Why light is a sinusoidal wave?

[lovetruth]

Light is always considered to be a sinusoidal or harmonic wave without giving any reason. Maxwell's equation says that light is an EM wave but does not say its a harmonic wave or even a periodic wave.
People say that every periodic or non-periodic function can be broken down as a sum of series of sine waves by using Fourier analysis. But any function can be broken down as sum of series of other function. For example, any function can be broken down into a sum of series of polynomial functions using taylor series. Sine wave can be broken down as a sum of polynomial functions using taylor series. A simple sine wave can also be broken down into a sum of series of other periodic waves with various frequencies and amplitudes. So why other periodic waves such as triangle, saw tooth, square waves or other periodic waves of arbitary shapes are not considered while talking about light.
While finding intensity in Young double slit experiment, the light is considered as a sine wave. If light is considered as pure triangle wave or other periodic wave, the formula of intensity as a function of distance on the screen will be very different.

 

[Dale]

People say that every periodic or non-periodic function can be broken down as a sum of series of sine waves by using Fourier analysis.

That is the only reason. There is nothing fundamental going on.

You could certainly use other basis functions, but they are usually less convenient.

While finding intensity in Young double slit experiment, the light is considered as a sine wave. If light is considered as pure triangle wave or other periodic wave, the formula of intensity as a function of distance on the screen will be very different.

This is factually incorrect. If you do the math right you will get the same answer in any basis. Of course, it is far easier to make a mistake working in a bad basis.

 

[lovetruth]

That is the only reason. There is nothing fundamental going on.

You could certainly use other basis functions, but they are usually less convenient.

This is factually incorrect. If you do the math right you will get the same answer in any basis. Of course, it is far easier to make a mistake working in a bad basis.

A simple sine wave of a certain frequency and amplitude when broken down in triangle wave results in sum of infinite number of triangle waves with various amplitudes and frequencies different than that of a single sine wave. So a single frequency sine wave should give a spectrum of triangle waves with various frequencies. This mean that the wavelength of light really depends on the type of wave being discussed. If all periodic waves were equivalent then a red color sine wave can be split into a infinite spectrum of triangle wave with varying wavelength.

 

[Dale]

A simple sine wave of a certain frequency and amplitude when broken down in triangle wave results in sum of infinite number of triangle waves with various amplitudes and frequencies different than that of a single sine wave. So a single frequency sine wave should give a spectrum of triangle waves with various frequencies. This mean that the wavelength of light really depends on the type of wave being discussed.

Sure. When people are talking about wavelengths and frequencies they are talking about sin waves, so doing that would cause all sorts of communication problems and confusion. But there is nothing physically wrong with what you are saying.

If all periodic waves were equivalent

What do you mean by "equivalent"?

 

[lovetruth]

What do you mean by "equivalent"?

By equivalent, I mean no bias for using sine wave when talking about light.

A triangle and sine wave of same wavelength are not similar mathematically. Yet we say color only depends upon the wavelength of the periodic wave. A triangle wave can be broken down into an infinite series of sine waves with varying wavelength.

 

[mathfeel]

Mathematically, there is really no distinction between a sinusoidal and a triangular basis. Any field of the form E(x,t) = f(k x - c k t) is a good solution to the wave equation in vacuum. Any set of basis is just as good. In practice, we have more tools to deal with sinusoidal basis with the whole subject of Fourier transform.

But when you are talking about color, you are talking about interaction with atoms (or color receptors in your eye). That corresponds to discrete atomic energy levels. So transition by light occurs for light is an eigenstate of the energy operator i\partial/\partial t, and that means a wave of the form: E(x, t) \propto e^{i (kx - ckt)}.

Plus, it is easier to generate a sinusoidal wave than a triangular wave.

 

[Dale]

By equivalent, I mean no bias for using sine wave when talking about light.

Not sure what you mean by "no bias".

Yet we say color only depends upon the wavelength of the periodic wave.

No, "color" refers specifically to the wavelength in the sinusoidal basis, not an arbitrary basis. A pure red filter passes only sinusoidal wavelengths of about 650 nm. A square or triangle wave of 650 nm will have some of the "edges" softened.

You cannot assume that filters or molecular transitions with a given spectrum in one basis will have the same spectrum in another basis.

 

[lovetruth]

Mathematically, there is really no distinction between a sinusoidal and a triangular basis. Any field of the form E(x,t) = f(k x - c k t) is a good solution to the wave equation in vacuum. Any set of basis is just as good. In practice, we have more tools to deal with sinusoidal basis with the whole subject of Fourier transform.

But when you are talking about color, you are talking about interaction with atoms (or color receptors in your eye). That corresponds to discrete atomic energy levels. So transition by light occurs for light is an eigenstate of the energy operator i\partial/\partial t, and that means a wave of the form: E(x, t) \propto e^{i (kx - ckt)}.

Plus, it is easier to generate a sinusoidal wave than a triangular wave.

We can talk about color without using eye or color receptor. We can instead use spectrophotometer or prism.
My question is that does a triangle wave of wavelength 700 nm will look red? A triangle wave of wavelength 700 nm can be broken down mathematically into sum of sine waves with wavelength varying from zero to infinity.

 

[lovetruth]

Not sure what you mean by "no bias".

No, "color" refers specifically to the wavelength in the sinusoidal basis, not an arbitrary basis. A pure red filter passes only sinusoidal wavelengths of about 650 nm. A square or triangle wave of 650 nm will have some of the "edges" softened.

You cannot assume that filters or molecular transitions with a given spectrum in one basis will have the same spectrum in another basis.

By bias I mean that light is always considered sinusoidal without giving an explanation

How can you prove that red color is sinusoidal?

 

[HallsofIvy]

By bias I mean that light is always considered sinusoidal without giving an explanation

Elementary texts don't because they don't want to go into Fourier Analysis. Advanced texts explain what you have been told here.

How can you prove that red color is sinusoidal?

?? No one has said it is. There is no such thing as "red color". There is a range of wavelengths for colors that we call "red". If we were to use some other basis rather than sinusoids, those same "colors" would have a different range of wave lengths.

 

[Dale]

By bias I mean that light is always considered sinusoidal without giving an explanation

In that case, there certainly is a bias. Most introductory physics textbooks will not bother to give an explanation about different basis functions.

I think that most textbook authors consider it a fairly minor point that would detract from the more important points, such as the fact that EM waves carry energy and momentum. They cannot cover everything, after all.

 

[lovetruth]

Elementary texts don't because they don't want to go into Fourier Analysis. Advanced texts explain what you have been told here.


?? No one has said it is. There is no such thing as "red color". There is a range of wavelengths for colors that we call "red". If we were to use some other basis rather than sinusoids, those same "colors" would have a different range of wave lengths.

By red color I mean 700 nm and not a range.

 

[lovetruth]

In that case, there certainly is a bias. Most introductory physics textbooks will not bother to give an explanation about different basis functions.

I think that most textbook authors consider it a fairly minor point that would detract from the more important points, such as the fact that EM waves carry energy and momentum. They cannot cover everything, after all.

What happens when triangle wave of wavelength 700 nm strikes a prism?

 

[Dale]

It gets separated into a rainbow of non-uniform intensity.

 

[nasu]

The frequency or period of the wave is independent of the basis.
The fact that you can decompose a triangular or square wave in sines does not mean you get a different frequency. The frequency of the wave is given by the fundamental.

If you have a triangular wave with frequency F0 and look at the Fourier decomposition the first component in the series will have frequency F0. You will have harmonics with frequencies larger than F0, true. What will be the effect of these? In the case of sound waves, these harmonics will determine the tone of the sound. A "triangular" wave will sound different than a sine wave of the same frequency. But they do have the same frequency and the same pitch. And this effect is independent of the basis.

For visible light, the harmonics may not have any influence as the range of visible frequencies is very narrow (less than an octave). The red receptors are sensitive to light of a specific frequency (actually a quite wide range). A wave with the frequency in this range will excite the red receptors. It does not matter if it is pure sine or it has many components. Suppose you take a triangular wave and decompose it in sines. If the fundamental is in the range of red receptors, the first harmonic will be outside the visible range. So it should look red and there will be no effect similar to the tone of the sound.

 

[nasu]

It gets separated into a rainbow of non-uniform intensity.

Shouldn't rather be a discrete series of lines? And all of them but one will be invisible. The first harmonic will have a wavelength of 350 nm which is in the near UV.

 

[lovetruth]

It gets separated into a rainbow of non-uniform intensity.

Why is the prism prejudiced for sine wave. Why can't it allows single frequency triangle wave to pass as single frequency triangle wave. But the prism allows single frequency sine wave to pass as single frequency sine wave.

 

[sophiecentaur]

If you pass what is defined as monochromatic light through a prism or diffraction grating then all you get is a single line spectrum. The photons of light will have a particular energy, given by hf. If the 'shape' of the waves were anything other than a sine wave, there would also be harmonics of this fundamental f. These would also have to be photons and would have energies of 2hf, 3hf, 4hf etc.. You will notice that these energies get bigger and bigger. The photons cannot exist with less energy than this - at least according to QM. Photons are released by transitions between certain energy levels in a charge system. We usually talk in terms of a gas, because its atoms have discrete energy levels. All this structure would go wonky if we were to have extra photons emitted, with even higher energies than the fundamental.
Edit: And I just realized that the amplitudes of these harmonics would need to be impossible values, too, because there would have to be photons of the harmonics for EVERY photon of the fundamental frequency. It just couldn't work.

So we have to assume that the emitted light does in fact have a sinusoidal wave shape. It's the only shape that QM will allow.
For em waves with frequencies, much lower than light (i.e. the more manageable RF frequencies), it is quite possible to produce a wave with any shape you want because the energy levels involved in an electronic circuit are spread into a continuous band and any frequency can be produced.

Another reason for favouring sine waves when studying waves in general is that a sine wave can be produced by the very simplest form of oscillator - the Simple harmonic oscillator - which oscillates in a 'sinusoidal' fashion. I think you can say that the sine wave is a pretty fundamental waveform, even though it is possible, as a mathematical exercise (and even in practice sometimes), to synthesise any repeating wave shape with many different families of harmonically related wave shapes.

 

[Dale]

Shouldn't rather be a discrete series of lines?

Only if it runs with perfect stability forever.

 

[mathfeel]

Why is the prism prejudiced for sine wave. Why can't it allows single frequency triangle wave to pass as single frequency triangle wave. But the prism allows single frequency sine wave to pass as single frequency sine wave.

The reason that prism is "prejudice" toward sine wave is that what scatters light into different direction in the prism are motions of charged particles such as atom or electron. The electric field of light drives them slightly away from their equilibrium position. And they responds by oscillating sinusoidally just like any small oscillator. These oscillating charge then omits sinusoidal EM waves.

When you send in a triangular wave, the charge particles that it interacts with do the same thing. But the superposition of the incoming wave and the induced wave will no longer be triangular.

 

[nasu]

Only if it runs with perfect stability forever.

If you mean the broadening due to the finite duration of the wave, then:
1. It applies to sine waves as well so it has no special relevance to the discussion about "triangular" waves.
2. Unless you look at very short pulses is quite small to talk about a "rainbow" as opposite to lines. A nanosecond pulse of light will have a natural band width of about 10^9 Hz, which is about 1 in 1 million (frequency of light is around 10^15 Hz).
You need to go down to femto-second laser pulses to have significant effect (rainbow)
An usual red laser sent through a prism will produce a narrow line rather than a rainbow.

The separation of harmonics in a hypothetical triangular wave would he much larger than the natural broadening and I don;t see how you can get a rainbow. Unless we are talking about a "triangular femto-laser".

 

[Dale]

If you mean the broadening due to the finite duration of the wave, then:

Yes, that is what I meant.

Whether the lines are discrete or continuous is not particularly relevant to the OP's question, I shouldn't have responded with irrelevancies, my apologies. The main point is that if the waves are anything other than an infinite pure sinusoid then the spectrum is not a single frequency.

 

[Dale]

Why is the prism prejudiced for sine wave. Why can't it allows single frequency triangle wave to pass as single frequency triangle wave. But the prism allows single frequency sine wave to pass as single frequency sine wave.

I like mathfeel's explanation. The fact is that sine waves show up naturally in a lot of phenomena, in particular any time there is a smooth potential around some stable equilibrium. Triangle waves do not show up so commonly.

 

[nasu]

Why is the prism prejudiced for sine wave. Why can't it allows single frequency triangle wave to pass as single frequency triangle wave. But the prism allows single frequency sine wave to pass as single frequency sine wave.

I would add to the previous answers and comments that this "prejudice" is called dispersion.
In a non-dispersive medium the triangular wave will remain triangular.
If the prism were not dispersive it won't do its job of separating spectral components.
On the other hand, in a dispersive medium even a sine wave (unless is infinite in time) will be distorted. So the prejudice is not specific to triangular waves.

 

[chingel]

Is my understanding of this discussion correct that light waves don't have harmonics and this is the reason why lightwaves are sinusoidal?

But do sine waves of sound have harmonics? If i generate two sine waves with my computer, being an octave+3 hz apart, I am pretty sure I hear a beat. Doesn't this mean that the lower sine wave has an harmonic that is an octave higher?

 

[Dale]

Is my understanding of this discussion correct that light waves don't have harmonics and this is the reason why lightwaves are sinusoidal?

Light waves aren't always sinusoidal. But it is a convenient basis.

 

[chingel]

Light waves aren't always sinusoidal. But it is a convenient basis.

Does that mean that they can have harmonics?

 

[Dale]

Yes, and they usually do.

 

[chingel]

Is it possible to see an harmonic wave of a light wave whose frequency is way below the visible range?

 

[Dale]

If by "see" you mean "detect", then yes. Obviously EM radiation outside the visible range is by definition not visible with just our eyes.

 

[sophiecentaur]

Light waves aren't always sinusoidal. But it is a convenient basis.

It has to depend how it was produced. As I said before, a light wave produced from gas atoms has to be a 'pure' sinusoid because it just cannot have photons of more than one energy.
But one could imagine an em signal with any, arbitrary, spectrum if you can think of a way to produce it.

 

[nasu]

But do sine waves of sound have harmonics? If i generate two sine waves with my computer, being an octave+3 hz apart, I am pretty sure I hear a beat. Doesn't this mean that the lower sine wave has an harmonic that is an octave higher?

The sound waves can have harmonics. They don't have to. It depends on how you produce them. With electronic devices you can produce quite pure sounds, without harmonics.
Sounds produced by musical instruments have various harmonic composition (this gives the specific tone of the instrument).
And there are sounds which have several sinusoidal components which are not harmonics (like some sounds produced by drums or bells).

Regarding your experiment, I suppose you mean adding two waves with frequencies 1000Hz and 2003 Hz, for example.
You don't need to have a harmonic of the lower frequency (2000 Hz) to get something like beats. If you add the two waves and plot the result you'll see that the amplitude changes slowly in time.

 

[sophiecentaur]

There are two issues here.
Firstly, you can make up any shape of electromagnetic wave (continuous or not) by adding together combinations of other waves. This is the idea of Fourier Analysis, which breaks a signal down into a combination of sinusoids. You can also do the same analysis in terms of square waves,pulses or triangles. Using sine waves is very convenient because of the Maths and for a number of practical reasons. But the original waveform stands on its own; you can describe it, simply as a time varying function of some variable such as Voltage. All this applies to waves and vibrations of all sorts.
The second issue is a practical one. Light is defined as the visible part of the spectrum and, as it only occupies an octave, the visible part of any em wave cannot contain harmonics. Also, the way it is often produced can dictate that there is only one frequency involved - for the QM reasons I have already stated.
A further point is that light from a group of atoms may well not be at precisely the same frequency - there is always a spread, even in 'monochromatic' light. A laser reduces this spread because of the way the photons are stimulated to leave the atoms very accurately in step / phase. Also, no wave is continuous; it is switched on and switched off and this introduces a finite bandwidth to the signal.

 

[chingel]

Regarding your experiment, I suppose you mean adding two waves with frequencies 1000Hz and 2003 Hz, for example.
You don't need to have a harmonic of the lower frequency (2000 Hz) to get something like beats. If you add the two waves and plot the result you'll see that the amplitude changes slowly in time.

I thought the reason beats are produced is because if they are only a few hz different, they will go out of phase and since they are almost the same frequency, they go up and down exactly oppositely for quite a long time, going through the cycle 1 times per second if they are 1 hz diffrent, cancelling each other out and then what we hear is the amplitude lowering, increasing, lowering etc.

But if they are a 1003 hz apart, how do they cancel each other out, if they aren't producing any harmonics? Since they wouldn't be going up and down together for too long, shouldn't the result be just some flickering? Would you like to expand on this a little or maybe give a link about it? If I play supposedly pure sine waves from my computer through my headphones, with frequencies 100 and 152, I hear a beat at 4 bps, despite the fact that they shouldn't be producing any harmonics. How does it work? At what frequency would the beating be heard in the case of the sine waves with no harmonics?

I don't want to hijack this thread, but another question I have about harmonics is about vowels. The vocal chords always produce the same sound, assuming you speak at the same tone, right? But when I look at a spectogram of the vowels, the harmonics are completely different.

Like this: http://en.citizendium.org/wiki/Vowel

I thought simple vibrating vocal chords always produce the same harmonics and you can simply change the volume of them by changing the shape of the resonating areas, eg your mouth, throat etc. But the harmonics themselves change drastically, fundamental going down while some harmonic is going up and various other combinations. How does it work?

 

[sophiecentaur]

I think that the terms are not being used correctly or consistently in this thread. It could be confusing. 'Harmonics' are just a way of describing a simple repeating waveform which is based on a single fundamental sinusoid with other components at multiples of the fundamental. Not all waveforms are the slightest bit repetitive and not all waveforms are made up of a single sinusoid plus harmonics. All waveforms can have a Fourier Analysis carried out on them but the result is not necessarily meaningful, as you always need to assume some repetition (i.e. you need to 'loop' it) But you can certainly come up with a Frequency Domain Analysis, which is what a spectrum analyser does.

Why do we hear 'beats'? Two or more signals will pass through a perfectly linear channel (amplifier / recording / radio link) and emerge from the other end exactly as they started. But, once a non-linearity is introduced (In which the result of adding the two signals is not exactly equal to their sum) a process called intermodulation takes place (a bit like multiplying rather than simple addition). The result of passing 1kHz and 1.03kHz signals through a nonlinearity will be to produce products at 1, 1.03, 0,03, 2, 1.06 2.03 kHz and many more, all at different levels. Our Ears are, bless-em, not very linear and produce a lot of intermodulation products so we hear beats between notes, particularly at high levels. This is not a matter just of Harmonics - the two input signals could be pure sinusoids - but harmonics and ip's are generated in any non-linear channel.

 

[olivermsun]

If I play supposedly pure sine waves from my computer through my headphones, with frequencies 100 and 152, I hear a beat at 4 bps, despite the fact that they shouldn't be producing any harmonics. How does it work? At what frequency would the beating be heard in the case of the sine waves with no harmonics?

You understand the general idea. Here is a good explanation of the modulation in the amplitude (the "envelope") which causes the perception of "beats," in terms of trig identities.
http://en.wikipedia.org/wiki/Beat_(acoustics )

 

[Nikitin]

Sinus wave or cosines wave it doesn't really matter does it? Cos and Sin are just nice commands that can graphically draw a wave.

You can also play around with the function and thus change the wavelength & amplitude of ur lil wave.

 

[olivermsun]

There are two issues here.
Firstly, you can make up any shape of electromagnetic wave (continuous or not) by adding together combinations of other waves. This is the idea of Fourier Analysis, which breaks a signal down into a combination of sinusoids.

Just to amend this, you cannot actually do this for non continuous waveforms. This is a fundamental result of Fourier theory which is often overlooked. You can make up something which approximates the waveform shape, but for a shape such as a square pulse, you cannot even get pointwise convergence.

It works just fine for a signal with limited bandwidth … but that is basically saying that the waveform can be represented by a finite Fourier series (i.e., frequencies only within some limits).

Why do we hear 'beats'? Two or more signals will pass through a perfectly linear channel (amplifier / recording / radio link) and emerge from the other end exactly as they started. But, once a non-linearity is introduced (In which the result of adding the two signals is not exactly equal to their sum) a process called intermodulation takes place (a bit like multiplying rather than simple addition)

You can hear beats as a result of linear theory, as seen from the trigonometric derivation in the Wiki article. There is no need to invoke nonlinearities.

 

[olivermsun]

Sinus wave or cosines wave it doesn't really matter does it? Cos and Sin are just nice commands that can graphically draw a wave.

You can also play around with the function and thus change the wavelength & amplitude of ur lil wave.

As sophiecentaur mentioned in a previous post, sinusoidal waves have some very special properties, making them convenient for Fourier transforms, etc. For example, sin mx and sin nx are orthogonal -- you don't need to do anything special to prevent "overlap" or "counting twice" when you decompose a signal into a series of sinusoidal frequency components. Also, the derivative of sin x is just cos x -- which is the same as sin plus a phase shift. This means that the derivative of a sinusoidal wave component only results in a phase shift, but the frequency is unchanged.

Finally, exp ix = cos x + i sin x , which nicely encapsulates those properties!

 

[Nikitin]

edit: nvm, I found out the answer to a question I was about to ask.
thx 4 correcting me anyhow

 

[chingel]

Why do we hear 'beats'? Two or more signals will pass through a perfectly linear channel (amplifier / recording / radio link) and emerge from the other end exactly as they started. But, once a non-linearity is introduced (In which the result of adding the two signals is not exactly equal to their sum) a process called intermodulation takes place (a bit like multiplying rather than simple addition). The result of passing 1kHz and 1.03kHz signals through a nonlinearity will be to produce products at 1, 1.03, 0,03, 2, 1.06 2.03 kHz and many more, all at different levels. Our Ears are, bless-em, not very linear and produce a lot of intermodulation products so we hear beats between notes, particularly at high levels. This is not a matter just of Harmonics - the two input signals could be pure sinusoids - but harmonics and ip's are generated in any non-linear channel.

Are you basically saying the beats are an illusion? If I play the 100 hz and 152 hz tones with no harmonics, I believe I hear a beat of 4 hz at the frequency of 100 hz, ie the loudness of the 100 hz tone goes up and down. Is this an illusion or is this actually happening? Does the 152 hz tone somehow interfere with the 100 hz tone or is it our ears playing tricks?

If we can detect the harmonics of photons, is it possible to literally see with human eyes the harmonic of a lightwave that is in infrared, assuming the person doesn't have glasses?

 

[sophiecentaur]

Forget about the harmonics. Even if the two input signals were pure enough to show no harmonic content then you just get products with frequencies which are combinations in the form nf1 + mf2 where m and n can be any integers, positive or negative. Depending on the 'law' of the non-linearity, the amplitudes of the even and odd components will be different.

As for "illusion", you could say that the shortcomings of our sensors cause illusions if you like - or just sensations. You could say that our awareness of harmony and discord are just illusions in the same way. If the input signal only contains two frequency components and we 'hear' more than that then I suppose it's an illusion. Just words.

 

[lovetruth]

The reason that prism is "prejudice" toward sine wave is that what scatters light into different direction in the prism are motions of charged particles such as atom or electron. The electric field of light drives them slightly away from their equilibrium position. And they responds by oscillating sinusoidally just like any small oscillator. These oscillating charge then omits sinusoidal EM waves.

When you send in a triangular wave, the charge particles that it interacts with do the same thing. But the superposition of the incoming wave and the induced wave will no longer be triangular.

Why does oscillating charge emits sine wave? Oscillation can mean non-harmonic waves too.
Also we do not know how the atom works exactly, how proton electron interacts. People will say QM explains atom. QM explains atom but not 100% correct. Has the QM able to derive Moseley law ab-initio? No, QM is not perfect.

 

[Dale]

Why does oscillating charge emits sine wave? Oscillation can mean non-harmonic waves too.

An oscillation means that you have some sort of restoring force, or in other words the potential has a minimum. If you do a Taylor series expansion of ANY smooth potential in a small region around a minimum then you get a constant term and a quadratic term. This leads to a sine wave oscillation. You only get deviations from a sine wave for large oscillations where higher order terms become important. This is not peculiar to light, but applies to all systems with a smooth potential.

 

[sophiecentaur]

Photons of harmonics need 2, 3, 4 times the energy. Not all light producing mechanisms have enough energy for this. This discussion would be better angled at lower frequency em radiations, which often have harmonics.

 

[sophiecentaur]

@ oliversum
My quote:
"Firstly, you can make up any shape of electromagnetic wave (continuous or not) by adding together combinations of other waves. This is the idea of Fourier Analysis, which breaks a signal down into a combination of sinusoids."

Your quote
"Just to amend this, you cannot actually do this for non continuous waveforms. This is a fundamental result of Fourier theory which is often overlooked. You can make up something which approximates the waveform shape, but for a shape such as a square pulse, you cannot even get pointwise convergence."

I was careful not to commit myself to saying that you can synthesise any waveform using continuous waveforms. Non- continuous waves can add together to form another non-continuous wave. That's just superposition and nothing very clever. At some stage I actually pointed out that you need to 'loop' a non continuous signal if you want to do a Fourier on it (forcing it to be continuous). Done 'tastefully' that is often a very satisfactory method of analysis - particularly if you use 'windowing'.

Your quote "You can hear beats as a result of linear theory, as seen from the trigonometric derivation in the Wiki article. There is no need to invoke nonlinearities."

I couldn't find this but, in any case, if you say that we can "hear" a beat, how can you insist that our hearing is linear? Certainly, if you add two sinewaves linearly, you get nothing but two sinewaves. The trig identity says nothing more than just expressing the sum of two trig functions in another way - nothing about our sensations.

 

[olivermsun]

I was careful not to commit myself to saying that you can synthesise any waveform using continuous waveforms. Non- continuous waves can add together to form another non-continuous wave.

For combinations of sines and cosines, which you mentioned, the difficulty is in synthesizing discontinuous functions. You are right in that, technically, one could "synthesize" any waveform by starting with a basis is specialized enough (e.g., the basis contains exactly that waveform). However, this is not a very helpful way of thinking about Fourier theory, since the concept is one of representing arbitrary functions using the same, generally useful basis.

At some stage I actually pointed out that you need to 'loop' a non continuous signal if you want to do a Fourier on it (forcing it to be continuous). Done 'tastefully' that is often a very satisfactory method of analysis - particularly if you use 'windowing'.

Whenever you have a finite record, then unless the signal is exactly periodic such that it matches the record length, you will introduce a discontinuity by "looping" (mapping the record to the unit circle). Windowing helps limit the frequency leakage that results, but this not the same as dealing with some inherent discontinuity in the function itself.

Put another way: How could you loop a wavetrain of square pulses so that you are "forcing it to be continuous"?

Your quote "You can hear beats as a result of linear theory, as seen from the trigonometric derivation in the Wiki article. There is no need to invoke nonlinearities."

I couldn't find this but, in any case, if you say that we can "hear" a beat, how can you insist that our hearing is linear?

I insisted nothing of the sort. What I said is that the beating phenomenon does not require nonlinearity, because the linear superposition of two very closely spaced frequencies is exactly the same as a waveform of the "average" frequency but with an amplitude varying at a much lower frequency. The effect of this low frequency amplitude modulation is what is commonly described as "beating."

Nonlinearities which create information at more frequencies could certainly allow more different "beats."

 

[sophiecentaur]

I see where you're coming from but the closeness or otherwise makes no difference to the sums.
The waveform is not the same as for amplitude modulation, with a real 'envelope'. The beating sensation is just that: a sensation. An electronic device would not hear anything other than two separate notes. If you disagree with that, I would ask you at what separation would it change its mind about what its input was?
I say that the beating is merely an artifact of our hearing.
I think we may have just to agree to disagree on this though.

 

[olivermsun]

I see where you're coming from but the closeness or otherwise makes no difference to the sums.

Of course it makes a difference (no pun intended). The beats themselves are interesting precisely because they appear on a time scale which seems so different from the waves which produce them. It's the closeness of the two frequencies which controls the slowness of the beats.

The waveform is not the same as for amplitude modulation, with a real 'envelope'.

It is exactly what happens in amplitude modulation. It's also the same phenomenon behind the spring and neap tides.

The beating sensation is just that: a sensation. An electronic device would not hear anything other than two separate notes. If you disagree with that, I would ask you at what separation would it change its mind about what its input was?

Again, the carrier and envelope are exactly the same thing as the superposition of separate frequencies. That's what the trigonometric identities tell us. An electronic instrument would be able to tell you the same information in two different ways: that you have the sum of two distinct frequencies and that the power in the signal has a slow "beating."

 

[sophiecentaur]

Look up "amplitude modulation". You will see that the spectrum consists of a carrier and Two sidebands. Also, the waveform is very different in that there is no phase change of the carrier nor a zero crossing of the envelope. You need a non linearity (diode detector etc.) to get the envelope signal from it.
AM is a good example of what I am saying in that it is detected using a non linearity.

Also, how would this machine measure the Power without using a non linear process? Power would be V², which looks pretty non-linear to me!