Why do sine waves define pure frequency?

thegreenlaser

Like the title says, I'm curious why sine waves are often referred to as "natural" or "pure" oscillations. Why not some other oscillating function? As an example of the type of idea I'm referring to, the wikipedia "Sine Wave" article says:

To me, the argument that it sounds pure because it's a single frequency with no harmonics is a little weak. Doesn't it only lack harmonics because we've defined the notion of harmonics using sine waves? I do understand that orthogonality means there's a certain mathematical convenience associated with sine waves as opposed to, e.g. triangle waves, but the claim seems to be that sine waves are more than just a mathematical convenience; they're somehow "natural" feeling.

So is there a physical reason to define everything based on sine waves, or is it purely mathematical convenience?

davenn

Hi there
sounding like a weak explanation to you doesnt take away from the fact that its correct check it out yourself on a spectrum analyser

if you wanna see serious harmonics being generated, start playing with square waves
a 10MHz squarewave oscillator from say a microprocessor circuit is rich in harmonics that can extend well past 100MHz in freq with still good strength.
one reason why there so much RF crud from PC's and other uPC equip.

cheers
Dave

K^2

It's not just matter of convenience. The reason is best described mathematically because we use mathematics to describe physics. And so you cannot really separate one from the other completely.

It probably helps if you consider how the human ear picks up the sound. Within the inner ear there is a structure called cochlea. It is filled with fluid and contains organ of Corti, which has thousands of hair cells. These cells respond to specific frequencies of vibration in the fluid. Because of how the whole thing is structured, different frequencies excite the hairs in different parts of chochlea. So which hair cells are excited determines the pitch you hear. Sine wave will, indeed, excite the narrowest region within cochlea, and so it will be interpreted as the purest tone.

All of that has to do with resonances, which leads you back to mathematical description where the spectrum of a pure sine wave gives you a single infinitely-narrow peak. The concept of harmonics comes from the fact that strings and pipes have multiple resonances, each corresponding to a wave with its own frequency.

AlephZero

The physical reason why sine waves are a good mathematical description is that linear systems naturally vibrate with the displacements, velocities, and accelerations as sine waves. Many "real world" systems behave in a way that is close to linear.

I suppooe you could argue that human hearing has evolved to discriminate sine waves because that's what the most of the sounds that are useful to discriminate consist of. The arguments about the structure of the cochlea etc are fairly weak, mainly because you hear with your brain, not with your ears, and also because simple versions of the theory don't explain very well what sounds humans can and can't discriminate between.

(As an example of the dfference between hearing with your ears and with your brain, think about people who literally can't "hear" the difference between "r" and "l" sounds in human speech, and their native languages don't include both sounds - but there is no evidence that their ears are anatomically any different from the rest of homo sapiens).

For mathematical signal processing, there are other ways to describe signals that are more useful than sine waves for some purposes - for eaxmple wavelets.

Hetware

Any force resulting from differentiable potential well will approach linearity near the minimum. That is to say ##F=-kx##. The general solution to that second order linear differential equation is ##A~ cos ~\omega t +B~ sin~ \omega t##. Any other cyclical motion is expressible as a sum of such solutions.

But calling it a "pure frequency" is incorrect. Frequency is how many times something happens, and has nothing directly to do with the shape of a wave. Pure oscillation would seem more appropriate.

Hetware

I should add that intermolecular forces are of the linear nature I described in the previous post when the neighboring molecules are near their equilibrium positions. That is why Hooke's law works. It's just a bunch of small displacements added together.

Studiot

First of all let me say that is a very good question, so don't be put off by those wanting to demonstrate how advanced they are.

So what is a pure tone?

Well the term was coined a very long time ago by musicians, not physicists to compare the same note played on different instruments.

Scientists later found that sound signals are very complicated and even a single note contains oscillations of many frequencies when played on an instrument.

Since speech is even more complicated, human hearing evolved to resolve these differences so it is not suprising that it can do so.

We call the pure tone or frequency the predominant one and (I think) it was Archimedes who first observed that the other tones present bear a simple integer relationship to the fundamental.

Notice so far I have not mentioned sinusoidal waves. Archimedes did not know about sine waves. So now let us enquire what sort of mathematical functions are useful in decribing repetitive waveforms such as tones.

First it must be repetitive ie repeat periodically.
Secondly it must remain bounded over the complete cycle.
Thirdly it should ideally be a simple a mathematical function as possible.

Well the sine wave certainly satisfies these requirements, unlike the tangent wave which, although repetitive goes to infinity at regular intervals so cannot be employed. The sine wave is bounded.

The sine wave is certainly very simple. It is simple because any given sine wave may be specified completely by the value of a single variable.
Given y = sin(x) we may draw or calculate y for every value of x.

Furthermore the sinewave is already normalised, since it varies between +1 and -1. So any wave size can be obtained by simple scaling (multiplication by a number or scalar).

However there are more properties required for some waveshapes.

A sine wave is symmetrical about the x axis. That is the second half cycle is an inverted version of the first.

A series of positive pulses form a repetitive waveform that does not have this property and cannot be generated by any number of sinewaves. To generate such waves you have to add a constant.

Such wavetrain can still be represented by the value of one single variable.

A more complicated wavetrain may need two (or more) variables to describe it.
For instance a 1 microsecond pulse repeated every millisecond require the pulse length and the pulse repetition frequency, since these may be varied independently.

rbj

it's not so much sine waves, but it's exponential functions of time. sinusoids are exponentials with imaginary argument. exponential functions are the eigenfunctions of linear, time-invariant systems. so in a sense, sinusoids are also eignefunctions of LTI systems. other families of functions do not have that property.

Hetware

"I do understand that orthogonality means there's a certain mathematical convenience associated with sine waves as opposed to, e.g. triangle waves,"

That indicates at least a passing familiarity with Fourier series, and therefore the ability to understand what a differentiable potential well is.

Feynman attributes that the Pythagoras. With the qualification that the Greeks were talking in terms of the length of the vibrating string, and not directly in terms of frequency.

DrZoidberg

Photons also show clearly that sine waves are the most natural waveform to use as a basis. If you have a large number of identical photons, lets say with a wavelength of 1 meter, all of which have the exact same energy, and you then send them towards an antenna, they get absorbed and produce an ac current in the antenna that follows a pure sine wave.

Studiot

Thank you for your information, Hetware, clearly it was one of the famous ancient Greeks.



That should not detract from the additional information I provided however.

Hetware

I was going to go there, but it smacks of some kind of Pythagorean Mysticism.

Yes, the fact that ##e^{i\omega t} = cos ~\omega t + i~sin ~\omega t##, along with the fact that we can get the Euclidean value for [itex]\pi[/itex] out of that does seem to have some fundamental deep meaning.

Feynman makes the same observation you did about exponential functions.

thegreenlaser

Thanks for all the responses! I do have enough background to understand all the responses, so everything was very helpful. Everything is much clearer now.

Hetware

I hope you construed that just about every generator of a wave has at its base a simple harmonic oscillator. I got that from Georg Joos, BTW.

thegreenlaser

Yeah, I actually did suspect it was something like that; I just wasn't confident enough to be satisfied.

Hetware

I'm not sure I really answered your question. My response does seem to provide some kind of "justification" for considering sine waves to be fundamental. One might equally argue that the fundamental nature of sine waves is related in some way to the Pythagorean theorem: ## cos^2 \theta + sin^2 \theta = 1^2=1 ##.

Khashishi

You can decompose a waveform into any set of complete orthogonal functions you choose. The reason sine waves are usually superior is because they are the solution to the simple harmonic oscillator. The simple harmonic oscillator is so common in physics because almost any real complicated oscillator behaves like a simple harmonic oscillator for sufficiently small oscillations. (This can be understood by taking the Taylor expansion of any physical function: almost always there will be a nonzero x^2 coefficient.) If you increase the size of the oscillation, it will start to become non-sinusoidal.

So, generally speaking, instruments sound more like a pure sine wave when they are played softer, since there are fewer nonlinear vibrations going on.