Constructive Interference of Sound

[lafalfa]

Hi,

Two speakers, A and B, are located at x = +0.5 m and x = -0.5 m. A 680 Hz signal is sent to both speakers. You then walk around the origin, x = 0, in a circle of radius 5.0 m. v_sound = 340 m/s

If you walk once around the complete circle, how many intensity maxima do you hear?

This is what I did: m*wavelength = path difference
Since the maximum path difference is 1.0 m and the wavelength is 0.5 metres, m can only be 0 or 1 or 2. This corresponds to 6 points on the circle (2 points per value of m).
Is this the correct way of doing this question?

The correct answer is 8, not 6, and I don't know where the extra 2 points come from!

Please help me. Thank you very much!

 

[SystemTheory]

If you are walking around a circle at 5m doesn't the path to each source depend on sine and cosine relationships, or some other variable geometry? In other words, I am asking, is the path difference to be considered as the distance of the listener to each source, and not the distance between the two sources?

 

[lafalfa]

The path difference is supposed to be the difference in distance that sound waves from each source must travel to get to you. So, for example, if you are 3 metres from a certain source and there is another source 1 metre directly behind the first source, the path difference would be 4-3=1 metre. I hope that makes sense.
I know for sure that the maximum path difference is 1 m because I had to determine that in another part of the question and the answer key agrees with me on that part.

 

[lafalfa]

Perhaps I have the entire concept wrong and there's some other mysterious formula that I am supposed to use...?

 

[Bob S]

Are the two speakers being driven in-phase or out-of-phase (polarity reversed)?
Bob S

 

[lafalfa]

I think we're supposed to assume that they're emitting sound waves in phase and that the only phase difference arises from the path difference.

 

[atyy]

Maybe there are 4 points corresponding to m=1? By circular symmetry?

 

[lafalfa]

How do I find the four points?

 

[atyy]

Draw the speakers and the circle. Draw the points where m=0 (no path difference), and the points where m=2 (maximal path difference). Presumably the m=1 points are between those.

To start, let's use x and y axes. I would guess that the two m=0 points are at (x=0, y=5) and (x=0, y=-5). Do you agree?

 

[lafalfa]

Yes, and the points where m=2 are at (5,0) and (-5,0)?

 

[atyy]

Yes. So if you try to place only two m=1 points between your m=0 and m=2 points, will you be able to respect the symmetry of the situation?

 

[atyy]

A bit more explicit, assume that an m=1 point lies on the circle between an m=0 and an m=2 point. Show by symmetry that there are 3 other points with the same path difference.

 

[lafalfa]

I see that by fitting two m=1 points between each m=0 and m=2 points, I would get a total of 8 intensity maxima positions, but I don't understand how we can assume that there are TWO instead of one m=1 points between an m=0 and an m=2 point.

 

[denverdoc]

Maybe I'm missing something, but I don't think that is necessary.

Start the circle on the positive x axis, speakers are situated at -0.5 and +0.5 also on the x-axis so you are in an m=2 mode. As I walk around the circle clockwise there is an m=0 maxima at 90 degrees. Presumably I have walked thruough an m=1 max on the way,. If you continue the circle tour there will be 7 maxima before retrurning to the starting point which will be eight.

 

[rl.bhat]

If O is the center of AB, OY the axis and P is the position of the observer, four position of P for m = 1 makes angles pi/4, 3pi/4, 5pi/4 and 7pi/4 with respect to Y-axis. So you get four points for maximum.

 

[atyy]

I see that by fitting two m=1 points between each m=0 and m=2 points, I would get a total of 8 intensity maxima positions, but I don't understand how we can assume that there are TWO instead of one m=1 points between an m=0 and an m=2 point.

It's a good question how many m=1 points lie between each pair of m=0 and m=2 points. But start by assuming that there is only one m=1 point between each pair, and ask, "how many quadrants to a circle are there?"

 

[lafalfa]

There are four quadrants in a circle, so I would say there should only be four m=1 points, that is, one m=1 point between each m=0 and m=2 points. I don't understand why there should be TWO m=1 points in each of the four quadrants of the circle.

 

[atyy]

draw the speakers and the circle. Draw the points where m=0 (no path difference), and the points where m=2 (maximal path difference). Presumably the m=1 points are between those.

To start, let's use x and y axes. I would guess that the two m=0 points are at (x=0, y=5) and (x=0, y=-5). Do you agree?

yes, and the points where m=2 are at (5,0) and (-5,0)?

there are four quadrants in a circle, so i would say there should only be four m=1 points, that is, one m=1 point between each m=0 and m=2 points. I don't understand why there should be two m=1 points in each of the four quadrants of the circle.

2+2+4=?

 

[lafalfa]

Eight! I forgot to add those initial 4 points. Haha thanks so much for helping me out and for the quick replies!

 

[atyy]

Eight! I forgot to add those initial 4 points. Haha thanks so much for helping me out and for the quick replies!

Actually, to be more careful, we should show using trigonometry and algebra that there is only one m=1 point between each pair of m=0 and m=2 points. But perhaps that can be left for another time.

 

[lafalfa]

I don't think they'll expect me to do that (hopefully) since they just say to "indicate the approximate position on the circle [which they have drawn] with the letter I." So, I should be okay. Thanks!