Ch. 15 - Wave Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 15 - Wave Motion

Problem 85a

Chapter 15, Problem 85a

Two strings on a musical instrument are tuned to play at 392 Hz (G) and 494 Hz (B). What are the frequencies of the first two overtones for each string?

Verified step by step guidance

Understand the concept of overtones: Overtones are the higher frequencies at which a string vibrates, in addition to its fundamental frequency. The first overtone is the second harmonic (2 times the fundamental frequency), and the second overtone is the third harmonic (3 times the fundamental frequency).

Identify the given fundamental frequencies: The first string has a fundamental frequency of 392 Hz, and the second string has a fundamental frequency of 494 Hz.

Calculate the first overtone (second harmonic) for each string: Use the formula \( f_{n} = n \cdot f_{1} \), where \( f_{1} \) is the fundamental frequency and \( n \) is the harmonic number. For the first overtone, \( n = 2 \). For the first string, \( f_{2} = 2 \cdot 392 \). For the second string, \( f_{2} = 2 \cdot 494 \).

Calculate the second overtone (third harmonic) for each string: Again, use the formula \( f_{n} = n \cdot f_{1} \). For the second overtone, \( n = 3 \). For the first string, \( f_{3} = 3 \cdot 392 \). For the second string, \( f_{3} = 3 \cdot 494 \).

Summarize the results: The first overtone for each string corresponds to the second harmonic, and the second overtone corresponds to the third harmonic. These frequencies are calculated by multiplying the fundamental frequency by 2 and 3, respectively.

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Key Concepts

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

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system oscillates, and it determines the pitch of the sound produced. In musical instruments, this frequency corresponds to the note being played. For the strings tuned to 392 Hz (G) and 494 Hz (B), these values represent their fundamental frequencies.

Overtones

Overtones are higher frequencies that occur at integer multiples of the fundamental frequency. The first overtone is typically the second harmonic, which is twice the fundamental frequency, while the second overtone is the third harmonic, three times the fundamental frequency. These overtones contribute to the timbre and richness of the sound produced by the strings.

Harmonic Series

The harmonic series is a sequence of frequencies that are integer multiples of a fundamental frequency. For a string fixed at both ends, the harmonics are produced by standing waves, where the first harmonic is the fundamental frequency, the second harmonic is the first overtone, and so on. Understanding this series is essential for calculating the frequencies of overtones for the given strings.

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