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 61c

Chapter 17, Problem 61c

Two loudspeakers emit sound waves of the same frequency along the x-axis. The amplitude of each wave is a. The sound intensity is minimum when speaker 2 is 10 cm behind speaker 1. The intensity increases as speaker 2 is moved forward and first reaches maximum, with amplitude 2a, when it is 30 cm in front of speaker 1. What is The amplitude of the sound (as a multiple of a) if the speakers are placed side by side?

Verified step by step guidance

Step 1: Understand the concept of interference. When two sound waves of the same frequency overlap, their amplitudes combine. Constructive interference occurs when the waves are in phase, leading to maximum amplitude, while destructive interference occurs when the waves are out of phase, leading to minimum amplitude.

Step 2: Analyze the given information. The sound intensity is minimum when speaker 2 is 10 cm behind speaker 1, indicating destructive interference. This means the path difference corresponds to half a wavelength (λ/2). The intensity is maximum when speaker 2 is 30 cm in front of speaker 1, indicating constructive interference, where the path difference corresponds to a full wavelength (λ).

Step 3: Calculate the wavelength (λ) of the sound waves. The difference between the positions of destructive and constructive interference is 40 cm (30 cm in front minus 10 cm behind). Since this corresponds to half a wavelength (λ/2 to λ), the wavelength is λ = 40 cm.

Step 4: Consider the scenario where the speakers are placed side by side. In this case, the path difference is zero, meaning the waves are perfectly in phase. Constructive interference occurs, and the amplitudes of the waves add together.

Step 5: Determine the amplitude of the sound when the speakers are side by side. Since the amplitude of each wave is 'a' and they are in phase, the total amplitude is the sum of the individual amplitudes: 2a.

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Key Concepts

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

Interference of Waves

Interference occurs when two or more waves overlap, resulting in a new wave pattern. Constructive interference happens when waves are in phase, leading to increased amplitude, while destructive interference occurs when waves are out of phase, reducing amplitude. The position of the speakers affects the phase relationship of the emitted sound waves, which is crucial for understanding the intensity variations described in the question.

Amplitude and Intensity Relationship

Amplitude refers to the maximum displacement of a wave from its rest position, while intensity is the power per unit area carried by the wave. The intensity of sound is proportional to the square of its amplitude. Therefore, when the amplitude of the sound waves from the speakers changes due to their positions, the resulting intensity also changes, which is key to solving the problem presented.

Phase Difference

Phase difference is the difference in the phase of two waves at a given point in time. It is crucial in determining whether the interference is constructive or destructive. In this scenario, the phase difference between the sound waves from the two speakers changes as their positions are altered, affecting the resultant amplitude and intensity of the sound when they are side by side.

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