# Let he who be without sine...June 28, 2008 7:00 PM   Subscribe

Why are sine waves considered "pure" tones? Why do we consider sinusoids the building blocks of periodic functions?

When analyzing physical, electronic, and acoustic/musical phenomena, it's often handy to view things in the frequency domain. I understand the basic idea of Fourier Analysis, namely breaking apart periodic functions in to the sums of sinusoids of given frequency and phase.

What I don't get is why or how sinusoids are "special". I know in the real world a great number of objects are essentially complicated dampened spring oscillators, so sinusoids are common in nature. Is that the end of it, or is there some deeper reason this function is good at representing periodic phenomena?

Isn't it possible to do a fourier analysis that breaks an arbitrary periodic function in to the sum of any other periodic function? Why not square waves, or a sawtooth?

While sine waves sound "pure" to me, it seems highly subjective. Is there something more than convention at work here? Some property of the ear?

The human ear can recognize single sine waves because sounds with such a waveform sound "clean" or "clear" to humans; some sounds that approximate a pure sine wave are whistling, a crystal glass set to vibrate by running a wet finger around its rim, and the sound made by a tuning fork.

This seems arbitrary though.

We say that the timbre of a violin is complex because it has many different harmonics. But the very idea of harmonic content assumes some basis function, right? Could we just as easily say a sine wave is complex by choosing square waves as our basis of spectral analysis?

So is this just a convention, and if so, where do we get it?
posted by phrontist to science & nature (35 answers total) 26 users marked this as a favorite

s that the end of it, or is there some deeper reason this function is good at representing periodic phenomena?

there was a brief moment of time in my life in college when I could see the full glory of the sinusoid, its mapping from the mathematical complex/imaginary space, the inter-relations of the transcendental functions, and fourier series decomposition.

To answer your question, square waves, triangular waves and sawtooth waves can all be composed from adding together the harmonics of sine waves.

The sinusoid, being the ratio of a uniformly rotating radius' components, is the purest repetitive series. Any other signal is not as "clean" ie. contains a degree of discontinuity in the signal.
posted by yort at 7:26 PM on June 28, 2008

I think it has something to do with the sine wave being the "smoothest" possible periodic function; both the function itself and all of its derivatives are continuous, whereas something like a sawtooth or square wave would have discontinuities in the first derivative.
posted by dixie flatline at 7:27 PM on June 28, 2008

The sine wave is arbitrary, other than yes its multitudinous physical analogues. Functional decomposition merely requires orthogonality of the basis functions of which there are numerous possible families, just few with as many close physical analogues.

From a computational standpoint for me at least the sawtooth is the purest waveform as its the easiest to synthesize.
posted by norabarnacl3 at 7:37 PM on June 28, 2008

It likely is the most efficient waveform for vibrating a string or membrane, like the eardrum.
posted by weapons-grade pandemonium at 7:37 PM on June 28, 2008

In physics, "periodic motion" is a term used to describe a vibration where an object moves back and forth. In particular, it describes a particular motion where the force on an object rises proportionally with how far away it is from the center point. Examples are the weight on the end of a pendulum, or a weight hanging from a spring.

Plotting the position of such an object against time, its movement is a sine wave. It turns out that the motion of the teeth of a tuning fork is also an example of periodic motion.

Other reasons: "resonance" refers to the way that vibrations can form standing waves. In a cone, a square wave or a sawtooth will form resonances in a large number of places, but a sine wave will only form a resonance in a single place.

And why is that interesting? Because the cochlea is such a cone, wrapped up into a spiral. The way our ears differentiate different frequencies is by little sensor nerves which have hair-like endings that stick up into that cone in different places. Whereever resonance forms, those hair endings get stimulated.

So a sine wave is a "pure tone" because it only registers at a single point in the cochlea. Other wave forms are not "pure" because they register at a lot of points.

Having said that, mathematically a sine wave is not unique. You can create arbitrary wave forms by adding together any particular shape -- and when we digitize music, we use square waves rather than sine waves.

Moving right along: another reason that sine waves are considered fundamental has to do with how low-pass filters work. If you feed a square wave or a sawtooth or a spike through an ideal low-pass filter, what you get out is a sine wave.)
posted by Class Goat at 7:39 PM on June 28, 2008 [2 favorites]

(...what I should have said...) and the reason that low pass filters are important is that essentially all real-time communications media (air, wire, radio) implicitly include low pass filters. In a communication channel with low bandwidth, a sine wave can be transmitted without distortion. Any other wave form will be distorted (because of loss of overtones). Perfect transmission of square waves and sawtooth waves requires infinite bandwidth.
posted by Class Goat at 7:45 PM on June 28, 2008 [1 favorite]

Class Goat: It was thinking about those phenomena that prompted me to wonder this in the first place. Your explanation of the functioning of the cochlea is great! I'll have to read up on that...

The sinusoid, being the ratio of a uniformly rotating radius' components, is the purest repetitive series. Any other signal is not as "clean" ie. contains a degree of discontinuity in the signal.

I think it has something to do with the sine wave being the "smoothest" possible periodic function; both the function itself and all of its derivatives are continuous, whereas something like a sawtooth or square wave would have discontinuities in the first derivative.

But there are other that are other functions that have those properties.

The sine wave is arbitrary, other than yes its multitudinous physical analogues. Functional decomposition merely requires orthogonality of the basis functions of which there are numerous possible families, just few with as many close physical analogues.

So in summary (and correct me if I'm wrong):

The sine wave is not mathematically special in terms of decomposing signals. It's useful though, because all sorts of physical systems (capacitors, walls, liquids, strings, our ear innards) react to sinusoidal forces in "special" ways (the unifying underlying behavior being distance-proportional restorative forces).
posted by phrontist at 7:53 PM on June 28, 2008

But there are other that are other functions that have those properties.

I'm pretty sure there is something that is mathematically most "simple" about a sine wave. The presence of discontinuities in other functions makes them more complex in some way (that I'm sure can be more precisely defined mathematically with the proper background).
posted by dixie flatline at 7:57 PM on June 28, 2008

The sine wave, at least in music, is considered a pure wave because a single, perfect sine wave produces no harmonics.

A sawtooth wave with a fundamental (f) 1 of amplitude (a) at 1 will produce harmonics at 2f@1/2a, 3f@1/3a, and so on.

A square wave will produce harmonics at 3f@1/3a, 5f@1/5a, and so on.

A triangle wave will produce harmonics at 3f@1/9a, 5f@1/25a, and so on.

But a sine wave with no distortion in the signal path doesn't produce any harmonics.

That's why it's pure.
posted by ochenk at 8:01 PM on June 28, 2008

It's not really that physical systems "react to sinusoidal forces"; it's that the behaviour of many, many systems is sinusoidal. It's the nature of the universe.
posted by ssg at 8:04 PM on June 28, 2008

(...and what I also should have said...)

The fact that the cochlea works the way it does has all kinds of interesting consequences for sound engineers. What that means is that our ears are real-time spectrum analyzers. The way we differentiate between the sound of a clarinet and the sound of a trumpet is by decomposing the sound into sine waves and analyzing the pattern of overtones. Even if they're playing the same note, they don't sound the same because their Fourier decompositions are different, and that's what our ears implicitly do in real time.

That's why MP3 can get the kind of compression it does without apparent distortion. The goal of the codec is not to reproduce the waveform. It is to reproduce the spectrum, because that's all we can hear. So an MP3 does a Fourier conversion on the sound and stores the coefficients.

But only half of them. It stores all the overtone amplitudes (subject to a quality cutoff) but it doesn't store any of the overtone phase relationships, because we can't hear that. That's one of the two biggest reasons why MP3 can get the kind of compression it does.

If you take the original source and and MP3 reproduction and feed them into a spectrum analyzer, they'll look nearly identical. But if you feed them into an oscilloscope, they won't bear any resemblance at all.
posted by Class Goat at 8:05 PM on June 28, 2008 [4 favorites]

Boy... this question is phrased differently than I have ever heard such a question.

My first reaction is to say "math is not nature". Periodicity of a huge number of natural things can be described by a sine function, but the sine function is ours. We are the ones using it to make sense of the world. With it appearing in so many places, why wouldn't our descriptive language address it predominantly? It seems sort of self evident.

I mean, our color vocabulary is restricted to visible wavelengths, right? We don't play music in ultrasonic ranges beyond our hearing.

It's a nice mental exercise to consider if there's something else out there that would work better than sines, and sums of sines. I am too simple minded to deal with anything other than sines, though. Fortunately, that works pretty well for my universe.

(I'm gonna mark this Q as favorite just to see if some brainier critter has a good answer for you.)
posted by FauxScot at 8:06 PM on June 28, 2008

ochenk, yes, but that is a circular argument, because "harmonics" are just projections onto sine waves. You could use triangular waves as your basis and define "triangular harmonics", and then you'd be able to say that a triangular wave is the only one with no triangular harmonics.
posted by dixie flatline at 8:06 PM on June 28, 2008

dixie flatline: Hmmm, perhaps I spoke to soon. My hunch is that you could find a (or more likely, a class of) polynomial function that, defined piecewise would be periodic, differential, and continuous at all points. Maybe norabarnacl3 can elaborate?

Assuming the sine/cosine functions are unique in that regard, I still don't understand what the significance of that would be.
posted by phrontist at 8:09 PM on June 28, 2008

You can build sines out of square waves (that's what really crappy DC to AC inverters look like), and vice versa (although you're getting asymptotically close in both cases), so there's no building block in terms of composition.

Part of it, I guess, is that when you take the derivative, nothing blows up on a sine function. With a sawtooth wave, at the tip of each, well ... there's no smooth transition in the math.

From the standpoint of physics, without damping, springs and whatnot oscillate in a sine wave. The ON OFF ON OFF of a square wave just isn't something you see a whole lot of in nature, nor the sawtooth wave. Nature is filled with examples of sinusoidal oscillations, but I'm scratching my head looking for natural incidents of sawtooth.
posted by adipocere at 8:14 PM on June 28, 2008

But there are other that are other functions that have those properties.

Arguably square waves (or more generally, step functions) are simpler in some ways than sine waves, since they're easier to store in digital form.

But a sine wave with no distortion in the signal path doesn't produce any harmonics.

Your statement begs the question. A "harmonic" is defined to be an overtone in a Fourier analysis, which is an analysis in terms of sine waves. By definition, a single frequency sine wave has no harmonics in a Fourier analysis.

There is no such thing as an overtone in a digitized waveform. The concept of "overtone" doesn't apply.
posted by Class Goat at 8:15 PM on June 28, 2008

ssg: It's not really that physical systems "react to sinusoidal forces"; it's that the behaviour of many, many systems is sinusoidal. It's the nature of the universe.

I think your phrasing is better, but I don't think our statements disagree.

ochenk: You are begging the question.
posted by phrontist at 8:17 PM on June 28, 2008

Periodicity of a huge number of natural things can be described by a sine function, but the sine function is ours. We are the ones using it to make sense of the world. With it appearing in so many places, why wouldn't our descriptive language address it predominantly? It seems sort of self evident.

Um, does mathematics exist independent of our ability to think about it? (How many angels can dance on the head of...)

If an object is moving in a circle at a constant velocity (which, for instance, is the case in a circular orbit) then its movement in one dimension will be a sine wave. In the other dimension it will be a sine wave one quarter of a cycle out of phase.

Sine waves have a unique property in calculus: the fourth derivative of a sine wave is identical to the original wave. (Which is to say, the derivative of a sine wave is an identical sine wave with a phase shift of one quarter.)
posted by Class Goat at 8:20 PM on June 28, 2008

Assuming the sine/cosine functions are unique in that regard, I still don't understand what the significance of that would be.

I'm just saying I believe there's a definite mathematical sense in which sine is the "simplest" (or "smoothest") periodic function (by nature of the continuity of its derivatives), and that this holds regardless of whether the sine wave does or does not appear in nature.

I think there's something that can be seen here without bringing any properties of nature into the argument. The sine wave would be the solution to a simple differential equation regardless of whether that equation refers to simple harmonic motion in physics. Square and sawtooth waves don't tend to "fall out" of the math that easily.

Arguably square waves (or more generally, step functions) are simpler in some ways than sine waves, since they're easier to store in digital form.

It could be that "simple" is not a good word (originally I said "smoothest"), but I was referring to all the derivatives being continuous.
posted by dixie flatline at 8:23 PM on June 28, 2008

adipocere: Heaviside functions can be said to represent some electrical phenomena (in reality, a totally discontinuous jump is not possible). The integral of that is a ramp function. That's basically how sawtooth wave generators work electrically. I'm not sure if that counts as "natural".

You can build sines out of square waves

If you squint! But yeah, mathematically, an infinitude of square waves can be made to add up to a sine (I haven't proved this myself, but mathematical sources I trust claim it to be the case).
posted by phrontist at 8:23 PM on June 28, 2008

I'm just saying I believe there's a definite mathematical sense in which sine is the "simplest" (or "smoothest") periodic function (by nature of the continuity of its derivatives), and that this holds regardless of whether the sine wave does or does not appear in nature.

Right. This is pretty much what I was trying to ask, and I'm inclined to agree with you, but I can't justify it.
posted by phrontist at 8:29 PM on June 28, 2008

The sine wave is not mathematically special in terms of decomposing signals. It's useful though, because all sorts of physical systems (capacitors, walls, liquids, strings, our ear innards) react to sinusoidal forces in "special" ways (the unifying underlying behavior being distance-proportional restorative forces).

That's mostly true but not really quite. I would use an analogy:

In physics as redesigned by Einstein, the formulas describing behavior depend entirely on the frame of reference you choose. It's a arbitrary point which for purposes of analysis is defined to be unmoving, so that all objects you are considering are measured in position and velocity relative to that arbitrary fixed point.

Under relativity, it's possible to describe a physical system based on any frame of reference. But some frames of reference result in fiendishly complicated measured behaviors and formulas, whereas other frames of reference can be quite simple and straightforward. Relativity says that none of them are special, because there is no universal frame of reference, but as a practical matter the difficulty of analyzing the system depends enormously on what frame of reference you choose.

That's the same thing here. Yeah, it's true that you can choose any arbitrary waveform to be your "pure" form, and to analyze more complicated forms in terms of that one, but how easy or hard you have made your work depends a lot on which precise waveform you've chosen as your "pure" one.

In many, many situations, the analysis is most simple if you base your analysis on sine waves. Partly that's because, as mentioned, many physical vibrations do behave sinusoidally, and partly that's because there's a large body of knowledge in mathematics involved in doing all this with sine waves.

On the other hand, there are situations where other shapes are more useful. The other important one is the step function, which became important with the development of digital computers.
posted by Class Goat at 8:30 PM on June 28, 2008

Wavelets are an alternative to Fourier analysis using sine waves:

The wavelet transform is often compared with the Fourier transform, in which signals are represented as a sum of sinusoids. The main difference is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. The Short-time Fourier transform (STFT) is also time and frequency localized but there are issues with the frequency time resolution and wavelets often give a better signal representation using Multiresolution analysis.

The discrete wavelet transform is also less computationally complex, taking O(N) time as compared to O(N log N) for the fast Fourier transform. This computational advantage is not inherent to the transform, but reflects the choice of a logarithmic division of frequency, in contrast to the equally spaced frequency divisions of the FFT.

What do you call an expert in the use of wavelets?

A shun sine superman, of course.
posted by jamjam at 8:35 PM on June 28, 2008

Imagine a point on a vertical circle that's turning at a steady speed. The shadow that the point casts on the ground is a sine wave (when the circle is moved horizontally at a steady speed - I wish it easier to describe.) It's a very simple curve, the most basic repetitive curve really.

(Thankyou jamjam, excellent! Disclaimer: I'm not an expert in the use of wavelets.)
posted by anadem at 8:55 PM on June 28, 2008

anadem: I always describe it as light put on the end of a helicopter rotor. If the helicopter moves vertically, it the light traces out a sine. Not all that related to the question though.
posted by phrontist at 9:03 PM on June 28, 2008

Standard deviations = sqrt(1/n * sum[(xi - mean(X))^2])

Mean absolute deviation
= 1/n * sum[ | xi ? mean(X) | ]

Everyone works with standard deviation. Why? The formula for mean absolute deviation is simpler, no?

Because the absolute value operator is a pain to work with. When writing a proof, each time you hit an absolute value you have to break into a case analysis. If those cases have absolute values themselves, then your case is that cases, and so on. It gets out of hand very quickly. Squaring and square roots are cakewalks in comparison.

Some people argue that mean absolute deviation is a better measure of dispersion than standard deviation. Squaring gives too much weight to outliers. They argue that we should stop being lazy, works through to hard math, and build an alternative set of statistical tools that works with mean absolute deviation.

Sine waves have no such downside. They capture physical phenomenons with unsurpassable precision, while still being a joy to work with algebraically. And I do mean joy, literally. They be interact with other math tools so beautifully, they make atheists wonder if there's a God. I mean, look at Euler's formula

e^(ix) = cos(x) +isin(x)

and it's corollary, Euler's identity, which is perhaps the most beautiful equation in all of mathematics,

e^(i*pi) + 1 = 0

doesn't that make you weep?

This is the tool you want to me give up? You want me to replace it with a sawthoot, which involves two absolute value operations per cycle? Over my dead body!
posted by gmarceau at 9:59 PM on June 28, 2008 [1 favorite]

For all of you questioning the existence of other periodic functions with lots of derivatives---there are as many as you could conceivably ask for. You can have a triangle wave with its tips rounded off---as small as you like, mind you---which will still be smooth, with all its derivatives existing.

If you're looking for analyticity, then it's another matter. But smooth? That's easy.

(Also, the sine function isn't unique in that its fourth derivative is equal to itself---there are at least three other such functions [ok, arguably only one other, depending on if you care about phase]. Can you recall which ones?)
posted by vernondalhart at 12:09 AM on June 29, 2008

Is it even possible to actually generate a non sinusoidal wave? If your system is trying to generate a square wave for example what you tend to end up with is a very close approximation made out of sine waves right? Isn't the capacitance of the medium going to smooth out your sharp transitions into smooth sine waves? Maybe if your entire system is super conducting? I Am Not A Signals Engineer.
posted by public at 4:54 AM on June 29, 2008

the sine function isn't unique in that its fourth derivative is equal to itself---there are at least three other such functions [ok, arguably only one other, depending on if you care about phase]. Can you recall which ones?

Well, exp, sin, and cos, but as pointed out by gmarceau, they're all the same :-)
Which one am I missing?
posted by lukemeister at 8:34 AM on June 29, 2008

If your system is trying to generate a square wave for example what you tend to end up with is a very close approximation made out of sine waves right? Isn't the capacitance of the medium going to smooth out your sharp transitions into smooth sine waves?

Not exactly. We can view that transition as being made up of component sinusoids (when thinking of things in the frequency domain), but in the time domain it's exponential.
posted by phrontist at 9:22 AM on June 29, 2008

Also, the sine function isn't unique in that its fourth derivative is equal to itself---there are at least three other such functions [ok, arguably only one other, depending on if you care about phase]. Can you recall which ones?

F(x)=e^x and F(x)=0 are also like that.

But F(x)=e^x isn't periodic. And F(x)=0 arguably is a degenerate sine wave with amplitude zero.

Also, for purposes of this discussion I consider cosine and sine to be the same thing, since the only difference is phase.
posted by Class Goat at 9:42 AM on June 29, 2008

sinusoidal functions are the simplest periodic function which form an orthogonal basis set, which is required for Fourier transforms. Although it is, of course, possible to approximate sinusoidals from other functions (it's an inverse Fourier transform to go from, say, a square wave or sawtooth to sinusoidals), it is significantly less straightforward, and you lose the mathematical beauty/power of using sinusoidal functions as a basis set.

Regarding the question about orthogonal polynomials: there are several (infinite?) sets, of which the Legendre polynomials are the most straightforward. They are sometimes used as a basis set, but they are not periodic.
posted by JMOZ at 12:03 PM on June 29, 2008

Without delving too deeply into the mysteries, in everyday parlance the word "pure" is simply referring to the fact that a sine wave is considered to consist of a single frequency.

When speaking of, say, a triangle wave, you wouldn't say it consists only of one frequency. You would say that it consists of various frequencies (of sine waves) added together in certain proportions. So it is not "pure"--meaning consisting of a single frequency--but is a compound, consisting of a whole variety of frequencies added together.

In general, any waveform other than a sine wave would be considered to consist of different frequencies of (sine) waves added together.

Now I know I'm just re-stating in more common terms what a lot of people have stated above in terms of Fourier series and all the rest. But that's the common parlance and the common way of thinking about waves, and so that is why we would use a word like "pure" to describe it--because in everyday thinking, a sine wave consists of one "pure" frequency.

Now (as discussed at length above) you can also decompose your waveforms using a lot of different basic waveforms--square, triangle, whatever. And let's say you were using a triangle wave as your basic waveform. In that case you would say a triangle wave is "pure", consisting of only a single frequency, and a sine would would be a complex composite of different triangle waves of various frequencies.

So why do we tend to favor the sine-wave breakdown? As noted in the discussion above, because of the way our ear works, the reduction of a complex sound into various constituent "simple" or "pure" sine-waves is closely analogous to the way our ear decomposes sound into frequencies.

You can choose to decompose waves using triangle waves or square waves or what-have-you, and these can certainly be extremely useful for various purposes, but there wouldn't be a similar correspondence between this mathematical analysis and how we as humans perceive sound. For that reason and others people have given in the discussion above, the sine-wave analysis is likely to be considered the norm.

FWIW the sort of ease and symmetry about the mathematics of the sine-wave decomposition of waves is likely reflected in the type of physics needed to do that type of decomposition. That is, the spiral shape of the cochlea is (in the grand scheme of things) pretty simple. What kind of shape would it take to deconstruct sound waves into constituent triangle or square waves? I don't know the answer to that but I'd be willing to wager it is not a very simple shape.

Similarly, as a musician I can say that the decomposition of sounds into overtones based on the sine wave isn't just imaginary or arbitrary, but something that musicians hear and perceive and work with day in and day out in performing on an instrument.

The overtone decomposition via "pure" sine waves isn't just an abstraction that's as good as any other--in fact the overtone series can be performed quite exactly on a brass instrument like the trombone or trumpet, as well as stringed instruments like the violin. It forms the basis for the operation of all woodwind instruments--for instance, you can use this type of analysis to determine where holes should be placed to obtain various pitches. Overtones are used in tuning, etc. etc.

You can teach pretty much anyone to hear the overtones within a complex tone. For example, touch the string lightly in the center to create an overtone and compare this to the original note a few times--you'll be able to hear the overtone ringing out within the sound of the original note.

Another simple demonstration--silently depress middle C on a piano. Now go down the keyboard, and for each note below middle C play it, moderately loud, for about a half second and then release. What you'll hear when you release the original note is that if middle C is an overtone of that note, it will resonate and you will hear it ringing after you release that note. If it's not an overtone of that note, there will be little or no ringing.

All these properties of "overtones" come from the basic properties of the resonating string or air column that makes the instrument work--and these correspond rather closely with the analysis of sound into frequencies that considers the sine wave a "pure" frequency.

(Interestingly, no real string or air column exactly corresponds to the ideal overtone series, but when you bow a string or use a reed or vibrating lips to drive an air column, it has the effect of "mode locking" the resonances so that they exactly match the harmonic ratios. So you could say that we have such a preference for these harmonic ratios that we even invent ways of playing instruments that bring their imperfect natural expression of the harmonic ratios into alignment with the ideal. In fact you could make a fair case that this effort to "reduce inharmonicity" has been one of the main driving forces behind music instrument development in the western European tradition over the past, say, 1000-1500 years.)

Now a further question could be asked--why do we seem to prefer instruments constructed this way, with the sine wave as their basis? Because it's certainly possible to construct instruments do not have this so-called "harmonic" structure. Lots of instruments lack it--snare drums, cymbals, gongs, wood blocks, etc.

That's not a very easy question to answer, partly because you could make a case that there are some human musical cultures around that do not have this preference. But insofar as we do have this preference, it probably goes back again to the way the human ear works and the way it decomposes sound into frequency--it does it by sine waves.
posted by flug at 3:16 PM on June 29, 2008

The reason physicists teach the mass on a spring problem, and indeed the reason sine waves pop up everywhere, is that sufficiently close to a minimum, virtually any potential is approximately parabolic. This is just a mathematical statement about Taylor expansions: unless your potential has absolutely no x^2 behavior (which most likely means you constructed it that way), then for small displacements from equilibrium, the leading contribution in the expansion goes as x^2. This is exactly the potential involved in the mass on a spring (1/2 k (x-x0)^2 for spring constant k). The solution is a sine.

Therefore, for virtually any arbitrary physical system displaced from a stable equilibrium point, the behavior is sinusoidal. This is why you use sines in the first place to discuss a tuning fork's signal. You can of course decompose the sine into other periodic functions (which don't necessarily, as far as I know, have to be orthogonal--although this certainly makes things easier--only complete), but why write down an infinite number of coefficients when you can write down 1?
posted by dsword at 5:12 PM on June 29, 2008

You have asked two separate but related questions.

A sine wave sounds like a "pure" tone because, as Class Goat has already said above, it stimulates a small number of nerve cells in the ear. Similarly a point of light (or "pixel") stimulates a small number of nerve cells in the eye. These choices have to do with the geometry and physics involved in the ear and eye.

Fourier analysis suggests that you could go back and forth, but because of Other Physical Effects (others have already mentioned low-pass filtering in the air, etc.) the correspondence isn't perfect. It's theoretically possible to reproduce the sound of a violin by adding a bunch of sine waves together, and this is how compressed recordings and synthesizers work. It's theoretically possible to take a large number of violin strings and play them at the same time so that all the overtones cancel and you're left with a sinusoidal waveform. But you would need an enormous number of violins, and the dispersion of the air (that is, different frequency components travel at different speeds) would break things into sine waves anyway. If you worked very hard you could maybe get a pure sinusoidal pressure wave over some small volume of air, but you couldn't control where very well.

I have actually gone the other way: I had a choral conductor who would have us hold a unison note until an overtone sounded, usually the fifth, usually sounding like a tenor. The people standing in the middle of the choir could hear it but those of us standing on the edge could not. We were able to add a large number of slightly different overtone series (our voices) to make the second harmonic carry an intensity comparable to the fundamental, in one direction.

Mathematically, the sine waves are a class of "special functions" which arise from solving the Poisson equation,
(∇2 - 2/∂t2)f = V,
in different coordinate systems. There are about a dozen coordinate systems where this equation is known to separate and you can write f as a product of functions that depend only on x,y,z,t; if V is easy to write down in one of these, then so is f. In rectangular coordinates, where x,y,z all get treated the same, f is a product of sinusoids. In cylindrical coordinates f is a sinusoid as you move parallel to or around the axis of the cylinder (though it has to be periodic as you go around), but f varies like a Bessel function as you move towards or away from the axis. In spherical coordinates the radial solution is a different subset of the Bessel functions, and the angular solutions are called the "spherical harmonics." In toroidal coordinates, where you label your location by two angles and a distance relative to some ring, the solutions have some other name. This is where the Chebyeshev and Hermite polynomials come from too, I think. Since computers have gotten cheap the more esoteric coordinate systems no longer offer much advantage over doing things numerically, so they don't get talked about much any more. But that's the history.

Usually you find expressions for the Bessel functions and the spherical harmonics written as polynomials or in terms of sines and cosines. But that isn't because sines and cosines are more fundamental; they just usually come up first, and people know them better, and sines are pretty easy to do with your hands if you forget how they go. I don't know of anything as pretty as e+1=0 involving Bessel or Hermite functions, but that doesn't mean it isn't there.
posted by fantabulous timewaster at 10:05 AM on July 1, 2008

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