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Ch 17: Superposition
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 17: SuperpositionProblema 8a
Capitolo 17, Problema 8a

What are the three longest wavelengths for standing waves on a 60 cm long string that is fixed at both ends?

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Understand the concept: For a string fixed at both ends, standing waves are formed at specific wavelengths. The wavelengths correspond to the harmonic series, where the length of the string is related to the wavelength by the formula: \( L = \frac{n \lambda}{2} \), where \( L \) is the length of the string, \( \lambda \) is the wavelength, and \( n \) is the harmonic number (\( n = 1, 2, 3, \dots \)).
Rearrange the formula to solve for the wavelength: \( \lambda = \frac{2L}{n} \). Here, \( L = 60 \; \text{cm} \) (convert to meters if needed, \( L = 0.6 \; \text{m} \)).
Identify the three longest wavelengths: The longest wavelengths correspond to the smallest harmonic numbers (\( n = 1, 2, 3 \)). Substitute \( n = 1 \), \( n = 2 \), and \( n = 3 \) into the formula \( \lambda = \frac{2L}{n} \) to calculate the wavelengths.
For \( n = 1 \) (fundamental frequency): \( \lambda_1 = \frac{2 \times 0.6}{1} \). For \( n = 2 \) (first overtone): \( \lambda_2 = \frac{2 \times 0.6}{2} \). For \( n = 3 \) (second overtone): \( \lambda_3 = \frac{2 \times 0.6}{3} \).
Conclude: The three longest wavelengths for standing waves on the string are \( \lambda_1 \), \( \lambda_2 \), and \( \lambda_3 \), corresponding to the fundamental frequency and the first two overtones.

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Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. In a fixed medium, such as a string, this results in specific points called nodes (where there is no movement) and antinodes (where the movement is maximum). The pattern of these waves is determined by the length of the string and the wavelength of the waves.
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Wavelength and Frequency Relationship

The wavelength of a wave is inversely related to its frequency, as described by the equation v = fλ, where v is the wave speed, f is the frequency, and λ is the wavelength. For standing waves on a string, the frequency is determined by the tension and mass per unit length of the string, which affects the wavelengths that can exist as standing waves.
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Harmonics

Harmonics refer to the specific frequencies at which standing waves can form on a string fixed at both ends. The fundamental frequency (first harmonic) corresponds to the longest wavelength, while higher harmonics (second, third, etc.) have shorter wavelengths. The wavelengths of these harmonics can be calculated using the formula λ_n = 2L/n, where L is the length of the string and n is the harmonic number.
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