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Chapter 14: Problem 92

The fundamental of an organ pipe that is closed at one end and open at the other end is \(261.6 \mathrm{Hz}\) (middle \(\mathrm{C}\) ). The second harmonic of an organ pipe that is open at both ends has the same frequency. What are the lengths of these two pipes?

Short Answer

Expert verified
The closed pipe is about 0.328 m long, and the open pipe is about 1.31 m long.

Step by step solution

01

Understanding Pipe Types

An organ pipe closed at one end is a closed pipe, while one open at both ends is an open pipe. Each type supports different harmonic series: closed pipes have odd harmonics, while open pipes have all harmonics.
02

Closed Pipe Fundamental Frequency

For a closed pipe, only odd harmonics are present. The fundamental frequency formula is \( f = \frac{v}{4L} \), where \( v \) is the speed of sound (approximately 343 m/s at room temperature) and \( L \) is the pipe length. In this case, \( f = 261.6 \mathrm{Hz} \).
03

Calculating Length of the Closed Pipe

Rearrange the fundamental frequency formula for \( L \): \( L = \frac{v}{4f} \). Substituting the given values, \( L = \frac{343}{4 \times 261.6} \approx 0.328 \mathrm{m} \). The length of the closed pipe is approximately 0.328 meters.
04

Open Pipe Harmonics

An open pipe supports harmonics at all integer multiples of the fundamental frequency. Here, the second harmonic frequency \( f_2 = 261.6 \mathrm{Hz} \), so \( f_2 = 2f_1 \), where \( f_1 \) is the fundamental frequency.
05

Finding Fundamental Frequency of Open Pipe

Since \( f_2 = 261.6 \mathrm{Hz} \), the fundamental frequency \( f_1 \) of the open pipe is \( \frac{261.6}{2} = 130.8 \mathrm{Hz} \).
06

Calculating Length of the Open Pipe

Use the fundamental frequency formula for an open pipe, \( f = \frac{v}{2L} \), rearranged for \( L \): \( L = \frac{v}{2f_1} \). Substituting \( 130.8 \mathrm{Hz} \) for \( f_1 \), \( L = \frac{343}{2 \times 130.8} \approx 1.31 \mathrm{m} \). The length of the open pipe is approximately 1.31 meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Harmonics
In the world of acoustics, harmonics play a crucial role in determining the natural frequencies at which objects vibrate. When sound waves travel within a medium, such as an organ pipe, they reflect and interfere, creating standing waves. These waves have specific frequencies called harmonics.

The harmonic series is a set of frequencies related by integer ratios, with each harmonic being a multiple of a fundamental frequency.
  • The first harmonic, or fundamental frequency, is the lowest frequency and determines the pitch of the sound.
  • The second harmonic is twice the frequency of the first, the third harmonic is three times, and so on.
Understanding harmonics is essential to predicting and analyzing the behavior of sound in different mediums, and they form the foundation of musical acoustics.
Closed Pipe
Closed pipes are unique in that they are open at one end and closed at the other. This configuration leads to a distinctive behavior in their harmonic series. Unlike open pipes, closed pipes can only support odd harmonics.

With odd harmonics, a closed pipe supports frequencies that are odd multiples of the fundamental, such as 1st, 3rd, 5th harmonics, etc. The formula used to find the fundamental frequency of a closed pipe is:\[f = \frac{v}{4L}\]where \(f\) is the frequency, \(v\) is the speed of sound, and \(L\) is the length of the pipe.

This equation reflects how the wavelength of the sound wave is divided in the pipe. Given the nature of closed pipes only supporting odd harmonics, they produce specific tonal qualities, which is why they are often used in certain musical instruments.
Open Pipe
An open pipe is one that is open at both ends. This type of pipe allows sound waves to reflect and create standing waves that accommodate all harmonics, making the entire harmonic series accessible: the fundamental frequency and its integer multiples.

The formula you use for an open pipe is:\[f = \frac{v}{2L}\]where \(f\) represents the frequency of the harmonics, \(v\) is the speed of sound, and \(L\) is the length of the pipe. As every harmonic is possible in an open pipe, these pipes produce a fuller range of tones.

This is why open pipes are often found in instruments seeking to produce a rich, melodious sound. The presence of multiple harmonics means they vibrate in a more complex manner, leading to a diverse array of sound frequencies.

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Most popular questions from this chapter

The Bullet Train The Shinkansen, the Japanese "bullet" train, runs at high speed from Tokyo to Nagoya. Riding on the Shinkansen, you notice that the frequency of a crossing signal changes markedly as you pass the crossing. As you approach the crossing, the frequency you hear is \(f\); as you recede from the crossing the frequency you hear is \(2 f / 3 .\) What is the speed of the train?

Two strings that are fixed at cach end are identical, except that one is \(0.560 \mathrm{cm}\) longer than the other. Waves on these strings propagate with a speed of \(34.2 \mathrm{m} / \mathrm{s},\) and the fundamental frequency of the shorter string is \(212 \mathrm{Hz}\). (a) What beat frequency is produced if each string is vibrating with its fundamental frequency? (b) Does the beat frequency in part (a) increase or decrease if the longer string is increased in length? (c) Repeat part (a), assuming that the longer string is \(0.761 \mathrm{cm}\) longer than the shorter string.

A surfer floating beyond the breakers notes 14 waves per minute passing her position. If the wavelength of these waves is \(34 \mathrm{m},\) what is their speed?

When guitar strings A and B are plucked at the same time, a beat frequency of \(2 \mathrm{Hz}\) is heard. If string \(\mathrm{A}\) is tightened, the beat frequency increases to \(3 \mathrm{Hz}\). Which of the two strings had the lower frequency initially?

Predict/Explain When you blow across the opening of a partially filled 2 - \(L\) soda pop bottle you hear a tone. (a) If you take a sip of the pop and blow across the opening again, does the tone you hear have a higher frequency, a lower frequency, or the same frequency as before? (b) Choose the best explanation from among the following: I. The same pop bottle will give the same frequency regardless of the amount of pop it contains. II. The greater distance from the top of the bottle to the level of the pop results in a higher frequency. III. A lower level of pop results in a longer column of air, and hence a lower frequency.
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