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Ch 16: Sound & Hearing
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 16: Sound & HearingProblem 40
Chapter 16, Problem 40

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.

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1
Understand that an organ pipe open at one end and closed at the other supports a fundamental frequency where the length of the pipe is one-fourth of the wavelength. This is because the closed end is a node and the open end is an antinode.
Calculate the fundamental frequency of the original pipe using the formula: \( f_1 = \frac{v}{4L} \), where \( v \) is the speed of sound in air (approximately 343 m/s) and \( L \) is the length of the pipe (1.14 m).
Calculate the new length of the second pipe after it is lengthened by 2.00 cm. Convert this length to meters and use it to find the new fundamental frequency \( f_2 \) using the same formula: \( f_2 = \frac{v}{4(L + 0.02)} \).
Determine the beat frequency produced when the two pipes are played together. The beat frequency is the absolute difference between the two fundamental frequencies: \( f_{beat} = |f_1 - f_2| \).
Summarize the process: Calculate \( f_1 \) and \( f_2 \) using the given lengths and the speed of sound, then find the beat frequency by taking the absolute difference between these two frequencies.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Fundamental Frequency of a Pipe

The fundamental frequency of a pipe open at one end and closed at the other is determined by the length of the pipe and the speed of sound. The formula is f = v / (4L), where v is the speed of sound and L is the length of the pipe. This frequency represents the lowest frequency at which the pipe resonates.
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Beat Frequency

Beat frequency occurs when two sound waves of slightly different frequencies interfere with each other. The beat frequency is the absolute difference between the two frequencies, given by f_beat = |f1 - f2|. This results in a periodic variation in sound intensity, perceived as beats.
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Effect of Length Change on Frequency

Changing the length of a pipe affects its fundamental frequency. For a pipe open at one end, increasing the length decreases the frequency, as the wavelength of the sound wave increases. The new frequency can be calculated using the modified length in the fundamental frequency formula.
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Related Practice
Textbook Question

A railroad train is traveling at 30.0 m/s in still air. The frequency of the note emitted by the train whistle is 352 Hz. What frequency is heard by a passenger on a train moving in the opposite direction to the first at 18.0 m/s and approaching the first?

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Textbook Question

A railroad train is traveling at 30.0 m/s in still air. The frequency of the note emitted by the train whistle is 352 Hz. What frequency is heard by a passenger on a train moving in the opposite direction to the first at 18.0 m/s and receding from the first?

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Textbook Question

Two small stereo speakers are driven in step by the same variable-frequency oscillator. Their sound is picked up by a microphone arranged as shown in Fig. E16.39. For what frequencies does their sound at the speakers produce constructive interference?

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Textbook Question

The siren of a fire engine that is driving northward at 30.0 m/s emits a sound of frequency 2000 Hz. A truck in front of this fire engine is moving northward at 20.0 m/s. (a) What is the frequency of the siren's sound that the fire engine's driver hears reflected from the back of the truck?

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Textbook Question

Two loudspeakers, A and B (Fig. E16.35), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. Consider point Q along the extension of the line connecting the speakers, 1.00 m to the right of speaker B. Both speakers emit sound waves that travel directly from the speaker to point Q. What is the lowest frequency for which constructive interference occurs at point Q?

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Textbook Question

Two loudspeakers, A and B (Fig. E16.35), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. Consider point Q along the extension of the line connecting the speakers, 1.00 m to the right of speaker B. Both speakers emit sound waves that travel directly from the speaker to point Q. What is the lowest frequency for which destructive interference occurs at point Q?

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