Ch 17: Superposition

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 17: Superposition

Problem 72c

Chapter 17, Problem 72c

Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have a frequency of 440 Hz and the note E should be at 659 Hz. The tuner starts with the tension in the E string a little low, then tightens it. What is the frequency of the E string when she hears four beats per second?

Verified step by step guidance

Step 1: Understand the concept of beats. Beats occur when two sound waves of slightly different frequencies interfere with each other. The beat frequency is equal to the absolute difference between the two frequencies: \( f_{\text{beat}} = |f_1 - f_2| \).

Step 2: Identify the given values. The frequency of the A note is \( f_A = 440 \ \text{Hz} \), and the tuner hears a beat frequency of \( f_{\text{beat}} = 4 \ \text{Hz} \). The frequency of the E string is initially lower than its proper value of \( f_E = 659 \ \text{Hz} \).

Step 3: Use the beat frequency formula to determine the possible frequencies of the E string. Since \( f_{\text{beat}} = |f_E - f_A| \), the possible frequencies of the E string are: \( f_E = f_A + f_{\text{beat}} \) or \( f_E = f_A - f_{\text{beat}} \).

Step 4: Substitute the known values into the equations. For the first case: \( f_E = 440 + 4 \ \text{Hz} \). For the second case: \( f_E = 440 - 4 \ \text{Hz} \).

Step 5: Determine the correct frequency of the E string. Since the tuner is tightening the string (increasing its frequency), the correct frequency is the higher value from the two possibilities. This will give the frequency of the E string when the tuner hears four beats per second.

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Key Concepts

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

Frequency and Beats

Frequency refers to the number of cycles of a wave that occur in one second, measured in Hertz (Hz). When two sound waves of slightly different frequencies interact, they produce a phenomenon known as 'beats,' which is the periodic variation in amplitude that occurs due to interference. The beat frequency is equal to the absolute difference between the two frequencies, allowing tuners to identify how close the notes are to being in tune.

Tension and Frequency Relationship

The frequency of a vibrating string is influenced by its tension, length, and mass per unit length. According to the wave equation, increasing the tension in a string raises its frequency, while decreasing tension lowers it. This relationship is crucial for piano tuners, as they adjust the tension of the strings to achieve the desired pitch.

Harmonics and Overtones

Harmonics are the integer multiples of a fundamental frequency produced by vibrating strings or air columns. When a string vibrates, it generates not only the fundamental frequency but also overtones, which contribute to the timbre of the sound. Understanding harmonics is essential for tuners, as they listen for the interaction between the fundamental frequencies and their harmonics to ensure the strings are properly tuned.

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Related Practice

Textbook Question

The three identical loudspeakers in FIGURE P17.71 play a 170 Hz tone in a room where the speed of sound is 340 m/s. You are standing 4.0 m in front of the middle speaker. At this point, the amplitude of the wave from each speaker is a. How far must speaker 2 be moved to the left to produce a maximum amplitude at the point where you are standing?

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Textbook Question

Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have a frequency of 440 Hz and the note E should be at 659 Hz. What is the frequency difference between the third harmonic of the A and the second harmonic of the E?

Textbook Question

The three identical loudspeakers in FIGURE P17.71 play a 170 Hz tone in a room where the speed of sound is 340 m/s. You are standing 4.0 m in front of the middle speaker. At this point, the amplitude of the wave from each speaker is a. When the amplitude is maximum, by what factor is the sound intensity greater than the sound intensity from a single speaker?

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Textbook Question

You have two small, identical boxes that generate 440 Hz notes. While holding one, you drop the other from a 20-m-high balcony. How many beats will you hear before the falling box hits the ground? You can ignore air resistance.

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Textbook Question

A flutist assembles her flute in a room where the speed of sound is 342 m/s. When she plays the note A, it is in perfect tune with a 440 Hz tuning fork. After a few minutes, the air inside her flute has warmed to where the speed of sound is 346 m/s. How far does she need to extend the 'tuning joint' of her flute to be in tune with the tuning fork?

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Textbook Question

When mass M is tied to the bottom end of a long, thin wire suspended from the ceiling, the wire's second-harmonic frequency is 200 Hz. Adding an additional 1.0 kg to the hanging mass increases the second-harmonic frequency to 245 Hz. What is M?