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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 71b
Capitolo 17, Problema 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?

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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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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.
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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.
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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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