Sound Wave Interference and Finding the Path Differences with Diagrams

[AN630078]

Homework Statement

Below I have attached a revision question on the wave interference of sound waves from two speakers. I have also attached the original question diagram in addition to my own diagrams to answer the questions. However, I feel this is one of my weaker topics and I am rather uncertain of the credibility of the diagrams I have drawn and my method to find the path difference in question 2. I would very much appreciate any feedback or improvements I could make, especially if tasked with drawing similar diagrams in the future. I am sorry this is rather a long thread, I only grouped the questions together since they are on a similar topic and I felt that any mistakes I may make in my earlier diagrams may continue into my subsequent solutions, so identifying their source may help in improving my approach.


Two speakers are being used to experiment with the superposition of waves, throughout which they are coherent.

Question 1; Initially the speakers are positioned 16 m apart.
a. Taking the speed of sound to be 340 ms-1 and the frequency of the source to be 85 Hz, calculate the wavelength.
b. Make a copy of the diagram and represent the compressions and rarefactions as whole and dashed lines respectively. Remember that the waves will radiate outwards, so you may want to use compasses to draw them. Mark several areas of constructive interference and destructive interference. Make your diagram to scale.

Question 2; Calculate the path difference for the second minimum from the centre. Use your
drawing to check.

Question 3; During the second experiment, the speakers are placed 4 m apart from each other. Now they oriented so that are both pointing in the same direction. The frequency is kept the same.
a. A person stands at point P 10 m from Speaker A and 8 m from Speaker B. Would the person hear a loud or quiet sound? Drawing a diagram may help.
b. If the person were to walk parallel to the speakers while someone increased the frequency, what effect, if any, would this have?

Relevant Equations

λ=v/f

Question 1:
a. λ=v/f
λ= 340/85
λ=4 m
b. Please see attached. Ihave tried to accurately and to scale construct a diagram representing the compressions and rarefactions of the sound waves. Since the wavelength of a wave is simply the length of one complete wave cycle, and I have found that the wavelength here is equal to 4m, I have drawn a compression every 4cm and a rarefaction every 4cm, so they are spaced with 2cm between them as a complete wavelength encompasses a compression and rarefaction, like a peak and trough in a transverse wave.
When two waves superpose in phase, they constructively interfere and produce a larger amplitude wave, which I have marked with dots in my drawing. When two waves superpose that are out of phase, they produce a smaller or zero-amplitude wave, these areas I have marked with crosses.
I have tried not to include too many overlapping compressions and rarefactions and must note that my diagram is already a little rough so I did not want to make it too messy.
I would very much appreciate any suggestions to improve my diagram and any comments as to whether my method of drawing it to scale would be correct.

Question 2:
I am not certain how to accomplish this, where the question states the "second minimum from the centre" does it mean the second trough from the centre, where a trough corresponds to a rarefaction? I have marked where this may be with a purple arrow on my diagram but I am not sure specifically where the question is referring to, which is causing me some confusion.
If this is the correct position, then to calculate the path difference (being the difference in distance traveled by the two waves from their respective sources to a given point on the pattern) would this be the difference between the wavelength of the wave emitted from the LH speaker and the difference between the wavelength of the wave from the RH speaker?
If this is the case, then the wavelength of the LH speaker at the purple arrow is 12cm=12m and the wavelength of the RH speaker is 4cm=4m (which hopefully you can see from my diagram).
Thus, the path difference would be equal to 12cm-4cm =8cm = 8m

Question 3:
a. Please see my diagram attached. I think that a person standing at point P will hear a quiet sound, as the wave from speaker (a compression) superposes the wave from B (a rarefaction) meaning the waves are out of phase, resulting in destructive interference to produce a smaller amplitude wave. If the two waves are exactly out-of-phase, meaning their phase difference is 180 degrees, then the resultant is a zero amplitude wave. Since the amplitude of a sound wave determines its loudness or volume, the decreased amplitude will reduce the noise heard by the person at point P from the speakers.

b.
The frequency is the number of complete wave cycles per second and corresponds to the pitch of the sound one would hear. Therefore, by increasing the frequency of the speakers a person walking parallel to the speakers would hear a higher pitched noise as the frequency was increased.
Moreover, the wave equation, v=f*λ shows that providing v remains constant, any increase in frequency must cause a reduction in wavelength, increasing the pitch. By increasing the frequency, on a time axis, the sound wave should compress and you can hear the sound at faster speed.

Do you think that the question was hinting about the changes to pitch here or have I answered along the wrong train of thought?

 

[BvU]

I would very much appreciate any suggestions to improve my diagram

so you may want to use compasses to draw them

Is the first that comes to mind

Question 2:
I am not certain how to accomplish this, where the question states the "second minimum from the centre" does it mean the second trough from the centre, where a trough corresponds to a rarefaction?

No. A minimum is where

they produce a smaller or zero-amplitude wave

like where you drew the crosses.

I am worried about your wording

these areas I have marked with crosses

first because a cross marks a point, not an area. but second because it gives the impression that they are areas. The are not: they are lines of points with the common characteristic that the distances to the speakers differ by $n\pm{1\over 4}$ wavelength

Third because "where a trough corresponds to a rarefaction" is not a stationary point ! Both waves move, in opposite directions on the axis.

Question 3:
a. Please see my diagram attached

Go find yourself compasses ! The dots on the symmetry axis should form a horizontal line.

I agree with "quiet sound".

b. Funny question. The pattern moves and the person moves too. More info is needed to give a decent answer.

 

[AN630078]

Is the first that comes to mind

No. A minimum is where
like where you drew the crosses.

I am worried about your wording
first because a cross marks a point, not an area. but second because it gives the impression that they are areas. The are not: they are lines of points with the common characteristic that the distances to the speakers differ by n±14 wavelength

Third because "where a trough corresponds to a rarefaction" is not a stationary point ! Both waves move, in opposite directions on the axis.

Go find yourself compasses ! The dots on the symmetry axis should form a horizontal line.

I agree with "quiet sound".

b. Funny question. The pattern moves and the person moves too. More info is needed to give a decent answer.

Thank you very much for your reply. I seem to have misplaced my compass which is why my diagrams are rather crude, I will keep searching and redraw them accordingly once I have found it.

Question 2; Oh, so is a minimum a point of destructive interference if it is like where I drew the crosses and is where a smaller or zero-amplitude wave is produced? I have attached the same diagram but with a pink blob of where I think the second minimum may be.
If it is here the the path difference would be the difference between the wavelengths of the speakers;
Wavelength from LH speaker to pink blob=7cm=7m
Wavelength from RH speaker to pin blob = 9cm=9m
Path difference: 9m=7m=2m or λ/2 wavelengths?

Ok, as mentioned I am still looking for a compass, I am not too sure what you mean by "The dots on the symmetry axis should form a horizontal line.", specifically with regards to the symmetry axis?

For question 3 b, perhaps an alternative answer would be that as a person walks parallel to the speakers they would hear a high volume then low, then high, etc. as they progressed. This would occur as they pass areas where the waves of the two speakers superpose and constructively then destructive interfere.
However, increasing the frequency of the waves will also increase the pitch, decreasing the wavelength of the waves. The sound waves will interfere more frequently and there will be more points of constructive and destructive interference as the wavelength is decreased. Moreover, the distance the person would have to travel between the louder and quieter areas would become shorter.

Would this be correct? Also, besides from my diagrams being rather rough would they otherwise be correct? Thank you again for taking the time to reply I am very grateful for your help.

 

[BvU]

a pink blob of where I think the second minimum may be

Would that be the intended

the second minimum from the centre.

with the centre point so close by ?
Your purple arrow was on the line connecting the two speakers; it is my impression the second minimum from the center on that connecting line is what is asked for.

Would this be correct?

I agree with your reasoning.

Already commented on the second picture:

Go find yourself compasses ! The dots on the symmetry axis should form a horizontal line.

The locus of other points with a nonzero equal phase difference are hyperbolae -- something that really requires very conscientious and accurate drawing with compasses to visualize.

 

[AN630078]

Would that be the intended
with the centre point so close by ?
Your purple arrow was on the line connecting the two speakers; it is my impression the second minimum from the center on that connecting line is what is asked for.

I agree with your reasoning.

Already commented on the second picture:The locus of other points with a nonzero equal phase difference are hyperbolae -- something that really requires very conscientious and accurate drawing with compasses to visualize.

Thank you for your reply. Oh ok so the purple arrow is the second minimum? Sorry I was confused because you said that a minimum is like where I drew the crosses at a point where a smaller or zero amplitude wave is produced?

Right, I will redraw the diagrams more accurately once I have found my compass and upload them for your thoughts!

 

[BvU]

Your purple arrow was on the line connecting the two speakers; it is my impression the second minimum from the center on that connecting line is what is asked for.

Please read more carefully.

Oh ok so the purple arrow is the second minimum?

It certainly is not.
Your purple arrow points to a maximum ! To be precise, the second maximum away from the central maximum on the connecting line.

The exercise asks for the second minimum

$\$

 

[AN630078]

Please read more carefully.

It certainly is not.
Your purple arrow points to a maximum ! To be precise, the second maximum away from the central maximum on the connecting line.

The exercise asks for the second minimum

$\$

Thank you very much for your reply, apologies for my delay, my new compass has arrived! Anyhow, I have redrawn both diagrams, rather roughly again, I did so quickly so that I was able to reply to you.
Sorry, I will pay more attention to what you have written. To be perfectly candid I am still confused regarding question 2, as to where the second minimum from the centre is situated. Would it be correct to assume that a minimum occurs at a point of destructive interference (marked by blue dots in my amended diagram). However, I am still unsure exactly where this is positioned, I apologise for taking a while to grasp this.
Furthermore, thank you very much for your exquisite diagram, it far surpasses my own I am very appreciative of it. I have drawn on it to shown lines of constructive interference in pink and destructive interference in blue.
I notice there is a central line between the two speakers, consequently I have found the four points which of destructive interference two from the centre and have circled these in black, would this be the second minimum from the centre?

Additionally I have reproduced my diagram to question 3 a, although I think I could improve upon it still I believe that it does substantiate that if a person is positioned at P they would hear a quiet noise, as this is a point of destructive interference: the wave from speaker A (a compression) superposes the wave from B (a rarefaction) meaning the waves are out of phase. This destructive interference would produce a smaller amplitude wave, since the amplitude of a sound wave determines its loudness or volume, the decreased amplitude will reduce the noise heard by the person at point P from the speakers.
Can I ask is my positioning of point P correct (being 10m from speaker A and 8m from speaker B) ?

Thank you very much again for all of your help

 

[BvU]

Sure looks a lot better !

Re q 2: I still think the exercise composer wants you to locate the second minimum on the line connecting the two speakers. You've marked five maxima on that line in question 1 b.jpg. It is clear there are seven of them in total. Where do you think the minima are (on that connecting line) ?

Do you understand why the maxima are 1/2 $\lambda$ apart ?

An animated version of approximately the situation on the connecting line is here
(difference: the central point is a minimum -- and I grant you that I find it hard to see why; must be the sources (4 $\lambda$ apart) are in counter-phase -- but I would need a static screenshot to confirm that )
[edit] And sure enough: static screenshot demonstrates it's hard to quantify a moving picture: the sources are 5 wavelengths apart ! So you need to remove $1/2 \lambda$ on both sides to get your exercise case.

The q 3 picture looks a lot better too, and I agree with the reasoning. I think you are grasping it ok !

is my positioning of point P correct

Looks good. Qualitatively comparable to second picture here (dazzling as well).

 

[AN630078]

Sure looks a lot better !

Re q 2: I still think the exercise composer wants you to locate the second minimum on the line connecting the two speakers. You've marked five maxima on that line in question 1 b.jpg. It is clear there are seven of them in total. Where do you think the minima are (on that connecting line) ?

Do you understand why the maxima are 1/2 $\lambda$ apart ?

An animated version of approximately the situation on the connecting line is here
(difference: the central point is a minimum -- and I grant you that I find it hard to see why; must be the sources (4 $\lambda$ apart) are in counter-phase -- but I would need a static screenshot to confirm that )
[edit] And sure enough: static screenshot demonstrates it's hard to quantify a moving picture: the sources are 5 wavelengths apart ! So you need to remove $1/2 \lambda$ on bothdides to get your exercicse case.

View attachment 268507The q 3 picture looks a lot better too, and I agree with the reasoning. I think you are grasping it ok !

Looks good. Qualitatively comparable to second picture here (dazzling as well).

Thank you very much for your reply. Splendid, I am glad you think my diagrams appear a little better, I will redo them today just to be a little neater.
Oh ok so the line connecting the two speakers would be the central pink line I drew on your diagram? If the maxima are the points of constructive interference where the waves of the speakers superpose would the minima be positioned half way between these points, I have marked these with green lines on the diagram attached.
Are the maxima 1/2 λ apart because a full wavelength consists of a peak and trough (here a compression and rarefaction in a longitudinal sound wave) positioned π radians apart?

Thank you for the link to the animated situation of the connecting line (is this a standing wave pattern?) Right, I think I understand what you are saying, how would I remove 1/2λ on both sides to the scenario in the exercise?

Thank you for evaluating my diagram and reasoning for question 3 also, and thank you for providing a link to a comparable image, indeed it is mesmerising!

 

[BvU]

is this a standing wave pattern?

According to the caption it is :

Animation of a standing wave (red) created by the superposition of a left traveling (blue) and right traveling (green) wave

I think I understand what you are saying, how would I remove 1/2λ on both sides to the scenario in the exercise?

I sense a little doubt. Quite rightly so, because I made one more mistake : in your exercise, the center point is a maximum. To get that, remove $3/4 \lambda$ on one end (in the picture below, I did that on the left), and $1/4 \lambda$ on the other. Now blue is coming from the right, at - amplitude, moving to the left and going up, and green is going to the right, also at - amplitude in the snapshot and also going up. So the sources are in phase. There is a maximum (red) in the middle and there are 8 (green) minima.

Are the maxima 1/2 λ apart because a full wavelength consists of a peak and trough (here a compression and rarefaction in a longitudinal sound wave) positioned π radians apart?

They are, but does that explain the $1/2 \lambda$ ?

 

[AN630078]

According to the caption it is :

I sense a little doubt. Quite rightly so, because I made one more mistake : in your exercise, the center point is a maximum. To get that, remove $3/4 \lambda$ on one end (in the picture below, I did that on the left), and $1/4 \lambda$ on the other. Now blue is coming from the right, at - amplitude, moving to the left and going up, and green is going to the right, also at - amplitude in the snapshot and also going up. So the sources are in phase. There is a maximum (red) in the middle and there are 8 (green) minima.

View attachment 268533

They are, but does that explain the $1/2 \lambda$ ?

Thank you for your reply. Sorry to sound like a stuck record but I am still a little confused. In the diagram you have provided I have circled in pink what I think the second minimum from the centre would be.
In which case would the path difference be 2 wavelengths which would be equal to 8m as λ was found to be 4m?
Why do I need to subtract 3/4λ on one side and 1/4λ on the other side? I think that this is perhaps more detail than the question requires?
I am very appreciative of all of your help and apologise that I am struggling to grasp these concepts!

Also, why are the maxima 1/2λ apart then?

 

[BvU]

Sorry to sound like a stuck record but I am still a little confused

No need to apologize -- things can be confusing, especially in the beginning

In the diagram you have provided I have circled in pink what I think the second minimum from the centre would be.

Maybe now I sound like a stuck record, but I still think the exercise composer is asking for the second minimum on the line connecting the two speakers.

In which case would the path difference be 2 wavelengths which would be equal to 8m as λ was found to be 4m?

In the center the difference is 0, so go 8 m to the left or right.

Why do I need to subtract 3/4λ on one side and 1/4λ on the other side? I think that this is perhaps more detail than the question requires?

I did that already for you in order to make the sitation in the screenshot similar to the situation in your exercise.

Also, why are the maxima 1/2λ apart then?

Because the two waves are traveling in opposite directions, a maximum only has to travel 1/2 λ to encounter another maximum.

In the diagram you have provided I have circled in pink what I think the second minimum from the centre would be...
struggling to grasp these concepts

I had missed this one, but it's rather important: in that picture you draw blue lines and pink lines; what are you suggesting with those ?

 

[AN630078]

No need to apologise -- things can be confusing, especially in the beginning

Maybe now I sound like a stuck record, but I still think the exercise composer is asking for the second minimum on the line connecting the two speakers.

In the center the difference is 0, so go 8 m to the left or right.

I did that already for you in order to make the sitation in the screenshot similar to the situation in your exercise.

Because the two waves are traveling in opposite directions, a maximum only has to travel 1/2 λ to encounter another maximum.I had missed this one, but it's rather important: in that picture you draw blue lines and pink lines; what are you suggesting with those ?

Thank you so much for your reply. Travelling 8m left or right would be a maximum point. Sorry, I had been confused about how to actually measure the path difference, I was doing so hypothetically from one of the speakers. So, the path difference for the second minimum from the centre would be 1.5 λ = 6m?

"I did that already for you in order to make the situation in the screenshot similar to the situation in your exercise."
Oh, thank you, I did not know whether it was a regular part of solving similar problems by subtracting various wavelengths.

"Because the two waves are traveling in opposite directions, a maximum only has to travel 1/2 λ to encounter another maximum."
Right, that makes complete sense now that you have said so!

I think with the blue and pink lines I was just trying to highlight the areas of constructive and destructive interference, however, would this be incorrect because for instance on the blue lines of "destructive interference" maximums occur between the blue minimum points? (and for the pink lines of "constructive interference" minimums would occur between the maximum pink points shown as the green lines)

 

[BvU]

however, would this be incorrect because for instance on the blue lines of "destructive interference" maximums occur between the blue minimum points? (and for the pink lines of "constructive interference" minimums would occur between the maximum pink points shown as the green lines)

Spot on !

The locus of other points with a nonzero equal phase difference are hyperbolae

perpendicular to your lines ...

 

[AN630078]

Spot on !

perpendicular to your lines ...

Thank you very much for your reply, yes I see my error. So would the path difference be 1.5 λ = 6m also?

 

[AN630078]

Spot on !

perpendicular to your lines ...

Sorry I realize I am actaually a little confused on how to find the path difference. Would this be the distance from one speaker to the second minimum from the centre, i.e. 1.5 λ = 6m.
Or rather would it be difference between the distance of the wave emitted from one speaker to the second minimum from the distance of the wave emitted from the other speaker to reach this second minimum, ie. 2.5 λ-1.5 λ =1 λ = 4m.

I realize that the path difference is the difference in distance traveled by the two waves from their respective sources to a given point in the pattern, so would it here be 4m?

I am sorry that I am still struggling with this.

 

[BvU]

how to find the path difference

You take the difference between

the path length from a point to A​

and

the path length from that same point to B​

Example: for all points on the green line the path difference is zero

For the line through the two speakers: (in units of $\bf \lambda$)

 

[AN630078]

View attachment 269374
You take the difference between

the path length from a point to A​

and

the path length from that same point to B​

Example: for all points on the green line the path difference is zero

Thank you for your reply. So would the path differnce to the second minimum be;
1.25 λ = 5m from A and
2.75 λ = 11m from B
11-5=6m which is equal to 1.5 λ?I have put pink dots on your diagram to shown the positions of the second minimums, would this be correct? Thank you again

 

[AN630078]

View attachment 269374
You take the difference between

the path length from a point to A​

and

the path length from that same point to B​

Example: for all points on the green line the path difference is zero

For the line through the two speakers: (in units of $\bf \lambda$)
View attachment 269376

Thank you for your reply

View attachment 269374
You take the difference between

the path length from a point to A​

and

the path length from that same point to B​

Example: for all points on the green line the path difference is zero

For the line through the two speakers: (in units of $\bf \lambda$)
View attachment 269376

Thank you for your reply and further diagram, so would this mean that the second minimums lie 0.75 λ from the centre green line.

So from A, 2.75 λ and from B 1.25 λ
2.75λ-1.25λ=1.5 λ
1.5λ=6m ?

 

[BvU]

So from A, 2.75 λ and from B 1.25 λ
2.75λ-1.25λ=1.5 λ
1.5λ=6m ?

Right !

 

[AN630078]

Right !

Splendid, thank you very much again for all of your help!