A question about wave motion and beat frequency

[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 ;)