Two in-phase loudspeakers are placed along a wall and are separated by a distance of 4.00 m. They emit sound with a frequency of 514 Hz. A person is standing away from the wall, in front of
One of the loudspeakers. What is the closest nonzero distance from the wall the person can stand
And hear constructive interference? The speed of sound in air is 343 m/s.
A) 1.2 m
B) 1.6 m
C) 0.73 m
D) 0.34 m

Question

Quizplus · Accepted Answer

The wavelength ($\lambda$) of the sound can be calculated using the formula $\lambda = v/f$, where $v$ is the speed of sound in air (343 m/s) and $f$ is the frequency (514 Hz). This gives $\lambda = 343 / 514 = 0.667 m$. For constructive interference, the path difference between the two speakers to the point of observation should be an integer multiple of the wavelength. However, since we're looking for the closest nonzero distance, we consider the first instance of constructive interference, which occurs at half the wavelength ($\lambda/2$). The geometry of the situation forms a right triangle, with the distance between the speakers as the base (4.00 m) and the distance from the person to the wall as the height. Using the Pythagorean theorem and considering the path difference as $\lambda/2 = 0.667/2 = 0.3335 m$, we find the height of the triangle. However, the calculation provided directly for the closest distance isn't straightforward from the given information without considering the specifics of the interference pattern and the geometry involved. The correct approach involves understanding that the path difference for the first constructive interference (not directly at the speaker) involves a more complex calculation involving the phase difference and the specific positioning of the speakers and the listener. Given the options and the typical approach to such a problem, the closest distance for constructive interference would be based on the geometry and the specific interference conditions, which might involve calculations not directly outlined in the response. The correct answer, considering typical interference patterns and distances involved, would be based on the calculation of the first point of constructive interference from the given setup.

Two In-Phase Loudspeakers Are Placed Along a Wall and Are