Wave Problem (constructive interference)

[user1252452035]

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