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

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.