Two adjacent sources each emit a frequency of 800 Hz in air where the velocity of sound is 340 m/s. How much farther back would source 1 have to be moved so an observer in front of the sources would hear no sound?

Question

BlackTom AI · Accepted Answer

For destructive interference the path difference between the two waves must equal (m + 1/2)λ (odd half-wavelengths), so the smallest backward shift that produces no sound is Δ = λ/2.

1) Relevant concepts
- Wave speed v, frequency f and wavelength λ are related by λ = v / f.
- Two coherent sources produce destructive interference at an observer when the difference in path lengths is (m + 1/2)λ (m an integer). The minimum nonzero shift corresponds to m = 0 → λ/2.

2) Step-by-step reasoning
- Compute the wavelength from the given speed and frequency.
- Determine the half-wavelength (λ/2) because shifting one source back by this amount produces the smallest path difference giving destructive interference.

3) Calculations (all intermediate steps)
- λ = v / f = 340 m/s ÷ 800 Hz = 0.425 m
- Minimum backward shift Δ = λ/2 = 0.425 m ÷ 2 = 0.2125 m ≈ 0.213 m

4) Why this is correct
- Initially the two adjacent sources are effectively at the same distance from the observer (path difference ≈ 0) and thus add constructively. Moving source 1 back by λ/2 introduces a path difference of half a wavelength so the two waves arrive out of phase by 180°, cancelling one another at the observer (destructive interference).

5) Why the other options are incorrect
- 0.123 m: not equal to λ/2 (0.2125 m); it does not produce the required half-wavelength path difference.
- 0.321 m and 0.312 m: neither equals λ/2 nor any odd multiple of λ/2 for m integer (they correspond to ≈0.75λ and ≈0.73λ respectively), so they will not produce complete cancellation.

6) Conclusion
- The correct minimum backward shift is 0.213 m, matching the provided answer.

Two adjacent sources each emit a frequency of 800 Hz in air where the velocity of sound is 340 m/s. How much farther back would source 1 have to be moved so an observer in front of the sources would hear no sound?单项选择题

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