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 71b

Chapter 17, Problem 71b

The three identical loudspeakers in FIGURE P17.71 play a 170 Hz tone in a room where the speed of sound is 340 m/s. You are standing 4.0 m in front of the middle speaker. At this point, the amplitude of the wave from each speaker is a. How far must speaker 2 be moved to the left to produce a maximum amplitude at the point where you are standing?

Verified step by step guidance

Step 1: Calculate the wavelength of the sound wave using the formula \( \lambda = \frac{v}{f} \), where \( v \) is the speed of sound (340 m/s) and \( f \) is the frequency of the sound (170 Hz).

Step 2: Determine the condition for constructive interference, which occurs when the path difference between the waves from the speakers is an integer multiple of the wavelength \( \lambda \). This ensures maximum amplitude at the point where you are standing.

Step 3: Analyze the geometry of the setup. Speaker 2 must be moved to the left such that the path difference between the waves from speaker 2 and the middle speaker matches the condition for constructive interference.

Step 4: Express the path difference as \( \Delta x = n \lambda \), where \( n \) is an integer (e.g., \( n = 1 \) for the first maximum). Substitute the value of \( \lambda \) calculated in Step 1 to find the distance \( \Delta x \).

Step 5: Conclude that the distance speaker 2 must be moved to the left is equal to \( \Delta x \). This ensures the waves from speaker 2 and the middle speaker constructively interfere at the point where you are standing.

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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, amplifying the resultant amplitude, while destructive interference occurs when waves are out of phase, reducing the amplitude. Understanding this concept is crucial for determining how the sound waves from the speakers interact at the listener's position.

Path Difference

Path difference refers to the difference in distance traveled by two waves from their sources to a common point. For constructive interference to occur, the path difference must be an integer multiple of the wavelength. In this scenario, calculating the path difference between the waves from the speakers will help determine how far speaker 2 needs to be moved to achieve maximum amplitude at the listener's position.

Wavelength and Frequency Relationship

The wavelength of a sound wave is inversely related to its frequency, as described by the equation v = fλ, where v is the speed of sound, f is the frequency, and λ is the wavelength. For the given frequency of 170 Hz and speed of sound at 340 m/s, the wavelength can be calculated. This relationship is essential for understanding how the distance between the speakers affects the interference pattern at the listener's location.

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Textbook Question

Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have a frequency of 440 Hz and the note E should be at 659 Hz. What is the frequency difference between the third harmonic of the A and the second harmonic of the E?

Textbook Question

The three identical loudspeakers in FIGURE P17.71 play a 170 Hz tone in a room where the speed of sound is 340 m/s. You are standing 4.0 m in front of the middle speaker. At this point, the amplitude of the wave from each speaker is a. When the amplitude is maximum, by what factor is the sound intensity greater than the sound intensity from a single speaker?

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Textbook Question

Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have a frequency of 440 Hz and the note E should be at 659 Hz. The tuner starts with the tension in the E string a little low, then tightens it. What is the frequency of the E string when she hears four beats per second?

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Textbook Question

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Textbook Question

Engineers are testing a new thin-film coating whose index of refraction is less than that of glass. They deposit a 560-nm-thick layer on glass, then shine lasers on it. A red laser with a wavelength of 640 nm has no reflection at all, but a violet laser with a wavelength of 400 nm has a maximum reflection. How the coating behaves at other wavelengths is unknown. What is the coating’s index of refraction?

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Scientists are testing a transparent material whose index of refraction for visible light varies with wavelength as n = 30.0 nm^1/2/λ^1/2 , where λ is in nm. If a 295-nm-thick coating is placed on glass (n=1.50), for what visible wavelengths will the reflected light have maximum constructive interference?

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