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Chapter 17: Problem 44

The range of human hearing is roughly from twenty hertz to twenty kilohertz. Based on these limits and a value of \(343\) \(\mathrm{m} / \mathrm{s}\) for the speed of sound, what are the lengths of the longest and shortest pipes (open at both ends and producing sound at their fundamental frequencies) that you expect to find in a pipe organ?

Short Answer

Expert verified
The longest pipe is 8.575 meters, and the shortest pipe is 8.575 millimeters.

Step by step solution

01

Understanding the Relationship Between Frequency and Wavelength

The fundamental frequency for an open pipe relates to the speed of sound and wavelength using the formula: \( f = \frac{v}{\lambda} \), where \( f \) is the frequency, \( v \) is the speed of sound, and \( \lambda \) is the wavelength. To find \( \lambda \), we transform the formula: \( \lambda = \frac{v}{f} \).
02

Calculate the Wavelength for Longest Pipe

To find the longest pipe, we use the lowest frequency in the human hearing range, which is \( 20 \) Hz. Using the formula \( \lambda = \frac{v}{f} \), substitute \( v = 343 \) m/s and \( f = 20 \) Hz:\[ \lambda = \frac{343}{20} = 17.15 \text{ meters} \]
03

Calculate the Length of the Longest Pipe

The pipe is open at both ends, so its length \( L \) will be half of the wavelength of the sound it produces at the fundamental frequency: \( L = \frac{\lambda}{2} \). Therefore, \( L = \frac{17.15}{2} = 8.575 \text{ meters} \).
04

Calculate the Wavelength for Shortest Pipe

To find the shortest pipe, we use the highest frequency in the human hearing range, which is \( 20,000 \) Hz. Using the formula \( \lambda = \frac{v}{f} \), substitute \( v = 343 \) m/s and \( f = 20,000 \) Hz:\[ \lambda = \frac{343}{20,000} = 0.01715 \text{ meters} \]
05

Calculate the Length of the Shortest Pipe

Using the same method for the longest pipe, the length \( L \) of the shortest pipe will also be half of the wavelength: \( L = \frac{\lambda}{2} \). Therefore, \( L = \frac{0.01715}{2} = 0.008575 \text{ meters} = 8.575 \text{ mm} \).

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Key Concepts

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

Frequency
Frequency is a core concept in acoustics, representing how often a sound wave repeats within a second. It is measured in hertz (Hz), with one hertz equaling one cycle per second. When we talk about the frequency range of human hearing, it spans from approximately 20 Hz to 20,000 Hz.
Younger individuals often hear higher frequencies better than older adults, whose hearing might decline with age.
In musical contexts like a pipe organ, frequency determines the pitch of the sound. The lower the frequency, the deeper the pitch, and vice versa.
Knowing the frequency is essential for calculating other sound wave properties, such as wavelength. This relationship is used to design musical instruments so that they produce the desired pitch.
Wavelength
Wavelength refers to the length of one complete cycle of a sound wave, measured from peak to peak or trough to trough. It's usually expressed in meters (m). In a pipe, the wavelength determines the length of the pipe needed to produce a particular sound.
For sound waves in the air, the wavelength can be calculated using the formula: \( \lambda = \frac{v}{f} \) where \( \lambda \) is the wavelength, \( v \) is the speed of sound, and \( f \) is the frequency of the sound wave.
Longer wavelengths correspond to lower frequencies and require longer pipes, while shorter wavelengths fit into smaller pipes.
Speed of Sound
The speed of sound is how fast sound waves travel through a medium. In air, the speed of sound is approximately 343 meters per second (m/s). However, this speed can vary based on factors like temperature, humidity, and air pressure.
Sound travels faster in warmer air because the molecules move more quickly. In musical instruments, understanding the speed of sound helps in designing them to produce the right pitch and sound.
Using the speed of sound, along with frequency, we can calculate the wavelength of a sound, fundamental for understanding how different sounds and notes are produced in physical spaces like a concert hall or in instruments like the pipe organ.
Pipe Organ
A pipe organ is a large musical instrument that produces sound by forcing air through pipes selected via a keyboard. Each pipe corresponds to a specific note, which is determined by the pipe's length.
Open pipes, a common choice for organs, produce their fundamental frequency when the pipe length is half the wavelength of the sound wave. This is why calculating the wavelength of a sound wave for a given frequency is crucial for determining the required pipe length.
  • Longer pipes are used for lower notes, corresponding to lower frequencies.
  • Shorter pipes produce higher notes, corresponding to higher frequencies.
This combination of pipe lengths allows the organ to cover a wide auditory range and produce complex compositions.

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Most popular questions from this chapter

Two tubes of gas are identical and are open only at one end. One tube contains neon (Ne) and the other krypton (Kr). Both are monatomic gases, have the same temperature, and may be assumed to be ideal gases. The fundamental frequency of the tube containing neon is 481 Hz. Concepts: (i) For a gas-filled tube open only at one end, the fundamental frequency \((n=1)\) is \(f_{1}=v /(4 L),\) where \(v\) is the speed of sound and \(L\) is the length of the tube. How is the speed related to the properties of the gas? (ii) All of the factors that affect the speed of sound in this problem are the same except for the atomic masses, which are given by 20.180 u for neon, and 83.80 u for krypton. Is the speed of sound in krypton greater than, smaller than, or equal to the speed of sound in neon? Why? (iii) Is the fundamental frequency of the tube containing krypton greater than, less than, or equal to the fundamental frequency of the tube containing neon? Explain. Calculations: What is the fundamental frequency of the tube containing krypton?

Two pianos each sound the same note simultaneously, but they are both out of tune. On a day when the speed of sound is \(343\) \(\mathrm{m} / \mathrm{s},\) piano \(\mathrm{A}\) produces a wavelength of \(0.769\) \(\mathrm{m},\) while piano \(\mathrm{B}\) produces a wavelength of \(0.776\) \(\mathrm{m} .\) How much time separates successive beats?

The E string on an electric bass guitar has a length of \(0.628\) \(\mathrm{m}\) and, when producing the note \(\mathrm{E},\) vibrates at a fundamental frequency of \(41.2\) \(\mathrm{Hz}\) Players sometimes add to their instruments a device called a "D-tuner." This device allows the \(\mathrm{E}\) string to be used to produce the note \(\mathrm{D},\) which has a fundamental frequency of \(36.7\) \(\mathrm{Hz} .\) The D-tuner works by extending the length of the string, keeping all other factors the same. By how much does a D-tuner extend the length of the E string?

A person hums into the top of a well and finds that standing waves are established at frequencies of \(42,70.0,\) and 98 Hz. The frequency of \(42\) \(\mathrm{Hz}\) is not necessarily the fundamental frequency. The speed of sound is \(343\) \(\mathrm{m} / \mathrm{s} .\) How deep is the well?

Sound enters the ear, travels through the auditory canal, and reaches the eardrum. The auditory canal is approximately a tube open at only one end. The other end is closed by the eardrum. A typical length for the auditory canal in an adult is about \(2.9\) \(\mathrm{cm} .\) The speed of sound is \(343\) \(\mathrm{m} / \mathrm{s} .\) What is the fundamental frequency of the canal? (Interestingly, the fundamental frequency is in the frequency range where human hearing is most sensitive.)
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