A question about wave motion and beat frequency

[harini07]

Homework Statement

3 tuning forks of frequencies 200, 203, 207 Hz are sounded together.find out the beat frequency.

Homework Equations

Beat frequency= n1-n2 (n=frequency).

The Attempt at a Solution

I know that beat frequency is the difference in the frequencies of two superposing notes. But here 3 wave frequencies are given the difference between the first 2 is 3 while between 203, 207 is 4 .How to find the beat frequency of all 3 waves together? I have even tried finding beat frequency of the lowest and highest frequency and got 7 but still it's not the answer.how to solve these kinds sum any little help will be greatly helpful :)

 

[haruspex]

Homework Statement

3 tuning forks of frequencies 200, 203, 207 Hz are sounded together.find out the beat frequency.

Homework Equations

Beat frequency= n1-n2 (n=frequency).

The Attempt at a Solution

I know that beat frequency is the difference in the frequencies of two superposing notes. But here 3 wave frequencies are given the difference between the first 2 is 3 while between 203, 207 is 4 .How to find the beat frequency of all 3 waves together? I have even tried finding beat frequency of the lowest and highest frequency and got 7 but still it's not the answer.how to solve these kinds sum any little help will be greatly helpful :)

You need to start with a general concept of what a beat frequency is. How would you define it?

 

[harini07]

Beats per second.i.e. difference between 2 interfering waves' frequency

 

[haruspex]

Beats per second.i.e. difference between 2 interfering waves' frequency

No, that doesn't define beat frequency.
Try defining it in terms of what you hear, or in terms of what you would see if you graphed the amplitude over time.

 

[harini07]

Let me say that it's the interference of the two waves with different frequency but same amplitude?!

 

[haruspex]

Let me say that it's the interference of the two waves with different frequency but same amplitude?!

No, you still miss the point of the question. Why are we interested in beat frequencies at all? Why do we call them beat frequencies? What do you hear when they occur? What do you notice if you look at a graph of interference between nearby frequencies?

 

[harini07]

Sorry I just couldn't figure out about what you are talking. And still I'm confused about that question :/ how to find the beat frequency of three superposed waves? Just today I found another question like that which says n+1, n, n-1 are the frequencies of three superimposed waves.now what will be their beat frequency? I guess I'm missing something cause I thought 1 to be it's answer which is contrast to the original answer I.e. 2 . Can you help me on how to find beat frequency when more than 2 waves say as here it's 3 is superimposed? This question is really depressing :(

 

[haruspex]

Sorry I just couldn't figure out about what you are talking.

If you do not know what "beat frequency" means, what hope do you have of calculating it?

The question I am asking is not that difficult. Look at this picture of two superimposed frequencies producing a beat:
https://en.wikipedia.org/wiki/File:Beating_Frequency.svg
In terms of the picture, what is the period of one beat?

 

[haruspex]

It "beats" me why you find this so hard to answer, but I'll explain: When a beat is produced by a combination of sound waves, it means that to the listener it seems like there is a rhythmic change in amplitude. Looking at the diagram, the beats are the highest peaks in amplitude.

The highest peaks occur, of course, when the individual waves come most closely into phase. With two pure sine waves, one with period 2 and one with period 3 (whatever time units), if they are perfectly in phase at time 0, when will they next be perfectly in phase?

 

[harini07]

It "beats" me why you find this so hard to answer, but I'll explain: When a beat is produced by a combination of sound waves, it means that to the listener it seems like there is a rhythmic change in amplitude. Looking at the diagram, the beats are the highest peaks in amplitude.

The highest peaks occur, of course, when the individual waves come most closely into phase. With two pure sine waves, one with period 2 and one with period 3 (whatever time units), if they are perfectly in phase at time 0, when will they next be perfectly in phase?

sorry for taking much time to reply. *assisgments -_- * coming back to the question they will be in phase after completing one cycle which will be equal to it's time period T

 

[haruspex]

sorry for taking much time to reply. *assisgments -_- * coming back to the question they will be in phase after completing one cycle which will be equal to it's time period T

Yes, but I gave you specific numbers for the individual periods: 2 and 3. So what is the beat period, as a number?

 

[harini07]

Yes, but I gave you specific numbers for the individual periods: 2 and 3. So what is the beat period, as a number?

one? they will meet after one period?

 

[haruspex]

one? they will meet after one period?

Draw a diagram. One peaks every 2 seconds (say) and the other every 3.

 

[harini07]

Draw a diagram. One peaks every 2 seconds (say) and the other every 3.

after T=2.5 sec?

 

[haruspex]

after T=2.5 sec?

Are you just guessing or did you draw a diagram? If you drew one, please try to post it.

 

[harini07]

Are you just guessing or did you draw a diagram? If you drew one, please try to post it.

 

[harini07]

View attachment 112073

I have included a rough graph.sorry for it's incoherence.i know it's too poorly picturised :/ but @haruspex (sorry if I spelt your name incorrectly) beg you please explain me how to do it at least now :(

 

[harini07]

I have included a rough graph.sorry for it's incoherence.i know it's too poorly picturised :/ but @haruspex (sorry if I spelt your name incorrectly) beg you please explain me how to do it at least now :(

And in that graph that I have drawn, it's clear that first they meet at 2.5 seconds and then at 4.5 I.e. again after 2.5 seconds.correct me if I am wrong? But how could I relate this to get the answer to the question that I've posted!

 

[haruspex]

And in that graph that I have drawn, it's clear that first they meet at 2.5 seconds and then at 4.5 I.e. again after 2.5 seconds.correct me if I am wrong? But how could I relate this to get the answer to the question that I've posted!

You appear to have drawn one of period 4 and one of period 5. They started 180 out of phase (one is going up and the other down).
At the 4.5 mark, they are not both back where they started. At a point where they have both completed a whole number of cycles, each should continue exactly as it did from time 0, i.e. both at the x axis, with the red line going down and the other up.

Mark 12 equal intervals on the x axis. Have both curves rising from the origin. The period 2 curve goes down through the x-axis at t=2, 6 and 10, and up through it at 0, 4 and 8. The period 3 goes down through the x-axis at 3 and 9, and up through it at 0, 6 and 12.

 

[haruspex]

Here are three sine waves of the same amplitude, all starting in phase. Can you figure out what the three periods are?
Can you find the time at which a pair of them are both back to their starting state? (There are two such pairs.)
Can you see how the time for two to get back into phase relates to their individual periods?
When will all three be back in phase?

 

[harini07]

View attachment 112077
Here are three sine waves of the same amplitude, all starting in phase. Can you figure out what the three periods are?
Can you find the time at which a pair of them are both back to their starting state? (There are two such pairs.)
Can you see how the time for two to get back into phase relates to their individual periods?
When will all three be back in phase?

so the period of blue wave is 2 (say n1) and the period of orange wave is 3 (n2), they are back at their original position at T=6s. (n1xn2).they will be again in phase 2(n1xn2) i.e T=12s and T=18s? but what about the third yellow wave? it's time period is T=5s and it doesn't seem to be in phase with the two of them at any point of time. now i would like you to correct me if I'm wrong. PS: thanks for the graph.it's really helpful! :)

 

[haruspex]

the third yellow wave? it's time period is T=5s and it doesn't seem to be in phase with the two of them at any point of time

It does get back into phase with one of them - just. But my graph only goes for 10 seconds. Maybe some resynchronisations take longer.
Try thinking of the periods of the waves as rods of different lengths, 2, 3 and 5. What is the shortest length that can be made entirely of rods length 3 or entirely of rods length 5?

 

[harini07]

It does get back into phase with one of them - just. But my graph only goes for 10 seconds. Maybe some resynchronisations take longer.
Try thinking of the periods of the waves as rods of different lengths, 2, 3 and 5. What is the shortest length that can be made entirely of rods length 3 or entirely of rods length 5?

the shortest length that can be made entirely of rod of length 3 or 5 is 1. how could i relate this with the question? should do the same for the given frequencies?

 

[haruspex]

the shortest length that can be made entirely of rod of length 3 or 5 is 1

How could you do that? Perhaps you are thinking 3+3-5? But for one thing I am not allowing subtractions, and for another I am not allowing mixtures. You have to make the same length entirely of 3s and entirely of 5s.

 

[harini07]

How could you do that? Perhaps you are thinking 3+3-5? But for one thing I am not allowing subtractions, and for another I am not allowing mixtures. You have to make the same length entirely of 3s and entirely of 5s.

then. i guess i didn't get your question properly! and still I'm not getting what you are asking!

 

[haruspex]

then. i guess i didn't get your question properly! and still I'm not getting what you are asking!

What is the smallest length that can be made by laying rods of length 3 end-to-end, and could also be made by laying rods of length 5 end to end.

 

[harini07]

What is the smallest length that can be made by laying rods of length 3 end-to-end, and could also be made by laying rods of length 5 end to end.

By laying down a rod of length 3 end to end ,the smallest length that can be made is 15 (by using 5 rods) and when we use 3 rods of length 5, we get that same length.

 

[haruspex]

By laying down a rod of length 3 end to end ,the smallest length that can be made is 15 (by using 5 rods) and when we use 3 rods of length 5, we get that same length.

Right.
Can you see how that connects with beat frequencies? If two waves have periods 3 and 5, the rod lengths represent the times for complete cycles. It takes a length of 15 for both to go through a whole number of cycles and be back in synch.
Now include a third wave, length 2. How long for all three to come back into synch?

 

[rude man]

Homework Statement

3 tuning forks of frequencies 200, 203, 207 Hz are sounded together.find out the beat frequency.

Say the three signals start in phase (all three peaks line up). How long before they line up again?

 

[harini07]

Right.
Can you see how that connects with beat frequencies? If two waves have periods 3 and 5, the rod lengths represent the times for complete cycles. It takes a length of 15 for both to go through a whole number of cycles and be back in synch.
Now include a third wave, length 2. How long for all three to come back into synch?

If a rod of length 2 is included then it will go upto 30. I mean here I took the greatest common multiple of that three.so should I do the same in case of this frequency of 3 waves?

 

[haruspex]

If a rod of length 2 is included then it will go upto 30. I mean here I took the greatest common multiple of that three.so should I do the same in case of this frequency of 3 waves?

Yes, except that you mean least common multiple.

 

[harini07]

Yes, except that you mean least common multiple.

So in case of 3,4 and 7 which is the given values of beat frequencies the least common multiple is 84.So the beat frequency of those three waves is 84? (Which is not the answer :/ )

 

[haruspex]

So in case of 3,4 and 7 which is the given values of beat frequencies the least common multiple is 84.So the beat frequency of those three waves is 84? (Which is not the answer :/ )

Sorry, but I'm coming to the conclusion the question makes no sense.
To respond to your last post first... What I led you to was that you need to take the LCM of the three beat periods, not of the frequencies.

But here's what it actually looks like:

I challenge anyone to decide where the beats are in that mess. The absolute max amplitudes are at 1 second intervals (indeed, it repeats at 1 second intervals, which should be obvious from the given frequencies), but there are lots of other local maxima at uneven spacings, some almost as large as the absolute maxima.
Maybe the reason I cannot find anything online about beats from three or more sources is that it is not well-defined.

Do you know what the answer is supposed to be?

 

[harini07]

Sorry, but I'm coming to the conclusion the question makes no sense.
To respond to your last post first... What I led you to was that you need to take the LCM of the three beat periods, not of the frequencies.

But here's what it actually looks like:
View attachment 112243
I challenge anyone to decide where the beats are in that mess. The absolute max amplitudes are at 1 second intervals (indeed, it repeats at 1 second intervals, which should be obvious from the given frequencies), but there are lots of other local maxima at uneven spacings, some almost as large as the absolute maxima.
Maybe the reason I cannot find anything online about beats from three or more sources is that it is not well-defined.

Do you know what the answer is supposed to be?

the answer is supposed to be 12. even I'm greatly confused on how they got it :/ but thanks for your efforts..you really tried a lot to get me understand things. may be it's my part that i failed to put all things together and decipher how to proceed!

 

[haruspex]

the answer is supposed to be 12.

That is demonstrably false. The pattern I posted repeats every 1 second ( which you can prove using the LCM of the three periods 1/3, 1/4, 1/7 of the three pairwise beat frequencies). Within that, you can see there are 7 more-or-less evenly spaced bulges each second, so you could make a case for the answer 7. One bulge is a lot smaller than the other 6, but if we discount that then the spacing becomes rather uneven, so no longer qualifies as a "beat".

 

[harini07]

That is demonstrably false. The pattern I posted repeats every 1 second ( which you can prove using the LCM of the three periods 1/3, 1/4, 1/7 of the three pairwise beat frequencies). Within that, you can see there are 7 more-or-less evenly spaced bulges each second, so you could make a case for the answer 7. One bulge is a lot smaller than the other 6, but if we discount that then the spacing becomes rather uneven, so no longer qualifies as a "beat".

Yes even I was thinking the same. It could be one 1 which is more convincing :/ may be the answer key is wrong.let me clear this one last thing.i just want to mark this question solved..It's.there for a long time [emoji39] let's say there are three waves with frequencies (sorry again) n+1,n,n-1 the beat frequency of the three waves will be 2 (by taking LCM ) right?

 

[haruspex]

Yes even I was thinking the same. It could be one 1 which is more convincing :/ may be the answer key is wrong.let me clear this one last thing.i just want to mark this question solved..It's.there for a long time [emoji39] let's say there are three waves with frequencies (sorry again) n+1,n,n-1 the beat frequency of the three waves will be 2 (by taking LCM ) right?

No, you keep taking LCMs of frequencies. If the LCM is relevant at all, it's the LCM of the periods.

 

[harini07]

No, you keep taking LCMs of frequencies. If the LCM is relevant at all, it's the LCM of the periods.

woah! yes, i was! but say th

No, you keep taking LCMs of frequencies. If the LCM is relevant at all, it's the LCM of the periods.

yes sorry :/ so the LCM of their time period would be 1. so beat frequency is 1 ?

 

[haruspex]

woah! yes, i was! but say th

yes sorry :/ so the LCM of their time period would be 1. so beat frequency is 1 ?

As I wrote in post 35, you can use the LCMs of the three pairwise beat frequencies to show that the pattern repeats after 1 second, but that is not the same as saying it has a beat frequency of 1 second. Beats are perceptual. If you were to hear the pattern of pulses depicted in post #33, I don't think you'd say it had a beat frequency of 1 second.

 

[harini07]

I've included the key that my textbook has given for this question.do look at that if it makes any sense :/ also I guess that's what you have told I.e. dividing the time period into smaller chunks it one gets the Maxima.see where I flawed. I'm being forced to give up on this question :(

 

[haruspex]

View attachment 112390 I've included the key that my textbook has given for this question.do look at that if it makes any sense :/ also I guess that's what you have told I.e. dividing the time period into smaller chunks it one gets the Maxima.see where I flawed. I'm being forced to give up on this question :(

Interesting.
As you can see from the image I posted at #33, the book answer is simply wrong. The question is why.
First, as I wrote at the outset, we need a definition of a beat. Originally it was certainly a matter of perception, and implied a regular pulse of sound. In the usual treatment of two waves, that is what you get. But three waves will not give a nice regular pulse.
The book appears to be defining it as the rate of local peaks in the combination of the three pairwise beats. From a theoretical perspective that feels odd. Why should it be broken into those two stages? We could more reasonably ask how many peaks there are in the amplitude altogether, but then we would answer 200x203x207.
From an auditory perspective it fails because some of these 11 (not 12) overlap so significantly you would not hear them as separate.

 

[harini07]

Interesting.
As you can see from the image I posted at #33, the book answer is simply wrong. The question is why.
First, as I wrote at the outset, we need a definition of a beat. Originally it was certainly a matter of perception, and implied a regular pulse of sound. In the usual treatment of two waves, that is what you get. But three waves will not give a nice regular pulse.
The book appears to be defining it as the rate of local peaks in the combination of the three pairwise beats. From a theoretical perspective that feels odd. Why should it be broken into those two stages? We could more reasonably ask how many peaks there are in the amplitude altogether, but then we would answer 200x203x207.
From an auditory perspective it fails because some of these 11 (not 12) overlap so significantly you would not hear them as separate.

yeah, 11 of them overlap. so you say questions like this can be never answered just by taking some theoretical assumptions? so there is no regular pulse which can be the ultimate answer to this question but simply there are set of pulses that get repeated over the period of time, right?

 

[haruspex]

11 of them overlap.

Well, post #33 shows, arguably, 7 pulses each second, so not all 11 overlap with others.

there is no regular pulse which can be the ultimate answer to this question but simply there are set of pulses that get repeated over the period of time, right?

Right.

 

[harini07]

Well, post #33 shows, arguably, 7 pulses each second, so not all 11 overlap with others.

Right.

FINALLY, :D thanks for the help @haruspex you are an awesome teacher (no flattering, my teacher got infuriated with me for this question and simply asked me to follow that flawed answer key). i will mark it solved ;)