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?

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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.