What Distance Should the Microphone Move Toward Speaker B for Minimum Intensity?

[nahanksh]

Homework Statement

Two audio speakers are facing each other, as shown in the picture:
https://online-s.physics.uiuc.edu/cgi/courses/shell/common/showme.pl?courses/phys214/spring10/hwb/01/05/speakers.gif

The speakers emit sound at 1300 Hz; initially the speakers emit in phase (φB = 0). Assume the speed of sound is 330 m/s. A special microphone that is in the middle, i.e, equidistant from the speakers, responds to the net sound intensity from the speakers. If only the left speaker (A) is on, the microphone registers an intensity IA = 2 Watts/m2. If only the right speaker (B) is on, the microphone registers an intensity IB = 4 Watts/m2.

Now we turn down the right speaker, so that both speakers individually produce intensity IA = IB = 2 Watts/m2. Also, assume the frequency is still 1200 Hz, and φB = 0. How far would you have to move the microphone toward speaker B to have a minimum in intensity?

Homework Equations

The Attempt at a Solution

I think sketching Imic(x), the net intensity as a function of distance from the original (central) location would help me a lot here to solve this. But as having the phase as 'zero', i have no idea how to proceed further..

Please could someone help me out here?

 

[anathema]

Hello! I would suggest using the concept of interference to solve this problem. Interference occurs when two or more waves overlap, resulting in either constructive or destructive interference depending on the phase difference between the waves.

In this scenario, we have two speakers emitting sound waves with a frequency of 1300 Hz and a speed of 330 m/s. The microphone is equidistant from the two speakers, so the waves from each speaker will reach the microphone at the same time. When the left speaker is on, the microphone registers an intensity of 2 Watts/m2. When the right speaker is on, the microphone registers an intensity of 4 Watts/m2.

When both speakers are on, the net intensity at the microphone will depend on the phase difference between the waves from the two speakers. In this case, the phase difference is 0, since both speakers are initially emitting in phase (φB = 0). This means that the waves will interfere constructively, resulting in a net intensity of 4 Watts/m2 (2 + 2 = 4).

Now, when we turn down the right speaker so that both speakers individually produce an intensity of 2 Watts/m2, the net intensity at the microphone will depend on the phase difference between the waves from the two speakers. In this case, the phase difference is still 0, since φB is still 0. However, the intensity from the right speaker is now half of what it was before, which means that the net intensity at the microphone will also be half of what it was before (2 + 1 = 3).

To find the minimum intensity, we need to find the point where the waves from the two speakers interfere destructively, resulting in a net intensity of 0. This occurs when the phase difference between the waves is π radians (180 degrees). So, we need to find the distance x from the original (central) location where the path difference between the waves is equal to half a wavelength (λ/2).

The path difference between the waves can be calculated using the formula Δx = λ/2 * sinθ, where λ is the wavelength and θ is the angle between the two waves. Since the frequency is still 1300 Hz, the wavelength is equal to 330 m/s divided by 1300 Hz, which is equal to 0.2538 m.

So, Δx = 0.2538 m/2 * sin

 

[kerimek]

I would approach this problem by first understanding the concept of interference of waves. When two waves of the same frequency and amplitude meet, they can either constructively interfere (increase in amplitude) or destructively interfere (cancel each other out). In this scenario, we have two audio speakers emitting sound waves at 1300 Hz, and we need to find the minimum intensity at a specific distance from the speakers.

To solve this problem, we need to consider the superposition of the sound waves from the two speakers. When the two speakers are emitting in phase (φB = 0), the sound waves will constructively interfere at the central location (where the microphone is placed). This results in an intensity of 2 Watts/m2. However, when we turn down the right speaker, the sound waves from the two speakers will no longer be in phase at the central location. This will result in interference patterns, where some points will experience constructive interference and others will experience destructive interference.

To find the minimum intensity, we need to find the point where the sound waves from the two speakers are exactly out of phase (φB = π). At this point, the waves will cancel each other out, resulting in a minimum intensity. To find this point, we can use the equation for the path difference between the two sound waves:

Δx = λ/2 = (v/f)/2 = (330 m/s)/(1300 Hz)/2 = 0.063 m

Therefore, the microphone needs to be moved 0.063 m towards speaker B to experience a minimum intensity. This is because at this distance, the path difference between the two sound waves will be half of the wavelength, resulting in destructive interference.

In summary, by understanding the concept of interference of waves and using the equation for path difference, we can determine the minimum intensity at a specific distance from the speakers. This approach can be applied to other scenarios involving interference of waves.