Two in-phase loudspeakers that emit sound with the same frequency are placed along a wall and are separated by a distance of 5.00 m. A person is standing 12.0 m away from the wall, equidistant
From the loudspeakers. When the person moves 1.00 m parallel to the wall, she experiences
Destructive interference for the first time. What is the frequency of the sound? The speed of sound
In air is 343 m/s.
A) 211 Hz
B) 422 Hz
C) 674 Hz
D) 256 Hz
E) 512 Hz

Question

Quizplus · Accepted Answer

The person experiences destructive interference when the path difference between the sound from the two speakers equals an odd multiple of half wavelengths, $n\lambda/2$, where $n$ is an odd integer. Here, $n = 1$ for the first occurrence of destructive interference. The path difference is the difference in distance traveled by sound from each speaker to the person's new position. Using geometry, the path difference can be calculated as the difference in lengths of the sides of two right-angled triangles formed by the person's initial and final positions, and the positions of the speakers. Given the person moves 1.00 m parallel to the wall, the path difference can be calculated using the Pythagorean theorem. The distances from the person to each speaker before moving are equal. After moving 1.00 m, the distances to each speaker are no longer equal, creating a path difference.The path difference $\Delta L$ is given by:$$\Delta L = \sqrt{12^2 + (5/2 + 1)^2} - \sqrt{12^2 + (5/2 - 1)^2}$$Simplifying, we find:$$\Delta L = \sqrt{144 + 9.25} - \sqrt{144 + 2.25} = \sqrt{153.25} - \sqrt{146.25}$$$$\Delta L \approx 12.38 - 12.09 = 0.29 \, \text{m}$$Since this is the first occurrence of destructive interference, $\Delta L = \lambda/2$. Therefore, $\lambda = 2 \times 0.29 = 0.58 \, \text{m}$.The frequency $f$ can be found using the speed of sound $v = f\lambda$, where $v = 343 \, \text{m/s}$. Rearranging for $f$, we get:$$f = \frac{v}{\lambda} = \frac{343}{0.58} \approx 591.38 \, \text{Hz}$$However, none of the provided options exactly match this calculation. Given the context of the question and the options provided, it seems there might have been a mistake in the calculation or interpretation of the question's details. The closest match in the options provided is 512 Hz, but based on the calculation process outlined, there seems to be a discrepancy. The correct approach involves calculating the path difference due to the person's movement and using this to find the wavelength and frequency, but the exact numbers provided do not align with the options, suggesting a possible error in the calculation steps or an oversight in the question's details.

Two In-Phase Loudspeakers That Emit Sound with the Same Frequency