Constructive Interference of Sound

In summary: The speakers 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. If you walk once around the complete circle, how many intensity maxima do you hear? In summary, you hear 6 intensity maxima.
  • #1
lafalfa
15
0
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!
 
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  • #2
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?
 
  • #3
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.
 
  • #4
Perhaps I have the entire concept wrong and there's some other mysterious formula that I am supposed to use...?
 
  • #5
Are the two speakers being driven in-phase or out-of-phase (polarity reversed)?
Bob S
 
  • #6
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.
 
  • #7
Maybe there are 4 points corresponding to m=1? By circular symmetry?
 
  • #8
How do I find the four points?
 
  • #9
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?
 
  • #10
Yes, and the points where m=2 are at (5,0) and (-5,0)?
 
  • #11
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?
 
  • #12
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.
 
  • #13
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.
 
  • #14
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.
 
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  • #15
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.
 
  • #16
lafalfa said:
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?"
 
  • #17
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.
 
  • #18
atyy said:
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 said:
yes, and the points where m=2 are at (5,0) and (-5,0)?

lafalfa said:
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=?
 
  • #19
Eight! I forgot to add those initial 4 points. Haha thanks so much for helping me out and for the quick replies!
 
  • #20
lafalfa said:
Eight! I forgot to add those initial 4 points. Haha thanks so much for helping me out and for the quick replies!

:smile: 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.
 
  • #21
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!
 

Related to Constructive Interference of Sound

1. What is constructive interference of sound?

Constructive interference of sound is when two or more sound waves combine and result in a higher amplitude or louder sound. This occurs when the crests and troughs of the waves align with each other, resulting in a stronger and more coherent sound.

2. How does constructive interference of sound occur?

Constructive interference of sound occurs when two or more sound waves with the same frequency and amplitude overlap in space and time. As the waves overlap, their amplitudes add together, resulting in a larger amplitude and thus a louder sound.

3. What is the difference between constructive and destructive interference of sound?

The main difference between constructive and destructive interference of sound is the resulting amplitude. In constructive interference, the amplitudes add together, resulting in a louder sound. In contrast, in destructive interference, the amplitudes cancel each other out, resulting in a quieter sound.

4. Can constructive interference of sound occur with more than two waves?

Yes, constructive interference of sound can occur with any number of waves as long as they have the same frequency and amplitude and overlap in space and time. The resulting amplitude will be the sum of all the individual amplitudes.

5. What are some practical applications of constructive interference of sound?

Constructive interference of sound is used in various technologies, such as noise-cancelling headphones and ultrasound imaging. It is also utilized in musical instruments, where multiple sound waves are combined to create a richer and fuller sound. Additionally, constructive interference of sound is important in acoustics and sound engineering for optimizing sound quality and amplification.

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