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Chapter 16: Problem 44

Find the greatest length of an organ pipe open at both ends that will have its fundamental frequency in the normal hearing range \((20-20,000 \mathrm{~Hz}) .\) Speed of sound in air \(=340 \mathrm{~m} \mathrm{~s}^{-1}\).

Short Answer

Expert verified
The greatest length of the organ pipe is 8.5 meters.

Step by step solution

01

Understand the Formulas

For a pipe open at both ends, the fundamental frequency is given by \( f = \frac{v}{2L} \), where \( f \) is the frequency, \( v \) is the speed of sound in air, and \( L \) is the length of the pipe. We are looking for the greatest length that results in a fundamental frequency within the range of 20 Hz to 20,000 Hz.
02

Set Up Inequality for Frequency

Since we need the fundamental frequency to be within the hearing range, we set up the inequality \( 20 \leq \frac{340}{2L} \leq 20000 \). We are interested in finding the maximum length \( L \), so let's consider \( f = 20 \) Hz.
03

Solve for Maximum Length L

Set the frequency equation to the minimum frequency: \( 20 = \frac{340}{2L} \). Solve for \( L \):\[ 20 = \frac{340}{2L} \]\[ 20 \times 2L = 340 \]\[ 40L = 340 \]\[ L = \frac{340}{40} \]\[ L = 8.5 \text{ meters} \]
04

Interpret the Result

The maximum length of the organ pipe that allows its fundamental frequency to remain within the audible hearing range is \( L = 8.5 \) meters. Any length greater than this would result in a fundamental frequency lower than 20 Hz, which is outside the normal hearing range.

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Key Concepts

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

Organ pipe acoustics
Organ pipe acoustics involves the study of sound waves within cylindrical pipes, which create musical notes. These pipes can be open at both ends, which is important for determining the types of sounds they produce. When a sound wave enters an open pipe, it travels back and forth inside. At each open end, the wave reflects back, setting up standing waves inside the pipe. These standing waves have nodes and antinodes. The nodes are points where the wave does not move, while the antinodes are points of maximum movement.
  • For pipes open at both ends, the fundamental frequency is the simplest vibration, creating a wave pattern with one antinode at each open end and a node in the middle.
  • The length of the pipe determines the frequencies of the sound waves that can form, affecting the pitch of the notes produced.
Understanding these principles is crucial to designing organ pipes that produce specific musical notes.
Speed of sound
The speed of sound is a key factor when dealing with sound waves in organ pipes. It is the speed at which sound waves travel through a medium, like air. In this scenario, we consider the speed of sound to be 340 meters per second in air.
  • This speed can vary depending on the medium, the temperature, and the composition of the air.
  • In general, sound travels faster in mediums with greater density and at higher temperatures.
  • Weather and environmental conditions may slightly change the speed of sound, affecting musical instruments like organ pipes.
Knowing the speed of sound helps in calculating other important properties, like the frequencies and lengths of organ pipes.
Audible hearing range
The audible hearing range refers to the spectrum of sound frequencies that can be heard by the average human ear. This range typically falls between 20 Hz to 20,000 Hz. Frequencies below or above this range might not be audible to humans.
  • Loudness and frequency both play roles in sound perception. Frequencies at the extremes (20 Hz and 20,000 Hz) are just at the edge of what most people can hear.
  • Organ pipes are often designed to produce sounds within this range, so they are audible to listeners.
  • Ensuring a pipe produces sound within the human hearing range is important for its functionality in music.
Understanding the range helps musicians and sound engineers craft sounds that are enjoyable and hearable.
Wave equation in open pipes
The wave equation in open pipes is central to understanding how sound frequencies are determined. For a pipe open at both ends, the wave equation to determine the fundamental frequency is given by:
  • \[ f = \frac{v}{2L} \]
where:
  • \( f \) is the fundamental frequency.
  • \( v \) is the speed of sound in the medium.
  • \( L \) is the length of the pipe.
This equation helps define the relationship between these factors:
  • As the length \( L \) increases, the fundamental frequency \( f \) decreases.
  • Conversely, a shorter pipe length results in a higher frequency.
  • This is crucial for tuning instruments to the desired note.
Using this equation, one can determine the greatest possible length of an organ pipe to stay within the desired frequency range, as was calculated in the exercise example.

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Most popular questions from this chapter

A traffic policeman sounds a whistle to stop a car-driver approaching towards him. The car-driver does not stop and takes the plea in court that because of the Doppler shift, the frequency of the whistle reaching him might have gone beyond the audible limit of \(20 \mathrm{kHz}\) and he did not hear it. Experiments showed that the whistle emits a sound with frequency close to \(16 \mathrm{kHz}\). Assuming that the claim of the driver is true, how fast was he driving the car? Take the speed of sound in air to be \(330 \mathrm{~m} \mathrm{~s}^{-1}\). Is this speed practical with today's technology ?

Two audio speakers are kept some distance apart and are driven by the same amplifier system. A person is sitting at a place \(6 \cdot 0 \mathrm{~m}\) from one of the speakers and \(6 \cdot 4 \mathrm{~m}\) from the other. If the sound signal is continuously varied from \(500 \mathrm{~Hz}\) to \(5000 \mathrm{~Hz}\), what are the frequencies for which there is a destructive interference at the place of the listener? Speed of sound in air \(=320 \mathrm{~m} \mathrm{~s}^{-1}\).

At what temperature will the speed of sound be double of its value at \(0^{\circ} \mathrm{C}\) ?

A person standing on a road sends a sound signal to the driver of a car going away from him at a speed of \(72 \mathrm{~km} \mathrm{~h}^{-1}\). The signal travelling at \(330 \mathrm{~m} \mathrm{~s}^{-1}\) in air and having a frequency of \(1600 \mathrm{~Hz}\) gets reflected from the body of the car and returns. Find the frequency of the reflected signal as heard by the person.

Two sources of sound, \(S_{1}\) and \(S_{2}\), emitting waves of equal wavelength \(20^{\circ} 0 \mathrm{~cm}\), are placed with a separation of \(20 \cdot 0 \mathrm{~cm}\) between them. A detector can be moved on a line parallel to \(S_{1} S_{2}\) and at a distance of \(20 \cdot 0 \mathrm{~cm}\) from it. Initially, the detector is equidistant from the two sources. Assuming that the waves emitted by the sources are in phase, find the minimum distance through which the detector should be shifted to detect a minimum of sound.
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