# 17.5 Sound interference and resonance: standing waves in air columns  (Page 9/14)

 Page 9 / 14

A guitar string of length L is bound at both ends. [link] shows the string’s harmonic frequencies when struck.

Harmonic Number Frequency
1 225/ L
2 450/ L
3 675/ L
4 900/ L
1. Based on the information above, what is the speed of the wave within the string?
2. The guitarist then slides her finger along the neck of the guitar, changing the string length as a result. Calculate the fundamental frequency of the string and wave speed present if the string length is reduced to 2/3 L .

## Section summary

• Sound interference and resonance have the same properties as defined for all waves.
• In air columns, the lowest-frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively, they are called harmonics.
• The resonant frequencies of a tube closed at one end are:
${f}_{1}$ is the fundamental and $L$ is the length of the tube.
• The resonant frequencies of a tube open at both ends are:

## Conceptual questions

How does an unamplified guitar produce sounds so much more intense than those of a plucked string held taut by a simple stick?

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?

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

## Problems&Exercises

A “showy” custom-built car has two brass horns that are supposed to produce the same frequency but actually emit 263.8 and 264.5 Hz. What beat frequency is produced?

0.7 Hz

What beat frequencies will be present: (a) If the musical notes A and C are played together (frequencies of 220 and 264 Hz)? (b) If D and F are played together (frequencies of 297 and 352 Hz)? (c) If all four are played together?

What beat frequencies result if a piano hammer hits three strings that emit frequencies of 127.8, 128.1, and 128.3 Hz?

0.3 Hz, 0.2 Hz, 0.5 Hz

A piano tuner hears a beat every 2.00 s when listening to a 264.0-Hz tuning fork and a single piano string. What are the two possible frequencies of the string?

(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?

(a) 256 Hz

(b) 512 Hz

If a wind instrument, such as a tuba, has a fundamental frequency of 32.0 Hz, what are its first three overtones? It is closed at one end. (The overtones of a real tuba are more complex than this example, because it is a tapered tube.)

What are the first three overtones of a bassoon that has a fundamental frequency of 90.0 Hz? It is open at both ends. (The overtones of a real bassoon are more complex than this example, because its double reed makes it act more like a tube closed at one end.)

180 Hz, 270 Hz, 360 Hz

How long must a flute be in order to have a fundamental frequency of 262 Hz (this frequency corresponds to middle C on the evenly tempered chromatic scale) on a day when air temperature is $20.0ºC$ ? It is open at both ends.

What length should an oboe have to produce a fundamental frequency of 110 Hz on a day when the speed of sound is 343 m/s? It is open at both ends.

1.56 m

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?

(a) Find the length of an organ pipe closed at one end that produces a fundamental frequency of 256 Hz when air temperature is $18.0ºC$ . (b) What is its fundamental frequency at $25.0ºC$ ?

(a) 0.334 m

(b) 259 Hz

By what fraction will the frequencies produced by a wind instrument change when air temperature goes from $10.0ºC$ to $30.0ºC$ ? That is, find the ratio of the frequencies at those temperatures.

The ear canal resonates like a tube closed at one end. (See [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 $37.0ºC$ , which is the same as body temperature. How does this result correlate with the intensity versus frequency graph ( [link] of the human ear?

3.39 to 4.90 kHz

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 $37.0ºC$ . 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.)

A crude approximation of voice production is to consider the breathing passages and mouth to be a resonating tube closed at one end. (See [link] .) (a) What is the fundamental frequency if the tube is 0.240-m long, by taking air temperature to be $37.0ºC$ ? (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) 367 Hz

(b) 1.07 kHz

(a) 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. 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)?

What frequencies will a 1.80-m-long tube produce in the audible range at $20.0ºC$ if: (a) The tube is closed at one end? (b) It is open at both ends?

(a) ${f}_{n}=n\left(47.6 Hz\right),\phantom{\rule{0.25em}{0ex}}n=1, 3, 5,..., 419$

(b) ${f}_{n}=n\left(95.3 Hz\right),\phantom{\rule{0.25em}{0ex}}n=1, 2, 3,..., 210$

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