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 22a

Chapter 17, Problem 22a

Two loudspeakers emit sound waves along the x-axis. The sound has maximum intensity when the speakers are 20 cm apart. The sound intensity decreases as the distance between the speakers is increased, reaching zero at a separation of 60 cm. What is the wavelength of the sound?

Verified step by step guidance

Understand the problem: The sound waves from the two speakers interfere with each other. Maximum intensity occurs when the waves are in phase (constructive interference), and zero intensity occurs when the waves are out of phase (destructive interference). The goal is to find the wavelength of the sound.

Recall the condition for destructive interference: Destructive interference occurs when the path difference between the two waves is an odd multiple of half the wavelength, i.e., \( \Delta x = (m + \frac{1}{2}) \lambda \), where \( m \) is an integer.

Identify the given data: The distance between the speakers changes from 20 cm (maximum intensity) to 60 cm (zero intensity). The change in distance corresponds to a transition from constructive to destructive interference. This means the path difference changes by half a wavelength.

Set up the relationship: The change in distance between the speakers is \( 60 \ \text{cm} - 20 \ \text{cm} = 40 \ \text{cm} \). This change corresponds to half a wavelength, so \( \frac{1}{2} \lambda = 40 \ \text{cm} \).

Solve for the wavelength: Multiply both sides of the equation \( \frac{1}{2} \lambda = 40 \ \text{cm} \) by 2 to find \( \lambda \). The wavelength \( \lambda \) is twice the change in distance.

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Key Concepts

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

Interference of Sound Waves

Interference occurs when two or more sound waves overlap, resulting in a new wave pattern. Constructive interference happens when waves are in phase, amplifying sound intensity, while destructive interference occurs when waves are out of phase, reducing intensity. The distance between the speakers affects the interference pattern, leading to regions of maximum and minimum sound intensity.

Wavelength

Wavelength is the distance between successive crests (or troughs) of a wave. It is a fundamental property of waves, including sound, and is inversely related to frequency. In the context of sound waves from the speakers, the wavelength can be determined by analyzing the distance at which constructive and destructive interference occurs.

Sound Intensity

Sound intensity is the power per unit area carried by a sound wave and is typically measured in watts per square meter. It is influenced by the amplitude of the sound wave and the distance from the source. In this scenario, the intensity varies with the distance between the speakers, indicating how sound waves interact and affect perceived loudness.

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