The human ear canal is about 2.8 cm long and can be regarded as a tube open at one end and close

step 1 In this problem, we have a human ear canal, and we are going to model it as a closed pipe, a pip

The human ear canal is about 2.8 cm long and can be regarded...

The human ear canal is about $2.8 \mathrm{cm}$ long and can be regarded as a tube open at one end and closed at the eardrum. What is the fundamental frequency around which we would expect hearing to be best when the speed of sound in air is $340 \mathrm{m} / \mathrm{s}$ ? (See Sample Problem 13 B.)

Key Concepts

Standing Waves in Air Columns

This concept involves the formation of stable wave patterns, known as standing waves, within a confined medium such as an air column. The wave reflects off boundaries and interferes with incoming waves, resulting in specific nodes and antinodes that dictate the resonant frequencies of the system.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which an air column resonates. It corresponds to the simplest standing wave pattern, typically having the longest wavelength. For a tube that is closed at one end and open at the other, the pattern has a node at the closed end and an antinode at the open end, leading to a wavelength that is four times the length of the tube.

Resonance in Tubes with One Closed End

In tubes with one end closed and the other open, the boundary conditions cause the formation of a quarter-wavelength standing wave for the fundamental frequency. This unique arrangement means that only odd harmonics are possible, and the relationship between the physical dimensions of the tube and the resonant frequencies becomes crucial in determining the sound characteristics.

Relationship Between Wavelength, Speed of Sound, and Frequency

The connection between wavelength, speed of sound, and frequency is a fundamental aspect of wave physics. This relationship is expressed as frequency equals the speed of sound divided by the wavelength. Understanding this equation allows one to calculate the resonant frequencies of an air column when given its dimensions and the speed of sound in the medium.

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