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Chapter 16: Problem 43

Two organ pipes, open at one end but closed at the other, are each 1.14 m long. One is now lengthened by 2.00 cm. Find the beat frequency that they produce when playing together in their fundamentals.

Short Answer

Expert verified
The beat frequency is 1.30 Hz.

Step by step solution

01

Calculate the fundamental frequency of the original pipe

The fundamental frequency of a pipe closed at one end is given by \( f = \frac{v}{4L} \), where \( v \) is the speed of sound in air (approximately 343 m/s at room temperature), and \( L \) is the length of the pipe. For the original pipe of length 1.14 m, the frequency is \( f_{1} = \frac{343}{4 \times 1.14} \approx 75.22 \) Hz.
02

Calculate the new length of the lengthened pipe

The second pipe is lengthened by 2.00 cm, which is equivalent to 0.02 m. Therefore, the new length \( L' \) of the second pipe is \( 1.14 + 0.02 = 1.16 \) meters.
03

Calculate the fundamental frequency of the lengthened pipe

Using the same fundamental frequency formula \( f = \frac{v}{4L} \), we calculate the frequency of the lengthened pipe. Thus, for \( L' = 1.16 \) m, the frequency is \( f_{2} = \frac{343}{4 \times 1.16} \approx 73.92 \) Hz.
04

Calculate the beat frequency

The beat frequency produced by two waves of slightly different frequencies \( f_1 \) and \( f_2 \) is given by the absolute difference between the two frequencies: \( |f_{1} - f_{2}| \). Therefore, the beat frequency is \( |75.22 - 73.92| = 1.30 \) Hz.

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Key Concepts

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

Organ Pipes
Organ pipes are fascinating instruments that produce sound through the vibration of air columns. In the context of physics, especially acoustics, an organ pipe can either be open at both ends or open at one end and closed at the other, also known as closed pipes. These variations affect how sound waves resonate within them, leading to different tonal characteristics.
  • Open pipes have both ends open to the air, allowing the sound waves to reflect and build constructive interference easily.
  • Closed pipes, like the ones in our exercise, are open at one end and closed at the other. These pipes resonate at odd harmonics of the fundamental frequency.
  • The length of the pipe directly influences the sound's pitch; longer pipes produce deeper tones.
Understanding these principles helps explain why alterations to the pipe's length, such as adding 2.00 cm, change the frequency of the sound produced.
Fundamental Frequency
The fundamental frequency is the lowest frequency produced by any vibrating object, including organ pipes. For a pipe closed at one end, this frequency is determined by the formula \( f = \frac{v}{4L} \), where \( v \) is the speed of sound and \( L \) is the length of the pipe.
  • In a closed pipe, only odd harmonics are present, which significantly impacts the sound it produces.
  • Reducing the length of a pipe raises the fundamental frequency, while increasing the length lowers it.
  • The fundamental frequency serves as the base tone from which other harmonics are generated.
Understanding how to calculate this frequency helps musicians and engineers design instruments and devices that rely on precise sound production.
Speed of Sound
The speed of sound is a critical factor in calculating frequencies in organ pipes. At room temperature, the speed of sound in air is approximately 343 m/s. This speed influences how quickly sound waves can travel through any medium, including air.
  • Sound speed can vary with temperature, humidity, and the medium through which it passes.
  • Higher temperatures generally increase the speed of sound, affecting the frequency calculations in practical scenarios.
  • Accurate measurements of sound speed are crucial for precise frequency calculations in musical acoustics.
In our exercise, using the standard speed of sound allows us to find the fundamental frequencies of the organ pipes accurately.
Wave Interference
Wave interference occurs when two or more waves overlap and combine to form a new wave pattern. This phenomenon can be constructive or destructive, impacting the sound perceived.
  • Constructive interference happens when waves align perfectly to amplify the resultant wave.
  • Destructive interference occurs when waves are out of phase, reducing the overall sound intensity.
  • In musical applications, wave interference can lead to beat frequencies, a result of slightly differing frequencies playing together.
In our organ pipe scenario, the two fundamental frequencies create a beat frequency due to their slight difference, resulting in a pulsating sound pattern.

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A swimming duck paddles the water with its feet once every 1.6 s, producing surface waves with this period. The duck is moving at constant speed in a pond where the speed of surface waves is 0.32 m/s, and the crests of the waves ahead of the duck are spaced 0.12 m apart. (a) What is the duck's speed? (b) How far apart are the crests behind the duck?

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You have a stopped pipe of adjustable length close to a taut 62.0-cm, 7.25-g wire under a tension of 4110 N. You want to adjust the length of the pipe so that, when it produces sound at its fundamental frequency, this sound causes the wire to vibrate in its second \(overtone\) with very large amplitude. How long should the pipe be?

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