Why light is a sinusoidal wave?

[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!

 

[chingel]

If I generate two sound waves with my computer with a few hz difference, the output volume meter definitely goes up and down.

I made a picture of what adding 100 and 152 hz sine waves together looks like:

http://www.freeimagehosting.net/newuploads/ab870.jpg

There is definitely the pattern of the amplitude going up and down 4 hz. Not as dramatic as adding sine waves 100 and 104, where the beat amplitude would go all the way, but it seems that just the addition of the sine waves creates the beating pattern.

Do I understand correctly that basically our ears do a Fourier transform on the sound? Then why don't they just break the sound down into two sine waves and hear them with a constant tone? Is it because the ears don't know they are supposed to do the transform on a 0,25 second chunk, but instead they do it in real time, and they don't know the waves are out of phase and it would look just like the waves have lower amplitudes, or is this a wrong idea?

 

[sophiecentaur]

Our ears detect the power spectrum, which loses the phase information. Part of our processing adds up the powers - hence we can tell the difference in loudness between one singer and a whole choir, which is obvious. But this power measurement involves a non linearity. I may be being picky (heaven forefend) but the fact is that there is NO low frequency component in the linear sum of two sine waves. There is NO low frequency component in an amplitude modulated signal either. (AM signals are all band pass filtered at many stages in the broadcasting process yet they still survive without any low frequency em waves getting to your house) The only way you can 'demodulate' any of these high frequency waveforms is with a non linear process. This may be hidden or 'implied' but it still has to be there.
Of course you can plot the voltage variations of a signal in time and show an overall variation in amplitude. When you look at the graph, you are performing all sorts of data manipulation of the information with your brain. That is all full of non-linear processes (decisions and pattern spotting), which are being ignored in your argument.
Either the LF signal is there or it isn't - and it isn't - so, if we hear it (or can see it on a graph) there must have been some non linear process going on. We can't always believe the evidence of our eyes or our ears.

 

[olivermsun]

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.

The traditional amplitude modulation leaves some constant (useful for some practical reasons), so the signal has the form
$$[ A + f(t) ] \cos(\omega_c t).$$
If the content signal to be encoded is
$$f(t) = \sin(\omega t) ,$$ then the modulated signal works out to be
$$A \cos(\omega_c t) + \sin(\omega t) \cos(\omega_c t)$$
where the first term is the carrier, one recognizes the second term as the quantity we have been discussing through this thread: the sum of two closely spaced frequencies which are the "sidebands". You may choose A = 0 as well and "suppress" the carrier (since it contains no information (other than, "there is a signal centered here") thereby saving a considerable amount of energy in transmission.

A quick lookup of "amplitude modulation" returns lots of pictures which show different types of modulated signals for various values of A.

You need a non linearity (diode detector etc.) to get the envelope signal from it.
...
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!

Your electronic detector would need nonlinearity to do a Fourier transform (or its equivalent) and decompose the signals into two linear waves in the first place, so I don't understand your point.

 

[olivermsun]

Do I understand correctly that basically our ears do a Fourier transform on the sound? Then why don't they just break the sound down into two sine waves and hear them with a constant tone?

There is a limit to our ears' resolution. If two tones are quite far apart, e.g., your 100 and 152 Hz example, then usually people do hear them as separate tones but sometimes also perceive one or more "interaction tones" or beats. If the tones are very close, however, most people can only perceive the slow beats.

 

[sophiecentaur]

My point was one of those 'what's really there' things. Some views seemed to be that the beat was really there because you can hear it. My view is that the whole of the signal (i.e. the two tones or whatever) is just those two tones and that any beat that you can 'see' or hear will only be there after some non-linear process. No piece of linear electronic gear could be 'aware' of or detect a beat, so I conclude that our senses must be non-linear. And, of course, with our basically logarithmic sensitivity to most inputs, they clearly are.
Perhaps I just find the Maths of this sort of thing so compelling that I just don't feel any need for the subjective approach. I may just have been involved in it for so long. . . .

As for the perception of beats vs tones, I think that we essentially hear an intermod tone as a tone if we would have heard a tone at that frequency. Beats are at sub-sonic frequencies which we are usually aware of as 'vibrations' via our touch senses, rather than our ears. In pre-history, we wouldn't have been exposed to many such beats (there are not many natural sources of cw tones) and the evolution of our hearing system could well have meant that beats confuse us.

 

[lovetruth]

Can anyone conclusively prove that light coming out of prism is sine wave and not any non-harmonic wave.

 

[sophiecentaur]

If there are no higher energy components, how can it be other than a sinusoid? I don't see where you are coming from about this.