Tone hearing frequency concept question

[Adam Rabe]

Homework Statement

Homework Equations

Dont know any sorry

The Attempt at a Solution

I tried imagining the ear as some sort of harmonic pipe with the sound coming inside as a sine wave. The question says the sine wave is not fully sinosoidal and is flattened at the top and bottom.
I don't know where to go from there sorry - any help would be greatly appreciated

 

[DrClaude]

As you are flattening the wave, you are getting closer to a square wave. You can use knowledge of how a square wave is represented as a sum of sine waves to help you.

http://mathworld.wolfram.com/FourierSeries.html

 

[Adam Rabe]

As you are flattening the wave, you are getting closer to a square wave. You can use knowledge of how a square wave is represented as a sum of sine waves to help you.

http://mathworld.wolfram.com/FourierSeries.html

Hello i don't fully understand what Fourier is, I know its that one with the flat topped/diagonal waves which you can get by adding up sine waves. I don't see how this applies to the question - can you give me another hint?

 

[berkeman]

Hello i don't fully understand what Fourier is, I know its that one with the flat topped/diagonal waves which you can get by adding up sine waves. I don't see how this applies to the question - can you give me another hint?

The key in this question is to see which sine waves make up the square wave. As you add more and more of the correct frequency sine waves (with adjusted amplitudes), you can turn a 300Hz sine wave into a 300Hz square wave.

You will learn more about this process in your classes, but for now, can you try a Google search on something like -- How to add up sine waves to make a square wave. Look at the frequencies of the lowest frequency used, and the relationship to the other sine waves of smaller amplitude that are added in.

Can you then say which one of the answers above is correct?

 

[Adam Rabe]

The key in this question is to see which sine waves make up the square wave. As you add more and more of the correct frequency sine waves (with adjusted amplitudes), you can turn a 300Hz sine wave into a 300Hz square wave.

You will learn more about this process in your classes, but for now, can you try a Google search on something like -- How to add up sine waves to make a square wave. Look at the frequencies of the lowest frequency used, and the relationship to the other sine waves of smaller amplitude that are added in.

Can you then say which one of the answers above is correct?

Hi the answer was E - can you maybe give an explanation as to why, I am really stuck

 

[berkeman]

Hi the answer was E - can you maybe give an explanation as to why, I am really stuck

Did you do the Google search I suggested? It would show you why the answer is E...

 

[DrClaude]

Let me recopy the formula for a square wave from the link I gave:
$$
\frac{4}{\pi} \sum_{n=1,3,5,\ldots}^{\infty} \frac{1}{n} \sin \left( \frac{n \pi x}{L} \right)
$$
Do you see the relationship between the frequencies of the different components?

Another to see it is that flattening the sine wave doesn't change the period over which the signal repeats. What frequencies can be included such that this period doesn't change?

 

[NTL2009]

"E" doesn't appear to be correct to me, depending on how strictly you read the question.

If a wave is flattened on both the top and bottom, by the same amount, you will have only odd harmonics. So 300 Hz + 900 Hz + 1500 Hz, etc.

To get an even harmonic, I think that requires non-symmetry above/below the X axis. You would need a significant difference in the flattening top/bottom to make a 2nd harmonic audible. I might play with this in Audacity (open-source sound editing program), for real life results. (a few minutes later... Yep, it showed what I thought it would)

 

[DrClaude]

"E" doesn't appear to be correct to me, depending on how strictly you read the question.

If a wave is flattened on both the top and bottom, by the same amount, you will have only odd harmonics. So 300 Hz + 900 Hz + 1500 Hz, etc.

Nowhere did the problem state that the modification was symmetric. It simply said that the wave isn't sinusoidal anymore (even the flattening is a "perhaps.")

 

[berkeman]

"E" doesn't appear to be correct to me, depending on how strictly you read the question.

If a wave is flattened on both the top and bottom, by the same amount, you will have only odd harmonics. So 300 Hz + 900 Hz + 1500 Hz, etc.

To get an even harmonic, I think that requires non-symmetry above/below the X axis. You would need a significant difference in the flattening top/bottom to make a 2nd harmonic audible. I might play with this in Audacity (open-source sound editing program), for real life results. (a few minutes later... Yep, it showed what I thought it would)

That's a fair point, but the other answers are clearly wrong, so only E is left as the "best" of the possible answers.

 

[NTL2009]

That's a fair point, but the other answers are clearly wrong, so only E is left as the "best" of the possible answers.

Absolutely agree with that, the other answers are way off. I guess I'm just saying the problem and answer could have been worded better.

 

[Merlin3189]

Perhaps the questioner was simply thinking that distortion produces harmonics and just offered some harmonics without worrying what sort of distortion was involved?

 

[NTL2009]

Perhaps the questioner was simply thinking that distortion produces harmonics and just offered some harmonics without worrying what sort of distortion was involved?

Yes, and maybe I'm making too big a deal about this (but it's not like I'm pounding my desk either! ). Let's just say they missed an opportunity, and they shouldn't be confusing students with what creates odd and what creates even harmonics. They sort of mish-mash the concepts here. But that got me thinking:

From memory (and playing with music synthesizers), I recall that a square wave has odd harmonics with amplitude of f/h (where h = freq of harmonic, f = freq of fundamental), so third harmonic is 1/3rd amplitude, fifth harmonic is 1/5th etc.

A triangle wave also has only odd harmonics, but the drop off is (f/h)^2. Third harmonic is 1/9th, fifth is 1/25th etc. That also makes sense when you consider a low-pass filtered square wave starts looking like a triangle wave.

And a sawtooth wave has both even and odd harmonics, at f/h.

But what I didn't recall is, what about only even harmonics? My analog music synths didn't have a waveform like that available. A little more playing in a graphing calculator and I get this, not sure how to describe it though? Do these regularly occur in nature, or maybe those combination of forces are not common?

https://www.desmos.com/calculator/sanwjjkkqu

 

[Merlin3189]

looks like a sawtooth to me.
If you alternate adding and subtracting harmonics and reduce them in amplitude by the square of their order, it gets much smoother -say sin(x)-sin(2x)/4+sin(4x)/16- sin(6x)/36 - etc..

 

[NTL2009]

A sawtooth has even and odd harmonics.

 

[Merlin3189]

You're right, of course.
I've actually been fooling myself trying to even out the bumps on nearly triangular waves. All the even harmonics of f are actually all the odd and even harmonics of 2f. So I was getting a perfect sawtooth by using all the even harmonics, then just removing the fundamental!

Anyhow, thanks for the link to the desmos calculator: it's great for graphing these functions.