# 17.4 Normal modes of a standing sound wave  (Page 6/9)

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## Conceptual questions

You are given two wind instruments of identical length. One is open at both ends, whereas the other is closed at one end. Which is able to produce the lowest frequency?

The fundamental wavelength of a tube open at each end is 2 L , where the wavelength of a tube open at one end and closed at one end is 4 L . The tube open at one end has the lower fundamental frequency, assuming the speed of sound is the same in both tubes.

What is the difference between an overtone and a harmonic? Are all harmonics overtones? Are all overtones harmonics?

Two identical columns, open at both ends, are in separate rooms. In room A , the temperature is $T=20\text{°}\text{C}$ and in room B , the temperature is $T=25\text{°}\text{C}$ . A speaker is attached to the end of each tube, causing the tubes to resonate at the fundamental frequency. Is the frequency the same for both tubes? Which has the higher frequency?

The wavelength in each is twice the length of the tube. The frequency depends on the wavelength and the speed of the sound waves. The frequency in room B is higher because the speed of sound is higher where the temperature is higher.

## Problems

(a) What is the fundamental frequency of a 0.672-m-long tube, open at both ends, on a day when the speed of sound is 344 m/s? (b) What is the frequency of its second harmonic?

What is the length of a tube that has a fundamental frequency of 176 Hz and a first overtone of 352 Hz if the speed of sound is 343 m/s?

0.974 m

The ear canal resonates like a tube closed at one end. (See [link]Figure 17_03_HumEar[/link].) If ear canals range in length from 1.80 to 2.60 cm in an average population, what is the range of fundamental resonant frequencies? Take air temperature to be $37.0\text{°}\text{C,}$ which is the same as body temperature.

Calculate the first overtone in an ear canal, which resonates like a 2.40-cm-long tube closed at one end, by taking air temperature to be $37.0\text{°}\text{C}$ . Is the ear particularly sensitive to such a frequency? (The resonances of the ear canal are complicated by its nonuniform shape, which we shall ignore.)

11.0 kHz; The ear is not particularly sensitive to this frequency, so we don’t hear overtones due to the ear canal.

A crude approximation of voice production is to consider the breathing passages and mouth to be a resonating tube closed at one end. (a) What is the fundamental frequency if the tube is 0.240 m long, by taking air temperature to be $37.0\text{°}\text{C}$ ? (b) What would this frequency become if the person replaced the air with helium? Assume the same temperature dependence for helium as for air.

A 4.0-m-long pipe, open at one end and closed at one end, is in a room where the temperature is $T=22\text{°}\text{C}\text{.}$ A speaker capable of producing variable frequencies is placed at the open end and is used to cause the tube to resonate. (a) What is the wavelength and the frequency of the fundamental frequency? (b) What is the frequency and wavelength of the first overtone?

a. $v=344.08\phantom{\rule{0.2em}{0ex}}\text{m/s,}\phantom{\rule{0.5em}{0ex}}{\lambda }_{1}=16.00\phantom{\rule{0.2em}{0ex}}\text{m,}\phantom{\rule{0.5em}{0ex}}{f}_{1}=21.51\phantom{\rule{0.2em}{0ex}}\text{Hz;}$
b. ${\lambda }_{3}=5.33\phantom{\rule{0.2em}{0ex}}\text{m,}\phantom{\rule{0.5em}{0ex}}{f}_{3}=64.56\phantom{\rule{0.2em}{0ex}}\text{Hz}$

A 4.0-m-long pipe, open at both ends, is placed in a room where the temperature is $T=25\text{°}\text{C}\text{.}$ A speaker capable of producing variable frequencies is placed at the open end and is used to cause the tube to resonate. (a) What are the wavelength and the frequency of the fundamental frequency? (b) What are the frequency and wavelength of the first overtone?

A nylon guitar string is fixed between two lab posts 2.00 m apart. The string has a linear mass density of $\mu =7.20\phantom{\rule{0.2em}{0ex}}\text{g/m}$ and is placed under a tension of 160.00 N. The string is placed next to a tube, open at both ends, of length L . The string is plucked and the tube resonates at the $n=3$ mode. The speed of sound is 343 m/s. What is the length of the tube?

$\begin{array}{}\\ \\ \\ {v}_{\text{string}}=149.07\phantom{\rule{0.2em}{0ex}}\text{m/s,}\phantom{\rule{0.5em}{0ex}}{\lambda }_{3}=1.33\phantom{\rule{0.2em}{0ex}}\text{m,}\phantom{\rule{0.5em}{0ex}}{f}_{3}=112.08\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \\ {\lambda }_{1}=\frac{v}{{f}_{1}},\phantom{\rule{0.5em}{0ex}}L=1.53\phantom{\rule{0.2em}{0ex}}\text{m}\hfill \end{array}$

A 512-Hz tuning fork is struck and placed next to a tube with a movable piston, creating a tube with a variable length. The piston is slid down the pipe and resonance is reached when the piston is 115.50 cm from the open end. The next resonance is reached when the piston is 82.50 cm from the open end. (a) What is the speed of sound in the tube? (b) How far from the open end will the piston cause the next mode of resonance?

Students in a physics lab are asked to find the length of an air column in a tube closed at one end that has a fundamental frequency of 256 Hz. They hold the tube vertically and fill it with water to the top, then lower the water while a 256-Hz tuning fork is rung and listen for the first resonance. (a) What is the air temperature if the resonance occurs for a length of 0.336 m? (b) At what length will they observe the second resonance (first overtone)?

a. $22.0\text{°C}$ ; b. 1.01 m

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