Final answer:
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.