Two speakers, both connected to the same audio source, are positioned 8 meters apart and face each other.A student is standing between the speakers, 3 meters away from one of them.Given that the wavelength of the sound is 0.4 meters, what is the path difference for the sound waves reaching the student from each speaker, and what type of interference does the student experience?

Question

BlackTom AI · Accepted Answer

1. Relevant knowledge points and concepts
- Path difference is the difference in distances from the two coherent sources (speakers) to the observer.
- Interference depends on the path difference Δ relative to the wavelength λ: constructive interference occurs when Δ = nλ (n an integer), destructive when Δ = (n + 1/2)λ.

2. Step-by-step reasoning and calculations
- Let speaker A be the one 3.0 m from the student. The speakers are 8.0 m apart, so the distance from the student to speaker B is 8.0 m − 3.0 m = 5.0 m.
- Path difference Δ = distance to B − distance to A = 5.0 m − 3.0 m = 2.0 m.
- Given wavelength λ = 0.4 m, compute Δ in units of λ: Δ / λ = 2.0 m / 0.4 m = 5.0.

3. Intermediate steps shown
- Distances: dA = 3.0 m, dB = 8.0 − 3.0 = 5.0 m
- Δ = dB − dA = 5.0 − 3.0 = 2.0 m
- Δ/λ = 2.0 / 0.4 = 5 → Δ = 5λ

4. Why the correct answer is correct
- Since Δ = 5λ and 5 is an integer (n = 5), the condition Δ = nλ is satisfied. That means the two waves arrive in phase and reinforce each other, producing constructive interference.

5. Why the other options are incorrect
- Options claiming a path difference of 5 m are wrong because they confuse the distance from one speaker (5 m) with the path difference; the path difference is 5 − 3 = 2 m.
- The option stating 2 m but destructive interference is incorrect because 2.0 m equals 5λ (an integer multiple), not a half-integer multiple like (n + 1/2)λ; destructive interference would require Δ = (n + 1/2)λ (e.g., 0.2 m, 0.6 m, ...), not 2.0 m.

Conclusion: The path difference is 2 meters and the student experiences constructive interference, matching the provided answer.

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