Talk:Electromagnetic wave equation

Equations for waves in a medium

I've noticed that the equations given in the article (including the ones in the introduction) are for waves in vacuum, but that is not clear. The article even states "The electromagnetic wave equation (...) describes the propagation (...) through a medium or in a vacuum." right before the equations. Is there a reason for this? Other articles (for example Maxwell's Equations) present first the general formulation and then the vacuum case. Shouldn't this be corrected? Arnoques (talk) 19:08, 21 November 2008 (UTC)

I agree. I came to this page to find the general solution and was surprised to not find it. Blablabliam (talk) 17:40, 21 January 2022 (UTC)

You missed that $v_{ph}$ depends on the media parameters. The equation is general. Constant314 (talk) 18:12, 21 January 2022 (UTC)

The difference between the B field and the H field

Q. Is there a reason why you chose to use the H vector for the magnetic field, rather than what I know to be the standard B vector? motorneuron

A. The difference between B and H can be traced back to Maxwell's concept of a sea of molecular vortices. Within that context, H represented pure vorticity (spin), whereas B was a weighted vorticity that was weighted for the density of the vortex sea. Maxwell considered magnetic permeability to be a measure of the density of the vortex sea. Hence the relationship,

Magnetic Induction Current

$\mathbf {B} \ =\ \mu \mathbf {H}$

was essentially an angular analogy to the linear electric current relationship,

Electric Convection Current

$\mathbf {J} \ =\ \rho \mathbf {v}$

B was seen as a kind of magnetic current of vortices aligned in their axial planes, with H being the circumferential velocity of the vortices.

You could view the electric current equation as a convective current of electric charge that involves motion. By analogy, the magnetic equation is an inductive current involving spin. There is no linear motion in the inductive current along the direction of the B vector. The magnetic inductive current represents lines of force. In particular, it represents lines of inverse square law force.

The extension of the above considerations confirms that where B is to H, and where J is to ρ, then it necessary follows from Gauss's law and from the equation of continuity of charge that D is to E. Ie. B parallels with D, whereas H parallels with E.

Having said that, in the context of the electromagnetic wave equation it doesn't actually matter whether we use magnetic induction vector B, magnetic field vector H, electric field vector E, electric displacement vector D, magnetic vector potential A, scalar potential Φ or indeed any function of the electromagnetic field.

The real question that we need to ask is 'why do we bother to present any particular pair of electromagnetic wave equations?'. Why not present just one example? They all represent different ways of measuring the same phenomenon.(203.115.188.254 06:13, 17 February 2007 (UTC))

Read the article on Maxwell's equations. B is standard notation for the magnetic flux density, whereas H is the proper notation for the magnetic field. B and H are not the same thing, although they are related. The B field arises in response to an applied H field. In vacuum, the B field is represented as

$\mathbf {B} \ =\ \mu _{o}\mathbf {H}$

I object to calling H THE magnetic field. First of all, it's extremely misleading to claim that there is no B field in a vacuum. It is B that couples to magnetic dipoles, provides the Lorentz force, and transforms with the electric field E within the field tensor. These functions make no reference to a medium. Perhaps H is what you apply and what you measure, but B controls the underlying physics. -- Melchoir 22:28, 11 November 2005 (UTC)

In a vacuum, exactly what are these magnetic dipoles that you are talking about?

Well, for example, a spinning charged particle constitutes a magnetic dipole. On more ordinary scales, you could have an isolated current loop. Magnetic dipoles don't need to occur in bulk, and they're simplest to analyze when they don't. By the way, please sign your posts. Melchoir 22:51, 11 November 2005 (UTC)

BTW, have you even looked at the articles on magnetic field and magnetic flux density? If you have, you should have seen a discussion about this exact issue. The bottom line is that there are two separate fields, one is the B field, and the other is the H field. They both exist. They are not the same thing. The are related to one another through something called a constitutive relation that depends on the medium. In vacuum, it really makes not a particle of difference whether you use the B field or the H field since one is proportional to the other by a well-known constant of nature. This entire debate is a total waste of time.

Yes, I've looked at those articles, and that's part of why I think we should standardize our notation across Wikipedia articles. Ideally we would choose only one field and call it the magnetic field in all articles; of course, you know which one I advocate. I am aware that B and H are different, which is why I felt it necessary to point out that B exists in a vacuum earlier. Of course, you're right that the constitutive relations in vacuum are linear and isotropic, so we can formulate the vacuum wave equations in terms of any combination of E and D, B and H. However, using the phrase "magnetic field" to describe H is a choice that should be uniform across all articles. The debate is not a waste of time if it helps us avoid confusion in the articles themselves. Melchoir 23:01, 11 November 2005 (UTC)

I agree about eliminating confusion. That is why I think it is very important to make the distinction between B and H extremely clear. If you call B the magnetic field, you are perpetuating the ambiguity, not helping to clear it up. It needs to be clear that the B field is not the magnetic field, but that it is in fact the magnetic flux density. Why? Because H is the magnetic field. How do we know? Because that's what Maxwell's equations tell us. The symmetry of the equations indicates the E corresponds to H, and D corresponds to B.

If you don't believe me, then go to Maxwell's equations and look at the form of Faraday's Law in comparison to Ampere's Law (assume that the current density J is zero). If electricity and magnetism are different forms of the same thing, then it should be clear by comparing Faraday's Law to Ampere's Law that H is the dual of E, and that B is the dual of D.

And you cannot simply choose one or the other. To describe the physical reality, it requires both B and H, just like it requires both E and D.

Well, it's an oversimplification to say that H is the magnetic field simply because the Maxwell equations tell us so; the Maxwell equations deal with physical quantities, not terminology. I've given the matter way too much thought for a Friday, and I've decided that the Maxwell equations in matter suggest a parallel between E and H because they are written in terms of free electric charge and current, so the field definitions must absorb the bound electric charge and current. If we allowed the possibility of magnetic charge, we would have to introduce more auxiliary macroscopic fields, and the resulting equations would draw a parallel between H and something else. In this case, the relation between E and H is an artifact of the absence of magnetic charges, a physical result which breaks the symmetry between electric and magnetic phenomena and makes duality arguments less than compelling.

To get back to my main point, though, you claim that calling B the magnetic field would be perpetuating an ambiguity. Why? The comment that started this whole topic was from someone who had only ever heard of B being called the magnetic field. I have no data on this claim, but it seems to me that the current trend in textbooks is to introduce B first, call it "the magnetic field", and state that it is fundamental. Textbook authors seem to be trying to break away from historical usage. If Wikipedia chooses the old convention, isn't that perpetuating the ambiguity? Melchoir 00:51, 12 November 2005 (UTC)

Who cares? I changed the wave equation to B instead of H because B is more general (it encompasses H), and it is less confusing to most readers. Enough said. -- Metacomet 17:48, 12 November 2005 (UTC)

Sorry, on second thought, I think it is better using H instead of B. The symmetry of the equations is much better when E is coupled with H instead of with B. Don't worry, though, I added the equation that shows the relationship between B and H. Perhaps that will help eliminate some of the confusion. -- Metacomet 00:47, 13 November 2005 (UTC)

Metacomet, In the context of the electromagnetic wave equation it doesn't actually matter whether we use magnetic induction vector B, magnetic field vector H, electric field vector E, electric displacement vector D, magnetic vector potential A, scalar potential Φ or indeed any function of the electromagnetic field.

The real question that we need to ask is 'why do we bother to present any particular pair of electromagnetic wave equations?'. Why not present just one example? They all represent different ways of measuring the same phenomenon. (203.115.188.254 05:59, 17 February 2007 (UTC))

Well you do sort of get the E and B wave equations at the same time, when doing the substitution as shown in the article. This is also the usual undergrad engineer's view of the matter. It would add some perspective and additional interest to show it for A and Φ. —Preceding unsigned comment added by 74.101.7.239 (talk) 04:02, 17 December 2009 (UTC)

Duality

Note that the dual of the electric field E (units of volts per meter) is the magnetic field H (in amperes per meter) and not the magnetic flux density B (in webers per square meter). So it would be misleading to have the wave equation for the electric field in terms of E and for the magnetic field in terms of B. The dual of the B field is the electric displacement D (in coulombs per square meter). Again, see the article on Maxwell's equations for a more complete explanation.

As for duality, which dual are you talking about? -- Melchoir 22:28, 11 November 2005 (UTC)

Duality: Faraday and Maxwell established that electricity and magnetism are really two different manifestations of a single underlying phenomenon, what we now know as electromagnetism. In an electric circuit, current and voltage are duals of one another because they form a power pair, meaning that that the electrical power flowing through the circuit is equal to the product of the current times the voltage. Likewise, the E field and the H field form a power density pair in an electromagnetic wave because the Poynting vector S, which gives the electromagnetic power per unit area, is the cross product between E and H:

$\mathbf {S} \ =\ \mathbf {E} \times \mathbf {H}$

You will notice that the Poynting vector depends on E and H but does not depend on B.

Discussion of nomenclature

Shouldn't we have a centralized debate over nomenclature somewhere? I haven't found it. (Disclaimer: I was raised on Purcell and Griffiths.) Melchoir 22:28, 11 November 2005 (UTC)

Actually it should be S=(1/2)Re[ExH] If you mean power as the real power, course
if you would like to include the Immaginary part of the Vector is allright.

Sinusoidal Solutions versus Arbitrary Pulse Shapes

Thanks. I think we need to do a little work on the solutions section. the wavelength emerges from a decomposition into sinusoids. there are some steps that come before this. also, i think we should move the solutions section after the derivation of the wave equations. this is because we will need to invoke some of maxwell's equations to get the relationship between electric and magnetic fields (which you allude to). I will check this out in a couple days when i have some more time.Complexica 01:50, 7 November 2005 (UTC)

Sorry, I do not agree. The solutions should come right after the presentation of the wave equation, and before the derivation. Also, the solution I presented was not at all related to sinusoids or Fourier decomposition. It was the most general form of the solution possible, which is to say any arbitrary function of $\mathbf {k} \cdot \mathbf {r} -\omega t$. I can show you the derivation, but I may not have time for several days or even a few weeks. Regardelss, the solution needs to be in terms of a dimensionless argument (radians), not in terms of time or distance or the speed of light. -- Rdrosson 02:48, 7 November 2005 (UTC)

I think you might find the plane wave solution I just put in what you are trying to say with $\mathbf {k} \cdot \mathbf {r} -\omega t$ argument. Check it out. -- Complexica 03:27, 7 November 2005 (UTC)

You are missing something very fundamental. The point is that the function $\ f(kr-\omega t)$ is not in any way restricted to sinusoids or any other functional form. This function can take on virtually any arbitrary shape (well-behaved), and as long as $k={\omega \over c}$, then the wave equation is satisfied. The only other issue is that the direction of the electric field must be orthogonal to both the magnetic field and the direction of propagation, which is parallel to k. -- Rdrosson 04:12, 7 November 2005 (UTC)

That's absolutely right and considering that k is a vector, this form suit enough while the medium is still Isotropic, becouse if not k cease to be a vector and start to be

a 9 component tensor, as a matter of fact the Wave is spitted in different direction. Typical example is light in a Quartz Christal.

Closed versus Open Surface Integrals

Also, I do not knoiw what a closed loop integral means for a surface integral. At any rate, it is not used consistently for surface integrals in the article. Why isn't Ampere's law a closed loop integral, for instance? -- Complexica 03:27, 7 November 2005 (UTC)

In Ampere's Law, the surface is not supposed to be a closed surface because Ampere's Law relates an integral across an open surface with the closed contour surrounding the open surface. There is a fundamental difference between Gauss's Law on the one hand, and Ampere's and Faraday's Laws on the other. Gauss's Law has to do with closed surfaces surrounding enclosed volumes, wheras Ampere and Faraday are related to closed contours surrounding open (meaning two dimensional) surfaces. This distinction is the reason why you need to use the divergence theorem to simplify Guass's Law, and Stokes Theorem to simplify Ampere and Faraday. It is also the reason why the differential form of Gauss's law is based on the divergence operator, whereas Ampere and Faraday are based on the curl operator.

If you want to understand these concepts, read the articles on Maxwell's equations, the divergence theorem, and Stokes theorem. These are fundamental ideas in vector calculus and electrodynamics. -- Rdrosson 03:41, 7 November 2005 (UTC)

OK. I see the distinction you are making regarding open and closed surfaces. -- Complexica 03:50, 7 November 2005 (UTC)

Also, if you have a chance, take a look at the book by H. M. Schey that I added to the Reference List under the heading Vector Calculus. The book, which is pretty short and concise, provides an excellenct discussion of the relationships between the three vector derivatives and the major multi-dimensional integrals. -- Rdrosson 03:53, 7 November 2005 (UTC)

Writing this article

Rdrosson, it has been a pleasure writing with you. Complexica 22:57, 6 November 2005 (UTC)

Keep up the good work. You have a lot of good ideas, and you have made some excellent contributions to improving this article. -- Rdrosson 23:47, 6 November 2005 (UTC)

General Solution

The following is a work-in-progress:

To find the general solution to the wave equation, we begin by assuming that E takes the following functional form:

$\mathbf {E} (\mathbf {r} ,t)=\mathbf {E} (\phi )$

where

$\phi \ =\ \phi (\mathbf {r} ,t)$

is some – unknown for now – scalar function of space r and time t.

If we can identify a simple functional form for the parameter $\ \phi (\mathbf {r} ,t)$, then we can simplify the wave equation from three partial differential equations in four independent variables, r = (x,y,z) and t, to three ordinary differential equations in a single variable, $\ \phi$. That would certainly make it much easier to find solutions to the wave equation.

Using the chain rule, we initially derive expressions for the first derivatives of E with respect to r and t:

$-(\nabla \times \mathbf {E} )={d\mathbf {E} \over d\phi }\times \nabla \phi$

and

${\partial \mathbf {E} \over \partial t}={d\mathbf {E} \over d\phi }\cdot {\partial \phi \over \partial t}$

Next, we use the product rule for differentiation to obtain the second derivatives of E with respect to r and t:

$\nabla ^{2}\mathbf {E} ={d^{2}\mathbf {E} \over d\phi ^{2}}\cdot \nabla \phi \cdot \nabla \phi \ +\ {d\mathbf {E} \over d\phi }\cdot \nabla ^{2}\phi$

and

${\partial ^{2}\mathbf {E} \over \partial t^{2}}={d^{2}\mathbf {E} \over d\phi ^{2}}\cdot \left({\partial \phi \over \partial t}\right)^{2}\ +\ {d\mathbf {E} \over d\phi }\cdot {\partial ^{2}\phi \over \partial t^{2}}$

to be continued...

Way too specific

You get more general wave equation whenever you consider Maxwell's equations for source-free and charge-free linear media at a frequency ω:

$\nabla \times {\frac {1}{\mu }}\nabla \times \mathbf {E} ={\frac {\omega ^{2}}{c^{2}}}\varepsilon \mathbf {E}$

with the constraint $\nabla \cdot \varepsilon \mathbf {E} =0$. Or, in terms of the magnetic field:

$\nabla \times {\frac {1}{\varepsilon }}\nabla \times \mathbf {H} ={\frac {\omega ^{2}}{c^{2}}}\mu \mathbf {H}$

with the constraint $\nabla \cdot \mu \mathbf {H} =0$.

For lossless materials, these have propagating solutions with a well-defined group velocity whenever you have at least one direction with translational symmetry, including discrete translational symmetry (periodicity, as in a photonic crystal). The solutions in such cases are given by Bloch waves.

(Note that it is easier to write in general as a function of frequency ω than as a function of time, to handle the case of dispersive materials. As a function of time, you need to convolve the susceptibility with the correpsonding field, which gets messy.)

It would be nice if the article weren't quite so narrow. Right now, I feel like there is so much focus on special cases and on reiterating stuff that appears in Maxwell's equations that you can't quite see the forest for the trees.

—Steven G. Johnson 20:42, 16 December 2005 (UTC)

GA Failed

This article failed WP:WIAGA 2b (lck of inline sitations. Tarret 22:12, 20 September 2006 (UTC)

div E = 0 ?

If you look at [Maxwell - First to propose that light is an electromagnetic wave], it would be helpfull, to introduce by defining ρ=0 in the objected area.

A little bit confusing if after having read about Maxwells div(D)=ρ, you see suddenly div(E)=0

LRJ 138.246.7.104 20:31, 15 March 2007 (UTC)

Inconsistencies between . . .

I think that the Inconsistencies section could use some more work. One thing that is not made clear is whether or not the changing electric field actually creates a change in the line integral of the magnetic field (it does, but this is not made clear.) Second, perhaps the relationship between $\nabla \cdot j$ and $j\,\!$ could be expanded. Finally, I don't see the relationship between conservation of charge and the displacement current. Perhaps substituting $\nabla \cdot j=-{\frac {\partial \rho }{\partial t}}$ back into the equations above would make this clearer.

Iain marcuson 20:03, 21 October 2007 (UTC)

Diagram misleading

I have explained (I hope) what seems to be wrong with the diagram Image:Spectre.svg at Talk:Terahertz radiation#Spectrum graph, where a related, derived, diagram was also removed. 84user (talk) 23:09, 27 May 2009 (UTC)

So should we put in the better diagram? The issues with the old diagram seem minor. Its purpose is to show a quick glance of the EM spectrum. Not to be a strict reference. Daniel.Cardenas (talk) 01:14, 28 May 2009 (UTC)

To me it doesn't matter, but I was not the one that removed the image and others may feel the same way (maybe something like "schematic artist's impression" in the caption?). I have proposed a fix at Commons:File talk:Spectre.svg if anyone wants to submit the request, as I'm too lazy at the moment. 84user (talk) 03:02, 28 May 2009 (UTC)

define symbols used in the equations?

In the text accompanying each equation, the symbols used in that equation should be defined so that someone not familiar with what they denote/mean will be able to know what those symbols denote/mean. The first two equations on this page fail to define three the symbols used therein. — Preceding unsigned comment added by 147.177.39.192 (talk) 13:24, 14 September 2011 (UTC)

I only see one symbol that might require being defined -- The time partial differential ${\frac {\partial }{\partial t}}$. What other symbols are you talking about? Dauto (talk) 17:35, 14 September 2011 (UTC)

I do not understand, in the introductory equation, why there is a "where c=..." since in the two equations proposed there is no "c". Is there an error in the equation? --201.204.200.18 (talk) 23:43, 1 November 2012 (UTC)

Many more wave solutions: evanescent waves and near fields, birefringence, ...

This article seems to intend to examine general electromagnetic waves, yet only discusses propagating waves in vacuum which are observed infinitely far away from radiation sources. This misses out on other important classes of waves that obey the same equation but occur nearer to sources or nearby material interfaces/surfaces. These other solutions cannot be described conveniently by superpositions of real plane waves, due to their divergent nature.

First of all, we can consider plane waves where the wavevector is complex. For example, an evanescently decaying wave near a surface, as in total internal reflection, is described by the in-plane wavevector being real-valued and exceeding $\omega /c$, however, the out of plane wavevector is imaginary so that ${\vec {k}}^{2}=\omega ^{2}/c^{2}$ (note in contrast that $|{\vec {k}}|\neq \omega /c$, for evanescent waves !).

In addition, there are solutions that obey the homogeneous Maxwell equations, corresponding to the fields in vacuum around some source. For example the field around a radiating dipole follows the wave equation everywhere, except at the position of the dipole itself. The solution includes near fields that are not conveniently described by superpositions of plane waves.

Another related limitation is that the entire article is focussed on a simple form of wave equation which only occurs in vacuum or in isotropic media, but which is not applicable to some other basic kinds of linear media. For instance the equations shown cannot describe propagation of light through a birefringent material like calcite. --Nanite (talk) 17:37, 1 October 2015 (UTC)

Improvements

What can I do to bring this to Good Article status? ScientistBuilder (talk) 20:34, 8 February 2022 (UTC)

Your enthusiastic efforts to improve Wikipedia are appreciated, but I think that you might misunderstand the process. You don't nominate a poor article to be a good article in the hope that someone will then convert it into a good article. Instead, you do the work to convert it into a good article and then nominate it. This particular article probably needs 100s of hours of work. Constant314 (talk) 03:47, 9 February 2022 (UTC)

@ScientistBuilder: One way you can improve the article is by adding citations to reliable sources! The second criteria states that most information in a good article must be verifiable to reliable, published sources and must have a reference list containing all such sources used. Right now, the article lacks inline citations and is thus unverifiable, and I can't find any reference lists either. The most I could find was this further reading section. You have quite a long way to go with bringing the article to GA, but with time and effort, I believe you can do it. Happy editing! 〜 ‍ ‍ ‍ elias. 🧣 ‍ ‍ ^{💬reach out to me!・📝see my work!} 05:25, 9 February 2022 (UTC)

What are some ways I can help make articles better that do not involve finding reliable sources?

I am more interested in rewriting and making articles clearer to read and understand than hunting down books, journals, and websites for articles. ScientistBuilder (talk) 13:37, 9 February 2022 (UTC)

If you are not going to refer to the reliable sources, you should probably stick to rearranging the words that are already here. There is plenty of valuable work to be done doing exactly that. Wikipedia policy is that all information in the article is either paraphrased from reliable sources or copied from reliable public domain sources. The policy is widely violated, but if you want to produce a good article, you should stick to it. The most common way to become proficient is to make edits and accept feedback when they get reverted or corrected. Proceed slowly and see what happens. You might also want to examine some good articles, such as negative resistance. Constant314 (talk) 01:11, 10 February 2022 (UTC)

Solutions to the homogeneous electromagnetic wave equation

The general solution given in this section is not quite correct, as I see it. The expressions for E(r, t) and B(r, t) are formulated in terms of exactly the same g, implying that E and B fields are the same (which they can't be already due to dimensions). Isn't it that one could salvage this by multiplying the expression for E by the speed of light and taking the cross product of one of the two with the normalized k vector? The further expressions below do not have this problem, as they are formulated in terms of independent E₀ and B₀. But I strongly think that this is also not the best way, as these two obviously are related, namely in the same way as above, the absolute values differ by a factor c and one is perpendicular to the other. This is implied in the statement "the magnetic field ... is related to the electric field by the relation", but I strongly think this should be made explicit. Seattle Jörg (talk) 09:45, 24 March 2022 (UTC)

I agree with your concern. I know what it means, but it could be misleading. Also, the solution is a plane wave. There are other solutions that are not plane waves. For example, there are also cylindrical wave and spherical wave solutions. Constant314 (talk) 16:03, 24 March 2022 (UTC)

No, it is not necessarily a plane wave, it is just a travelling wave with any functional form along its propagation direction. For very mild assumptions, you can write this travelling wave as a (possibly infinite) sum of plane waves, that is, essentially a Fourier transform. For any of those summands the thing with multiplication by c and cross product holds, you can pull those through the summation, so it holds also for the general travelling wave. Specifically, it holds also for circularly polarized waves (which are the sum of two plane waves with 90° phase shift). Seattle Jörg (talk) 15:20, 4 April 2022 (UTC)

The solution shown is a plane wave because the solution is everywhere the same on any plain perpendicular to the wave vector. That is not the case with cylindrical solutions, which are also homogeneous solutions. I believe that it is true that you can decompose a cylindrical wave, at a point, into a sum of plane waves, I also believe that that sum will not converge to the cylindrical wave everywhere. You can certainly improve what is written, just please do not imply that all the homogeneous solutions have form g(k·x - ωt), unless you have a reliable source that says that. Constant314 (talk) 15:49, 4 April 2022 (UTC)

Don't cylindrical waves require a source at the axis, and therefore are not homogeneous? And re plane wave, do some sources take that to mean sinusoidal, or not? Dicklyon (talk) 16:30, 4 April 2022 (UTC)

Cylindrical solutions to boundry value problems require axial symmetry. A practical example is a round fiber optic, a coaxial cable, and a round wave guide. In an unbounded vacuum, cylindrical waves are just as good as plane waves for being homogeneous.Constant314 (talk) 17:14, 4 April 2022 (UTC)

Then spherical waves are homogeneous solutions, too? Singularities at axis or center are OK? Dicklyon (talk) 17:58, 4 April 2022 (UTC)

I believe that spherical waves are also homogeneous solutions. Homogeneous solutions are the source free solutions. So, point source, line source, or plane source doesn't matter. As far as I know, there are no physical source distributions that can produce a plane wave, cylindrical wave, or a spherical wave. They are all equal in that way. Constant314 (talk) 18:11, 4 April 2022 (UTC)

Sorry, perhaps this discussion got a bit off track because my initial comment was not formulated perfectly accurately: the first lines of the section we are talking about say that the general solution can be written as a linear superposition of waves of the form... And this is true. It is even true if you restrict g to be a sine function, because any (or nearly any) function can be Fourier-transformed. So the point I am making is that the g of E and B field are not the same, but related in a known way which we should make explicit. So no, not all homogeneous solutions are travelling waves, but all can be written as superpositions of travelling waves. Does anybody doubt that? Seattle Jörg (talk) 10:53, 5 April 2022 (UTC)

Article issues and classification

The B-class criteria #1 states; The article is suitably referenced, with inline citations. It has reliable sources, and any important or controversial material which is likely to be challenged is cited. The "More citations needed" tag at the top of the page is untypical of a B-class article. Considering the enormous amount of unsourced content, which includes many equations, the article fails the criteria so I am reassessing to C-class.

The "See also" section and three subsections have thirty-one links which appear to be rather long. When this is added to the "Further reading" section, with four subsections and fifteen links, they could both be examined for trimming. Especially when "Undergraduate-level textbooks" and Graduate-level textbooks" are included since Wikipedia is not a textbook or repository. -- Otr500 (talk) 13:22, 1 March 2023 (UTC)