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Ch 35: Interference
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 35: InterferenceProblem 5a
Chapter 35, Problem 5a

Two speakers, emitting identical sound waves of wavelength 2.0 m in phase with each other, and an observer are located as shown in Fig. E35.5. At the observer's location, what is the path difference for waves from the two speakers?
Illustration of two sound sources 6m apart with an observer 8m away, depicting wave interference patterns.

Verified step by step guidance
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Step 1: Identify the positions of the speakers and the observer. From the diagram, the two speakers are separated by a horizontal distance of 6.0 m, and the observer is located 8.0 m vertically above the midpoint between the speakers.
Step 2: Calculate the distance from the observer to each speaker. Use the Pythagorean theorem to find the distance from the observer to each speaker. For the left speaker, the horizontal distance is 3.0 m (half of 6.0 m), and the vertical distance is 8.0 m. Similarly, for the right speaker, the horizontal distance is also 3.0 m, and the vertical distance is 8.0 m.
Step 3: Apply the Pythagorean theorem to calculate the distances. For the left speaker, the distance is \( \sqrt{(3.0 \text{ m})^2 + (8.0 \text{ m})^2} \). For the right speaker, the distance is \( \sqrt{(3.0 \text{ m})^2 + (8.0 \text{ m})^2} \).
Step 4: Determine the path difference. The path difference is the absolute difference between the distances from the observer to each speaker. Since the distances are symmetric, the path difference will be zero.
Step 5: Relate the path difference to the wavelength. The wavelength of the sound waves is given as 2.0 m. Since the path difference is zero, the waves from both speakers will arrive at the observer in phase, resulting in constructive interference.

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Key Concepts

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

Path Difference

Path difference refers to the difference in distance traveled by two waves from their respective sources to a common point, such as an observer. In the context of wave interference, this difference determines whether the waves will constructively or destructively interfere, affecting the resultant amplitude of the sound heard by the observer.
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Wave Interference

Wave interference occurs when two or more waves overlap and combine to form a new wave pattern. This can result in constructive interference, where wave amplitudes add together, or destructive interference, where they cancel each other out. The type of interference depends on the path difference between the waves at the observer's location.
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Wavelength

Wavelength is the distance between successive crests (or troughs) of a wave, which in this case is given as 2.0 m for the sound waves emitted by the speakers. The wavelength is crucial for determining the conditions for constructive and destructive interference, as it relates directly to the path difference needed for the waves to be in phase or out of phase at the observer's position.
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Related Practice
Textbook Question

Two slits spaced 0.450 mm apart are placed 75.0 cm from a screen. What is the distance between the second and third dark lines of the interference pattern on the screen when the slits are illuminated with coherent light with a wavelength of 500 nm?

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Textbook Question

Coherent light with wavelength 450 nm falls on a pair of slits. On a screen 1.80 m away, the distance between dark fringes is 3.90 mm. What is the slit separation?

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Textbook Question

Coherent light of frequency 6.32 × 1014 Hz passes through two thin slits and falls on a screen 85.0 cm away. You observe that the third bright fringe occurs at ±3.11 cm on either side of the central bright fringe. (a) How far apart are the two slits? (b) At what distance from the central bright fringe will the third dark fringe occur?

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Textbook Question

Two speakers that are 15.0 m apart produce in-phase sound waves of frequency 250.0 Hz in a room where the speed of sound is 340.0 m/s. A woman starts out at the midpoint between the two speakers. The room's walls and ceiling are covered with absorbers to eliminate reflections, and she listens with only one ear for best precision. How far from the center must she walk before she first hears the sound maximally enhanced?

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Textbook Question

Two radio antennas A and B radiate in phase. Antenna B is 120 m to the right of antenna A. Consider point Q along the extension of the line connecting the antennas, a horizontal distance of 40 m to the right of antenna B. The frequency, and hence the wavelength, of the emitted waves can be varied. What is the longest wavelength for which there will be destructive interference at point Q?

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Textbook Question

Two radio antennas A and B radiate in phase. Antenna B is 120 m to the right of antenna A. Consider point Q along the extension of the line connecting the antennas, a horizontal distance of 40 m to the right of antenna B. The frequency, and hence the wavelength, of the emitted waves can be varied. What is the longest wavelength for which there will be constructive interference at point Q?

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