A person stands between two speakers driven by the same source. Each speaker produces a tone with a frequency of 200 Hz on a day when the speed of sound is 330 m/s. The person is 1.65 m from one speaker and 4.95 m from the other. What type of interference does the person sense?

Question

BlackTom AI · Accepted Answer

Constructive interference occurs when the path difference between two waves equals an integer multiple of the wavelength: Δ = nλ (n = 0, 1, 2, ...). Destructive interference occurs when Δ = (n + 1/2)λ.

1. Relevant knowledge points and concepts
- Superposition principle: two waves from the speakers add; the listener senses interference determined by their relative phase.
- Wavelength formula: λ = v / f, where v is sound speed and f is frequency.
- Path difference Δ = |r2 − r1| determines the phase difference at the listener: if Δ is an integer multiple of λ, the waves arrive in phase (constructive); if Δ is a half-integer multiple, they arrive out of phase (destructive).

2. Complete step-by-step reasoning and calculations
- Given: f = 200 Hz, v = 330 m/s, distances r1 = 1.65 m and r2 = 4.95 m.
- Compute wavelength: λ = v / f = 330 m/s ÷ 200 s⁻¹ = 1.65 m.
- Compute path difference: Δ = |r2 − r1| = |4.95 m − 1.65 m| = 3.30 m.
- Compute Δ relative to λ: Δ / λ = 3.30 m ÷ 1.65 m = 2.0.

3. Intermediate steps shown
- λ = 330/200 = 1.65 m
- Δ = 4.95 − 1.65 = 3.30 m
- n = Δ/λ = 3.30/1.65 = 2 (an integer)

4. Why the correct answer is correct
Because the path difference equals exactly 2λ (n = 2), the two waves arrive in phase at the person’s position, reinforcing each other and producing constructive interference (maximum amplitude).

5. Why the other options are incorrect
- Destructive: would require Δ = (n + 1/2)λ (e.g., 0.5λ, 1.5λ, 2.5λ...), which is not the case here.
- Both constructive and destructive: impossible at a single fixed location for a single-frequency, steady source and fixed distances; only one interference type (here constructive) occurs.
- Neither constructive nor destructive: incorrect because the waves do interfere; the condition corresponds to constructive interference.

6. Conclusion
Since Δ = 2λ, the person senses constructive interference. This matches the provided answer: constructive.

A person stands between two speakers driven by the same source. Each speaker produces a tone with a frequency of 200 Hz on a day when the speed of sound is 330 m/s. The person is 1.65 m from one speaker and 4.95 m from the other. What type of interference does the person sense?Single choice

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