determine the wavelength at the fundamental mode, which then helps characterize the acoustic behavior of the system.

Resonance in Open and Closed Tubes

Open and closed tubes support standing waves with different boundary conditions, leading to different relationships between the tube’s length and the wavelength of the fundamental frequency. An open tube (open at both ends) supports antinodes at the ends and has a fundamental wavelength equal to twice the tube’s length (? = 2L), while a tube closed at one end has a node at the closed end and an antinode at the open end, resulting in a fundamental wavelength equal to four times the tube’s length (? = 4L). Identifying the type of tube in a problem often involves comparing these relationships along with other given data.

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