A tube is open only at one end. A certain harmonic produced...

Name: Problem 47, Chapter 17: Physics
Uploaded: 2020-04-21T08:55:08-08:00
Duration: 2 min 11 s
Channel: Farnaz Mohseni
Description: A tube is open only at one end. A certain harmonic produced by the tube has a frequency of 450 Hz. The next higher harmonic has a frequency of 750 Hz. The speed of sound in air is 343 m/s. (a) What is the integer n that describes the harmonic whose frequency is 450 Hz? (b) What is the length of the tube?

A tube is open only at one end. A certain harmonic produced by the tube has a frequency of 450 Hz. The next higher harmonic has a frequency of 750 Hz. The speed of sound in air is 343 m/s. (a) What is the integer n that describes the harmonic whose frequency is 450 Hz? (b) What is the length of the tube?

Key Concepts

Wave Speed and Frequency Relationship

The relationship between wave speed, frequency, and wavelength is fundamental to acoustics and is given by the equation f = v/?, where f is frequency, v is the speed of sound, and ? is the wavelength. This relationship is used to connect the physical dimensions of a resonating tube with the observed frequencies.

Determining Physical Dimensions from Resonant Frequencies

Once the harmonic number is identified, the length of the tube can be calculated using the boundary condition for a tube closed at one end. The formula L = n·(?/4) or equivalently L = n·(v/(4f)) links the tube’s length to the speed of sound and the resonant frequency, thereby allowing the determination of physical dimensions from acoustic measurements.

Resonance in Open-Ended Air Columns

In a tube that is open at one end and closed at the other, standing waves form in such a way that there is a displacement antinode at the open end and a displacement node at the closed end. This configuration supports only odd harmonics of the fundamental frequency. Thus, the resonant frequencies are given by f_n = n·(v/4L) with n being odd (e.g., 1, 3, 5, …).

Odd Harmonic Series

Since a tube open at one end only supports odd harmonics, the sequence of resonant frequencies does not increase in equal intervals as in tubes open at both ends. Understanding this concept is crucial when determining which harmonic corresponds to a given frequency because it restricts the possible values of n to odd integers.

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