Wikipedia:Reference desk/Archives/Science/2021 November 13

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November 13

How can a wave produce even without current?

This page says: When the current is removed, the field collapses which again sends a wave.

How can a wave produce even without current? Rizosome (talk) 00:05, 13 November 2021 (UTC)

It is the collapsing field, caused by the change in current, that sends a momentary wave.--Shantavira|^{feed me} 08:34, 13 November 2021 (UTC)

The page says that the current flow builds an electromagnetic field around the wire, but for a steady direct current a better description is that it builds a magnetic field around the wire. This magnetic field contains magnetic energy. A variable magnetic field induces an electromagnetic wave (see Maxwell's equations), so when the field collapses, a wave is produced; its energy is the energy that was stored in the magnetic field but is now released. In general, when the current varies, so does the magnetic field, and a wave is emitted. It does not have to be an on–off situation; an alternating current produces a continuous wave.  --Lambiam 13:02, 13 November 2021 (UTC)

Additionally, there can exist a displacement current - a current that does not involve any movement of charge - that is a mathematical representation of the change in electric displacement over time. This "theoretical" current has identical effect on the electric and magnetic field, just as though a true charge was moving. A time-varying displacement current can induce an electromagnetic wave, with- or without- physical movement of charge. Understanding and modeling this effect, and its impact on the full wave equation, becomes critical for the practical engineering design of high-quality antennas, evaluations of interference, design of microelectronics, and so on.

This is in addition - literally, actually added - to the current (the change in charge per unit of time). This key insight is described mathematically, and in its historical context, in many of our articles including the extensded Ampère's Law; also sometimes known as the completed, or corrected, law.

Nimur (talk) 13:45, 13 November 2021 (UTC)

Lambiam says: an alternating current produces a continuous wave. But where exactly this continuous wave produce in radio circuits which was the main topic of this question? Through radio speakers ? Rizosome (talk) 09:06, 14 November 2021 (UTC)

The original question was about producing a wave without current. The answer to the new question is that the radio waves used in communication are produced by a transmitter; our articles Radio wave and Transmitter give further explanations.  --Lambiam 10:49, 14 November 2021 (UTC)

The line: "the radio waves used in communication are produced by a transmitter" solved my doubt. Rizosome (talk) 00:10, 15 November 2021 (UTC)

Resolved

Why are very complex things like geoids modeled as a weighted sum of millions of spherical harmonics?

Why not just make a table? Pixel number 1 is 3, pixel number 2 is 4, pixel number 3 is 4 and so on. If you have enough pixels (RAM is cheap) you don't even need to care about distributing them evenly, it could just be x pixels per circumference in both latitude and longitude.

Sure with the sum of unrelated harmonic shapes way all ∞ points have defined values, to any false precision you want but if you must have that then just define the model value of points not listed as linearly interpolated.

I read that they're only up to the 360th level of spherical harmonics till 2008 (>130,000 coefficients giving only 55-110km resolution) and are barely at 10 nautical mile resolution now in spherical harmonics, can that even show Mount Everest? It must take a painfully long time to brute force 4 million (!) coefficients till it's not worse than the actual data, can someone explain why they bother? For 4 million bytes they could make a lookup table even better than 10 nautical miles to a height precision they can't even measure. Sagittarian Milky Way (talk) 04:21, 13 November 2021 (UTC)

You write "bubbles that are repulsed by each other and attracted to the origin and maybe have size limits or surface tension(?)". I think you misunderstand the nature of mathematical functions. The sine and cosine functions are neither repulsed by or attracted to each other or anything else, and the same holds for the spherical harmonics. Where did you read that the spherical harmonics representation of the geoid only gives you barely a 10-mile resolution? This statement does not make sense. To cover the surface of the geoid with pixels about the size of a circle whose diameter is 10 nautical miles, you already need some 60 million pixels. A geoid is supposed to model what is basically a gravitational equipotential surface, on which Mount Everest does not make a bulge that is noticeable without magnification. EGM2020 gives a 10 cm (3.9 in) accuracy. What more do you want?  --Lambiam 12:43, 13 November 2021 (UTC)

I know what basic trig is, I use 90° minus inverse tangent all the time. Cause I don't have a cotangent button. It's the sphere harmonics I'm very fuzzy on. My chem teacher showed electron cloud shapes and said the shape is cause they can't collapse from attraction cause quantitized energy level reasons and the negative charges repel. For all I know they rarely look like sphere harmonics cause clouds of different energy levels distort each other or something. So what is a circle or sphere harmonic? Why are these shapes the harmonics of a sphere and no others? Shouldn't there be one with 4 eggs in a tetrahedral orientation? Why is there a double tetrahedral thing with 8 eggs? Why is 6 eggs in a plane on the same level as that? Why is level two 3 copies of the same thing, just aligned with x, y or z? Why do some shapes get cloned (even down to orientation and polarity)? It's right there in the section, it says 1/6th degree horizontal resolution. Which would only need 2160*1080 pixels by the way, not 60 million. Or ~41,252*36 if you make them equal area. There are only 200 million square miles on the entire Earth. Sagittarian Milky Way (talk) 00:41, 14 November 2021 (UTC)

This is a why question; the answer is astonishingly easy! People who use geoids prefer using harmonic series. It is literally a matter of preference, colored by the practical ease of use. If you work the math using these mathematical forms, the math is easier. We can use more difficult math to describe the same shape - but this kind of infinite series is the best and easiest math for certain jobs. It is the math we are familiar with, and that we train for, and that we use in many other places, so it works well.

When I learned Calculus, using this famous and popular book, the first several chapters were about sequences and series. My inclination was to skip these and go "straight to the integration and derivatives." I specifically recall one very smart professor telling me that I would not be very good at most of mathematics if I didn't spend a little time learning to love the methods of studying infinite sequences... it is such a succinct and accurate statement, and it rings true. Series representations show up everywhere and they are very helpful for getting real work done.

Nimur (talk) 13:55, 13 November 2021 (UTC)

What kinds of things do model users need to do that's inconvenient even with today's fast computers if done with a lookup table but less so if done with a sum of 4 million individually bad electron cloud shapes? (or large numbers of other shapes in other fields, it's cool that if you had computing time to spare you could find a sum of trig curves that reproduce heyjude.mp3 to better than 1-sigma but the file already exists, would the trig equations be helpful for anything like frequency analysis?)

If millions of mutually-correcting terms is the easiest way to represent a geoid or Google Earth's exact world elevation map with math (I suspected that) then math is just not good at representing it. Especially compared to many other shapes like conic sections of revolution or Pringles®.

Another question: If you used a better resolution model of a hairless standing nude man to model his outer surfaces (skin, eyes, nostrils or ≤~1 nostril diameter into the nose etc) with 4 million 3D graphics triangles and 4 million trig curves and 4 million sphere harmonics how would the 3 fidelities compare? Sagittarian Milky Way (talk) 16:13, 13 November 2021 (UTC)

As it happens, approximating the audio signal as a series of trig functions is exactly how a MP3 file works… because that’s a very efficient and meaningful representation of the data. Your ear does something similar before sending the nerve signals to your brain. —Amble (talk) 17:17, 13 November 2021 (UTC)

I guess that makes sense since a sine wave is the simplest sound and each mp3 curves are apparently only fitted to very short blocks, not the entire song (how hard would it be to brute force a set of sine waves that play for the entire song?) Sagittarian Milky Way (talk) 18:56, 13 November 2021 (UTC)

That would be the DFT of the entire time sequence. Not difficult, but the block structure makes it much easier to do things like on-the-fly encoding and decoding, streaming, and seeking, and it avoids the infinite impulse response (IIR) problem where a big transient like a pop or bang could result in “ringing” through the entire song. —Amble (talk) 19:21, 13 November 2021 (UTC)

Even more than that, a single continuous DFT would not be an efficient representation of the song Bea use it would require a large bit depth (very high numeric precision on every sine and cosine) to avoid degrading the audio. So you would either get a much larger file, or a worse-sounding one, or (most likely) both. A format like MP3 is based on a sophisticated psychoacoustic model so that the digital representation of the signal in both time and frequency is similar to how you perceive it in time and frequency. That way, the file can keep the components you hear, and drop other components that you would not notice anyway. That’s why it is able to reduce the file size and still sound good (or at least OK). —Amble (talk) 19:38, 13 November 2021 (UTC)

• The answer is harmonics are easily modeled using fourier series. Indeed, any function in any number of dimensions can be modeled as an infinite series of harmonic functions. Calculating these by hand is hard, but not for computers. --Jayron32 12:04, 15 November 2021 (UTC)

I think our OP might benefit from reading about convergent series, about sampling, and about coordinate transforms. Raster formats (the sort of things we commonly intuitively understand as "pixels") are only one way of thinking. We can be much more efficient - whether we're working by hand or with a computing machine - by being creative and opening up our minds a little wider. If the connections between these concepts, and the OP's questions about geoids, still aren't obvious - I'd recommend stepping away from the computer, grabbing a copy of the book I recommended earlier, and drawing some curves on paper until the first glimmer of insight emerges. The opening question asks, "why not just make a table?" To this, all I can answer is... why not?

Nimur (talk) 17:50, 16 November 2021 (UTC)

I'm not smart enough to deduce what a lot of the math symbols I never learned stand for (like u and counterclockwise δ), so the alienese can't help me know how the fuck they make these interesting shapes. In fact the lead image made me think it looked like lobes, which is surely a useful interpretation of the equations for some purposes but if it had been the simplest possible line drawing of the pyramid of spheres cut into bands and gores, with a pyramid of spheres with the red/white/blue shading underneath then I would've understood instantly what the fuck these are. Not totally of course but a core idea. And buried in the alienese (I never did like learning languages) is a sentence or two saying these are also sphere vibration modes and "string/sine vibration modes but 1-dimension higher" other very interesting ways of saying what these are that I couldn't have otherwise figured out except possibly after going down a rabbit hole of links full of dense math. Eventually I plan to learn some of the math I never did to understand a bit more of the alienese in science but not yet. Having finally learned what sphere harmonics are the truncated series of the 4.6 million simplest of the infinity sphere harmonics converging on partially non-symmetric geoid (or other semi-random sphere maps like Big Bang radiation) data thing seems a bit more "pure" to me. Sagittarian Milky Way (talk) 19:52, 16 November 2021 (UTC)

If you're not interested in learning the answers to your questions, you should stop asking them. You not wanting to put in the work to understand the answers is not our problem. --Jayron32 13:07, 17 November 2021 (UTC)

Wouldn't it take me at least ~a year to learn the math scientists know? Sometimes I want to know things like "is there an equation to assign a strength of continentality number to a climate?" or "how quickly would the ground track of a plane aligned with the Leaning Tower of Pisa deviate from the ellipsoid geodesic with the same initial azimuths?" but the math looks like hieroglyphics to me. I'm too uneducated, I left before calculus and didn't retain everything after math quickly went from easy (signed PEMDAS, slope intercept, FOIL, basic trig..) to painful stuff I only retained enough to avoid summer school without cheating. Well I "cheated" with $x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\ \$ ≥90% of the time just to finish homework faster.

If you know what's the closest thing to a flowchart or categorization system of alienese by visual appearance that would be nice. Especially if doesn't just say e.g. "it's vector calculus" but also shows a "tech tree" or "Mohs scale" of narrow subfields you must know to comprehend, e.g. it could mention vector calculus, integration and/or differentiation, vector conventions etc. If there are any common uses for something (e.g. find area under a curve) or possibilities for what a symbol stands for (e.g. if this equation has a τ it could mean time) listing those would be nice too). (Sub)categories might have names like "equation with lone big Greek letter on left edge", "equation with lone big uppercase pi on left edge (signature of a ... equation)", "symbols resembling Latin letters but not close enough to be mistaken for them (listed alphabetically by resemblance)" and "symbols that either are Latin letters or resemble them close enough to possibly be mistaken for them" Sagittarian Milky Way (talk) 21:20, 17 November 2021 (UTC)

Don't get too hung up on the "lobes" – the spherical harmonics only look like that when plotted individually, and they look like that because they are zero in some directions. When you're modelling the geoid (or the CMB temperature fluctuations) you are starting with the sphere (the lowest-order spherical harmonic) and then you're adding small amounts of the higher harmonics to that. Take the next row of harmonics, these are the dipoles, there's three of them for the spatial directions – by summing them with the correct coefficients you can make dipoles in any direction. If you add a bit of one of them to the sphere, you increase the radius of the sphere in one direction, and decrease it in the opposite direction. Where the dipole is zero (call that the equator) the sphere is left unchanged. As I said above by using all three basis dipoles you can distort the sphere in any direction you want. The next row is the quadrupoles, add one of them again to the sphere (remove the dipoles for the moment) and the sphere will bulge out in the direction where the quadrupole is blue, bulge in where the quadrupole is yellow, and be unchanged in those directions where the quadrupole is zero (for the left one that would be in longitude 0, 90W, 90E and 180, and of course in the direction of the north and south poles). It's not as easily explained why there are five basis quadrupoles, but as with the dipoles you can create any quadrupole with any orientation by suitably combining those five. Do the same for the higher spherical harmonics, the octupoles etc., and you distort the sphere on ever smaller scales. The whole process is analogous to a Fourier series, there's a similar convergence theorem (any function on the sphere, i.e. any function of two angles, say longitude and latitude, can be written as an infinite sum of spherical harmonics), but for modelling you always use a finite number of terms, so a finite sum, no need to worry about convergence. Spherical harmonics have mathematical properties that make their use very convenient - the basis harmonics as they are shown in the article are orthogonal to each other, i.e. maximally independent. They are actually an orthogonal basis of a infinitely dimensional vector space (the three dipoles for instance are the basis vectors of a three-dimensional vector space). I don't think it's necessary to learn a lot of calculus to get a bit of an intuitive feel for how this works... --Wrongfilter (talk) 18:19, 17 November 2021 (UTC)

Yes I agree, it's a lot simpler than the math looks. I thought lots of square degrees' radials pierced the lobe twice which confused me. And negative lobes means you shave the sphere there, now I see what the lobe representation means. And the coefficient can be negative which is why you need 3 dipoles, not 6. Sphere harmonics are more aesthetic than they seemed! This must also be why deflection of the plumb bob is given as deflection N-S & E-W but no actual deflections or azimuths anywhere: math people must like splitting up stuff into a sum of orthogonal subcomponents Sagittarian Milky Way (talk) 22:15, 17 November 2021 (UTC)

Is this [1] what we're talking about? 86.188.26.117 (talk) 17:59, 20 November 2021 (UTC)