Wave Problem (constructive interference)

[tubaplaya76]

Hey everyone,

here's a problem that's been troubling me all day and I really have no idea what else to do:

"Speakers A and B are vibrating in phase. They are directly facing each other, are 8.0 m apart, and are each playing a 76.0-Hz tone. The speed of sound is 343 m/s. On the line between the speakers there are three points where constructive interference occurs. What are the distances of these three points from speaker A?"

I've found the wavelength (using v=343 and f=76) to be 4.513.

I then set up three equations, that I think in theory will give me the right answer:
(w = wavelength)
(l = length (8m))

x=w
l-x=3w

y=2w
l-y=2w

z=3w
l-z=w

However, when I submitted this each answer online, I got an incorrect answer. Any help would be appreciated. thanks in advance,

Sam

 

[nrqed]

Hey everyone,

here's a problem that's been troubling me all day and I really have no idea what else to do:

"Speakers A and B are vibrating in phase. They are directly facing each other, are 8.0 m apart, and are each playing a 76.0-Hz tone. The speed of sound is 343 m/s. On the line between the speakers there are three points where constructive interference occurs. What are the distances of these three points from speaker A?"

I've found the wavelength (using v=343 and f=76) to be 4.513.

I then set up three equations, that I think in theory will give me the right answer:
(w = wavelength)
(l = length (8m))

x=w
l-x=3w

...

There is no reason to impose that you are at a distance w from one speaker! You could be at 1.3 w, or 0.668w or any other distance!
(btw, notice that your solution would work only if l = 4 w which is not the case here).
What you must do is to impose that the difference of distance traveled by both waves is either 0 or w or 2w and so on.

Call x the distance from the first speaker. Then l-x is the distance from the second speaker (as you already had). Then impose that the difference between the two distances is 0 or w or 2 w, etc.

so

l-x - x = 0, w, 2w , ect.

The first possibility gives you the obvious solution: right in the middle. Then you will get the others. Of course, no solution is possible when n w gets above l, which happens for the first time when n =3. so l- 2x = 3 w has no solution in your case. This shows that there are only 3 solutions (corresponding to 0, w and 2w)

Pat

 

[kyle8921]

Dear Sam,

Thank you for sharing your problem with us. From your calculations, it seems like you have a good understanding of the concept of constructive interference. However, there are a few things that might have caused your incorrect answer.

Firstly, in your equations, you have used the wavelength (w) as the distance between the speakers, but it should actually be the distance between the speakers divided by the number of wavelengths between them. So for example, for the first equation, it should be x = 8m/1w = 8m/4.513 = 1.772m. This should give you a more accurate answer.

Secondly, it is important to note that the distance between the speakers is 8.0 m, but the distance between the speakers and the three points of constructive interference will be different. This is because the sound waves travel at different distances from the speakers before they reach these points. So, for the first equation, you should have x = 8m - 1.772m = 6.228m. Similarly, for the other two equations, you should subtract the distance traveled by the sound waves from the total distance of 8m.

I hope this helps you solve the problem. If you have any further questions, please feel free to reach out. Keep up the good work!

Best,

Scientist