Question

Two side-by-side loudspeakers at the origin emit 684 Hz sound waves on a day when the speed of sound is 342 m/s. A microphone 1.00 m away on the x-axis detects a maximum in the sound intensity. Then one of the speakers is moved slowly along the y-axis.

How far does it move before the microphone first detects a minimum in the sound intensity?
Express your answer with the appropriate units.

Answer 1

The minimum distance between the two speakers is 0.34 meters.

Step-by-step explanation:

The distance between two speakers producing sounds that arrive at the ear noticeably different times can be calculated using the formula:

Δd = vΔt,

where Δd is the minimum distance between the speakers, v is the speed of sound, and Δt is the maximum time difference noticeable by the ear. In this case, Δt is 1.00 ms (0.001 s) and v is 340 m/s. Plugging in the values, we get:

Δd = (340 m/s)(0.001 s) = 0.34 m.

Therefore, the minimum distance between the two speakers is 0.34 meters.

Answer 2

The speaker needs to move 0.5 m before the microphone detects a minimum in sound intensity.

Step-by-step explanation:

When two sound waves from two speakers interfere with each other, they can create areas of constructive interference, where the sound is louder, and areas of destructive interference, where the sound is quieter. In this case, the microphone detects a maximum sound intensity when the waves combine constructively. When the speakers move, the distance between them changes, causing the interference pattern to shift. To find the distance the speaker moves before the microphone detects a minimum in sound intensity, we can use the concept of path difference.

Path difference is the difference in distance travelled by waves from two sources. For constructive interference, the path difference is a whole number of wavelengths. For destructive interference, the path difference is a whole number of wavelengths plus half a wavelength.

In this scenario, the microphone is located 1.00 m away from the speakers, so the path difference for constructive interference would be an even number of wavelengths. The wavelength of the sound wave can be found using the formula:

wavelength = speed of sound/frequency

Given that the frequency is 684 Hz and the speed of sound is 342 m/s, the wavelength would be 342/684 = 0.5 m.

To find the distance the speaker moves before the microphone detects a minimum in sound intensity, we need to determine the path difference for destructive interference. This can be achieved by introducing a shift of half a wavelength to the path difference for constructive interference. Therefore, the path difference for destructive interference would be an odd number of wavelengths plus half a wavelength.

Since the microphone is 1.00 m away from the speakers, the speaker would need to move a distance of 0.5 m (half the wavelength) to introduce a path difference of 1.5 wavelengths (1.0 + 0.5). Hence, the speaker moves 0.5 m before the microphone detects a minimum in sound intensity.