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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 is 0.008575 meters.

Step by step solution

01

Understanding Pipe Organ Acoustics

In a pipe organ, open pipes resonate at their fundamental frequency, producing a standing wave with a wavelength equal to twice the length of the pipe. This means that for a pipe open at both ends, the length of the pipe, denoted as \( L \), is half the wavelength of the sound produced.
02

Relationship Between Speed, Frequency, and Wavelength

The speed of sound \( v \), frequency \( f \), and wavelength \( \lambda \) are related by the equation \( v = f \lambda \). Therefore, the wavelength \( \lambda \) of sound is \( \lambda = \frac{v}{f} \).
03

Calculate Wavelength for Low Frequency Limit

Using the lower frequency limit of 20 Hz, the wavelength is \( \lambda = \frac{343}{20} = 17.15 \text{ meters} \). Since the length of the pipe is half the wavelength, \( L = \frac{17.15}{2} = 8.575 \text{ meters} \).
04

Calculate Wavelength for High Frequency Limit

Using the upper frequency limit of 20,000 Hz, the wavelength is \( \lambda = \frac{343}{20000} = 0.01715 \text{ meters} \). Therefore, the length of the pipe is \( L = \frac{0.01715}{2} = 0.008575 \text{ meters} \).

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Key Concepts

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

Speed of Sound
The speed of sound is a fundamental concept when it comes to understanding how sound waves travel through different mediums. In air, the speed of sound is approximately 343 meters per second (m/s) at room temperature. This speed can change based on factors like temperature, pressure, and the medium through which sound is traveling. Generally, sound travels faster in solids than in liquids, and faster in liquids than in gases.
  • Speed of sound in air: around 343 m/s
  • Can vary based on environmental conditions
  • Faster in denser mediums
Understanding the speed of sound is crucial in various applications, such as calculating the distance of a thunderstorm, designing musical instruments, and even in medical ultrasonography.
Frequency
Frequency refers to how often a wave oscillates as it travels through a medium. It is measured in hertz (Hz) and indicates the number of cycles of a wave per second. In sound, a higher frequency means a higher pitch, and a lower frequency means a lower pitch. Humans can hear a range of sounds from 20 Hz to 20,000 Hz (or 20 kHz).
  • Frequency = Number of wave cycles per second
  • Measured in hertz (Hz)
  • Audible human range: 20 Hz to 20 kHz
Understanding frequency helps us design instruments like the pipe organ, where different frequencies produce different musical notes. Musicians and audio engineers often work with frequency to create harmonious sounds.
Wavelength
Wavelength is the distance between consecutive points of a wave, such as from peak to peak or trough to trough. In sound waves, wavelength is crucial for determining how sound behaves in a space. Wavelength is inversely related to frequency, which means as frequency increases, wavelength decreases, and vice versa. This relationship is captured by the formula: \[ \lambda = \frac{v}{f} \]where \( \lambda \) is the wavelength, \( v \) is the speed of sound, and \( f \) is the frequency.
  • Wavelength = Distance between wave peaks
  • Inversely proportional to frequency
  • Calculated using speed of sound and frequency
Understanding wavelength is imperative for applications like acoustics, where different wavelengths can affect how sound is heard in a room or concert hall.
Pipe Organ Acoustics
Pipe organ acoustics explore how sound waves resonate inside pipes to produce musical notes. Each pipe in an organ is designed to vibrate at a specific fundamental frequency, producing a sound that corresponds to a musical note. For a pipe open at both ends, like those addressed in the original problem, the fundamental wavelength is twice the length of the pipe.
  • Open pipe resonance: Fundamental frequency
  • Wavelength = Twice the length of the pipe
  • Longer the pipe, lower the frequency
In practical terms, the length of the pipe organ pipe determines the pitch of the sound it produces. A longer pipe will resonate with a lower fundamental frequency, resulting in a deeper sound. Understanding these principles allows organ builders to design instruments capable of producing a wide range of harmonics.

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Most popular questions from this chapter

Two waves are traveling in opposite directions on the same string. The displacements caused by the individual waves are given by \(y_{1}=(24.0 \mathrm{mm}) \sin (9.00 \pi t-1.25 \pi x)\) and \(y_{2}=(35.0 \mathrm{mm}) \sin (2.88 \pi t+\) \(0.400 \pi x\) ). Note that the phase angles \((9.00 \pi t-1.25 \pi x)\) and \((2.88 \pi t+\) \(0.400 \pi x\) ) are in radians, \(t\) is in seconds, and \(x\) is in meters. At \(t=4.00 \mathrm{s}\) what is the net displacement (in \(\mathrm{mm}\) ) of the string at (a) \(x=2.16 \mathrm{m}\) and (b) \(x=2.56 \mathrm{m} ?\) Be sure to include the algebraic sign \((+\) or \(-)\) with your answers.

One method for measuring the speed of sound uses standing waves. A cylindrical tube is open at both ends, and one end admits sound from a tuning fork. A movable plunger is inserted into the other end at a distance \(L\) from the end of the tube where the tuning fork is. For a fixed frequency, the plunger is moved until the smallest value of \(L\) is measured that allows a standing wave to be formed. Suppose that the tuning fork produces a \(485-\mathrm{Hz}\) tone, and that the smallest value observed for \(L\) is \(0.264 \mathrm{m} .\) What is the speed of sound in the gas in the tube?

Sound (speed \(=343 \mathrm{m} / \mathrm{s})\) exits a diffraction horn loudspeaker through a rectangular opening like a small doorway. A person is sitting at an angle \(\alpha\) off to the side of a diffraction horn that has a width \(D\) of \(0.060 \mathrm{m} .\) This individual does not hear a sound wave that has a frequency of \(8100 \mathrm{Hz} .\) When she is sitting at an angle \(\alpha / 2,\) the frequency that she does not hear is different. What is this frequency?

A pipe open only at one end has a fundamental frequency of 256 Hz. A second pipe, initially identical to the first pipe, is shortened by cutting off a portion of the open end. Now, when both pipes vibrate at their fundamental frequencies, a beat frequency of \(12 \mathrm{Hz}\) is heard. How many centimeters were cut off the end of the second pipe? The speed of sound is \(343 \mathrm{m} / \mathrm{s}\)

A \(440.0-\mathrm{Hz}\) tuning fork is sounded together with an out-of-tune guitar string, and a beat frequency of \(3 \mathrm{Hz}\) is heard. When the string is tightened, the frequency at which it vibrates increases, and the beat frequency is heard to decrease. What was the original frequency of the guitar string?
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