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 8a

Chapter 17, Problem 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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Key Concepts

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

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.

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.

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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Textbook Question

The two highest-pitch strings on a violin are tuned to 440 Hz (the A string) and 659 Hz (the E string). What is the ratio of the mass of the A string to that of the E string? Violin strings are all the same length and under essentially the same tension.

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Textbook Question

FIGURE EX17.7 shows a standing wave on a string that is oscillating at 100 Hz. How many antinodes will there be if the frequency is increased to 200 Hz?

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Textbook Question

Standing waves on a 1.0-m-long string that is fixed at both ends are seen at successive frequencies of 36 Hz and 48 Hz. Draw the standing-wave pattern when the string oscillates at 48 Hz.

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Textbook Question

FIGURE EX17.6 shows a standing wave oscillating at 100 Hz on a string. What is the wave speed?

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A carbon dioxide laser is an infrared laser. A CO₂ laser with a cavity length of 53.00 cm oscillates in the m=100,000 mode. What are the wavelength and frequency of the laser beam?