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Chapter 16: Problem 48

A bugle can be represented by a cylindrical pipe of length \(L=1.35 \mathrm{~m} .\) Since the ends are open, the standing waves produced in the bugle have antinodes at the open ends, where the air molecules move back and forth the most. Calculate the longest three wavelengths of standing waves inside the bugle. Also calculate the three lowest frequencies and the three longest wavelengths of the sound that is produced in the air around the bugle.

Short Answer

Expert verified
Answer: The three longest wavelengths inside the bugle are 2.70 meters, 1.35 meters, and 0.90 meters. The three lowest frequencies are approximately 127.04 Hz, 254.07 Hz, and 381.11 Hz. The three longest wavelengths of the sound produced in the air around the bugle are also 2.70 meters, 1.35 meters, and 0.90 meters.

Step by step solution

01

Understand the characteristics of open pipes

In an open pipe, both ends have antinodes. The simplest standing wave can be created, when we have only one antinode more than the endpoints, which is half a wavelength (λ/2). In the case of open pipes, the fundamental frequency or first harmonic has a length of L = λ/2. For higher harmonics, the length is given by L = n(λ/2), where n is the harmonic number.
02

Calculate the three longest wavelengths inside the bugle

Let's find the longest three wavelengths inside the bugle by using the formula L = n(λ/2), and solve for λ. We will do this for n = 1, 2, and 3, as they represent the three lowest harmonics. λ1 = 2 * L / 1 λ2 = 2 * L / 2 λ3 = 2 * L / 3 Plug in the given length, L = 1.35 meters: λ1 = 2 * 1.35 / 1 λ1 = 2.70 meters λ2 = 2 * 1.35 / 2 λ2 = 1.35 meters λ3 = 2 * 1.35 / 3 λ3 = 0.90 meters The three longest wavelengths inside the bugle are 2.70 meters, 1.35 meters, and 0.90 meters.
03

Calculate the three lowest frequencies

Now, let's find the three lowest frequencies of standing waves inside the bugle. We will use the equation f = v / λ, where f is the frequency, v is the speed of sound (approximately 343 meters/second for air), and λ is the wavelength. f1 = v / λ1 f2 = v / λ2 f3 = v / λ3 Use speed of sound in air, v = 343 meters/second: f1 = 343 / 2.70 f1 ≈ 127.04 Hz f2 = 343 / 1.35 f2 ≈ 254.07 Hz f3 = 343 / 0.90 f3 ≈ 381.11 Hz The three lowest frequencies are approximately 127.04 Hz, 254.07 Hz, and 381.11 Hz.
04

Calculate the wavelengths of the sound produced in the air around the bugle

The wavelengths of the sound produced in the air around the bugle are the same as the wavelengths inside the bugle. Therefore, the three longest wavelengths of the sound produced in the air around the bugle are 2.70 meters, 1.35 meters, and 0.90 meters.

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Most popular questions from this chapter

At a distance of \(20.0 \mathrm{~m}\) from a sound source, the intensity of the sound is \(60.0 \mathrm{~dB}\). What is the intensity (in \(\mathrm{dB}\) ) at a point \(2.00 \mathrm{~m}\) from the source? Assume that the sound radiates equally in all directions from the source.

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