# Talk:Plane wave

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## What is dω/dk?

What does it mean to divide a scalar by a vector? Or an infinitesimal scalar by an infinitesimal vector? An explanation is needed here.

201.215.210.189 (talk) 10:02, 15 February 2011 (UTC)

I think the answer is in Directional_derivative:Derivatives_of_scalar_valued_functions_of_vectorsConstant314 (talk) 15:50, 15 February 2011 (UTC)

It means the gradient ∇kω, i.e. the gradient of ω with respect to the k variables. — Steven G. Johnson (talk) 05:08, 16 February 2011 (UTC)
The explanation is too involved to be included here. The interested reader can click the group velocity link. - Dave3457 (talk) 01:56, 28 February 2013 (UTC)

## Regarding the picture text

Regarding the picture text that says "The real part of a plane wave travelling up.", does that mean that the real part a plane wave is travelling up, or that the plane wave itself is travelling up and that it's real part is shown?

Yes to both. Daniel.Cardenas (talk) 02:22, 27 December 2008 (UTC)

## Regarding the definition ==

I am highly suprised that a plane wave needs to be monochromatic! A plane wave is a wave with a PLANE wavefront, hence the name. Guy Drijkoningen (TUD) —Preceding unsigned comment added by 131.180.60.61 (talk) 06:52, 7 May 2009 (UTC)

Yes, this is wrong and needs to be fixed. A plane wave does not have to be a cosine or sine. We just teach that to students early on because its easy to understand. A plane wave could have just about any functional shape in the direction of propagation and still be planar in the transverse direction. For instance, sunlight from one spot on the sun is essentially a plane wave since the sun is so far away. But sunlight is definitely not monochromatic and does not have a cosine waveform. I will try to fix this. Please comment below if you disagree and want to change it back 129.63.129.196 (talk) 19:11, 26 February 2013 (UTC)
According to Saleh and Teich (among other sources), Fundamentals of Photonics, a "planewave" is defined as a sinusoidal wave, not just any wave solution with planar wave fronts. Please have an authoritative source before you put in any alternate definition. — Steven G. Johnson (talk) 19:20, 26 February 2013 (UTC)

## Creation of Polarized electromagnetic plane waves section

If you are short on time, leave any criticisms here and I will try to deal with them. April 9th 2010
Dave3 (talk) 22:07, 9 April 2010 (UTC)

## Near Field Plane Wave Creation

Is it possible to actually create a 2.4GHz RF (Sinusoidal or other) Planewave in the near field of the transmitter(s) and if so how would one go about doing it?

Thanks,

kdavis45@comcast.net —Preceding unsigned comment added by 67.160.220.64 (talk) 17:29, 13 June 2010 (UTC)

## Image descriptions

It would be nice if someone can elaborate on the animations, what the colors represent, etc... Thanks. —Preceding unsigned comment added by 216.239.66.193 (talk) 15:53, 24 October 2010 (UTC)

## Engineering Time Convention

Engineers use the same time as physicists.

But they (at least the electrical engineers) do use a different definition phase which accounts for the negative sign.

See The interpretation of phase for mathematics vs. engineering Constant314 (talk) 05:07, 9 January 2011 (UTC)

I reverted that. If it's true, cite a source that says so. Or perhaps a pair of sources that are unambiguous about the different fields. Dicklyon (talk) 05:29, 9 January 2011 (UTC)

## Concerning revert of new section Acoustic plane waves

Martarius, I reverted your new section. Of course feel free to un-revert it but I would have to ask you justify its inclusion.

I can’t help but feel that a section called Acoustic plane waves should talk about Acoustic plane waves and not a specific speaker system.

Your article sounds like something that an employee of the Magnepan speaker manufacture would try to sneak in. However because you have made many positive contributions to Wikipedia I’m ruling that out. I’m assuming that just bought one and really like it or something :)

In short I don’t see that the Plane waves article is the place for this.
By the way, agree with your views about Deletionists. Dave3457 (talk) 20:29, 19 March 2011 (UTC)

## Simplification of equations

I have simplified the equations describing plane waves, as I felt that they were unnecessarily complicated. For instance, why was the amplitude A in the equation for U(r,t) complex?

I also moved the paragraph which talks about approximations to plane waves to a new section, as its inclusion in the initial discussion again makes the discussion harder to follow. Epzcaw (talk) 13:46, 16 April 2011 (UTC)

Touch up of recent changes
With regards to your comment on my talk page which I copied below...
I did not delete the section about approximations to plane waves which you 're-instated' - I moved it to a separate section entitled 'Approximations to plane waves' where I felt it was more appropriate, since the initial discussion relates to 'true' plane waves. At the moment, it appears twice in the article - you should decide which one should stay and which one should go.
Sorry for the over site, the talk page was not on my watch list and your multiple edits made it hard to follow what happened.
Since you left the decision to me, I moved the added section, which you improved, back up to the lede. I can't say I agree that it makes things harder to follow. My main concern with what was there was that the lede jumped to quickly into the equations. It is my sense that alot of people, like me, often just read the lede with the desire to get an idea of the subject matter. I feel that paragraph goes along way towards that end. It also seemed appropriate given that plane waves don't actually exist. I left in your sentence " Laser beams are also approximately planar." as is, because I couldn't work it into the other paragraphs but I think it stands out a bit.
I also created a new section to isolate the lede. But I’m not crazy about my choice of heading however..."Mathematical formalism"
By the way I changed EXP to e^ to match the other pages in Wikipedia. I’ve never been a fan of EXP so the consensus suits me.
I also changed the order of the equations with the thinking that it is better to go from the more concise to the less concise for comprehension purposes. - I also moved the imaginary i to the front of the expressions. Let me know if you have a problem with that. - I re-arranged the definitions a bit, added one and converted the symbols to LaTeX notation.
I hope you don’t feel that I’m “messing in you business”. Dave3457 (talk) 09:00, 20 April 2011 (UTC)
This is much better now - good idea to separate out the mathematics which, as you say, many people would prefer not to read. I want to be able to refer to the equations in this page elsewhere, so perhaps got carried away with the mathematics to the detriment of general understanψψding.

I will think about laser beam bit, and maybe expand or else remove.

You are absolutely not 'messing in my business - I am very happy to have comments/criticism!

I am still struggling wiht LaTeX - it seems to have a mind of its won about what it does, so all contributions welcome Epzcaw (talk) 10:36, 20 April 2011 (UTC)

Requiring the amplitude to be real is wrong. It restricts planewaves to have a particular phase at the origin of your coordinate system, which is not true in general. — Steven G. Johnson (talk) 20:11, 20 April 2011 (UTC)
The amplitude has not been 'required' to be real. The value of the real part is given. Epzcaw (talk) 22:51, 20 April 2011 (UTC)
Until Dave fixed them a little while ago, the equations silently assumed that A is real, which is not generally true of planewaves. There are still two problems. First, saying that A is the "peak-to-peak amplitude" is inconsistent with allowing it to be complex (since you don't get the phase by looking at peak-to-peak amplitude alone). Second, saying that A is "generally" chosen to be real is hogwash in my experience, and would need a reference even if it were true. (As long as you are using complex exponentials anyway, there is no reason to avoid complex numbers for the amplitude, and in most practical contexts, e.g. Fourier analysis, you have nonzero phases.) The previous version of the article, which allowed the amplitude to be complex, is better. Please revert to the complex formula. — Steven G. Johnson (talk) 23:49, 20 April 2011 (UTC)
Right now it says "Note that the resulting function U(r,t) may be real or complex. Generally one selects a complex value of A so that U(r,t) is real and has the desired phase shift." Saying that a function is both real and has a phase shift is almost a contradiction. But consider what happens at the origion at time zero. The exponent of the exponential is 0 so that U(0,0) = A. So, if U(0,t) is a pure cosine then A is real. But if U(0,t) is not a pure cosine then then A must be complex to account for the sine component.Constant314 (talk) 01:53, 21 April 2011 (UTC)
It was more wrong than that. That expression can't give a real wave. See if you agree with my fixes. Dicklyon (talk) 05:03, 21 April 2011 (UTC)
Yes I goofed, sorry about that. As it happens I was in the process of correcting it when, Dicklyon beat me to it resulting in an editing conflict. I was imagining the addition of two complex waves with opposite angular frequencies when I wrote it. Please don't hate me :) He also beat me to the Peek-to-peek error resulting in a second editing conflict, BUT THAT ERROR WASN'T MY FAULT! :) Dave3457 (talk) 05:22, 21 April 2011 (UTC)
Drat. Who can I hate then? Dicklyon (talk) 06:04, 21 April 2011 (UTC)

I find it very confusing to have the same letter used to denote both a complex and a real quantity, and suggest the following form for the first set of equations instead:

The complex amplitude of a plane wave at time t, and position vector r can be represented by the following equivalent equations:

 ${\displaystyle u(\mathbf {r} ,t)}$ ${\displaystyle =Ue^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}=Ae^{i(\mathbf {k} \cdot \mathbf {r} -\omega t+\phi )}}$ ${\displaystyle =Ue^{i2\pi (\mathbf {n} \cdot \mathbf {r} /\lambda -ft)}=Ae^{{i[2\pi (\mathbf {n} \cdot \mathbf {r} /\lambda -ft)}+\phi ]}}$ ${\displaystyle =Ue^{i(2\pi /\lambda )(\mathbf {n} \cdot \mathbf {r} -ct)}=Ae^{{i[(2\pi /\lambda )(\mathbf {n} \cdot \mathbf {r} -ct)}+\phi ]}}$

where

• ${\displaystyle U\,\!}$ is the complex amplitude of the wave.
• ${\displaystyle A\,\!}$ is the peak-to-peak amplitude of the wave
• ${\displaystyle \phi \,\!}$ is the phase at t =0
• ${\displaystyle \mathbf {k} }$ is the wave vector, ${\displaystyle \mathbf {k} =(2\pi /\lambda )\mathbf {n} }$
• ${\displaystyle \omega \,\!}$ is the angular frequency, ${\displaystyle \omega =2\pi f\,\!}$
• ${\displaystyle \mathbf {n} }$ is a unit vector in the direction of travel
• ${\displaystyle \lambda \,\!}$ is the wavelength
• ${\displaystyle f\,\!}$ is the frequency
• ${\displaystyle c\,\!}$ is the velocity

of the wave.

Actually, using two different symbols the represent what can easily be represented by one symbol is more confusing. The easy answer to your complaint is to just always consider A to be complex and sometimes the imaginary part is zero. I do not see anything wrong with your math, but it is customary to let A take care of the phase and amplitude.Constant314 (talk) 12:38, 21 April 2011 (UTC)
What bothered me was not that A was sometimes real and sometimes complex, but just that we use it to represent two different values; you did fix that, but didn't show the relation between U and A; another way is to use complex A and then use |A| and Arg(A) in the real equations. I'd follow Constant314's suggestion and look at sources for how they do this. And there is an error in the above equations: the i needs to multiply the phi. Dicklyon (talk) 15:31, 21 April 2011 (UTC)

Born & Wolf states 'In the case of a plane wavefront, one often separates the constant factor e -iδ, and implies by the complex amplitude only the variable part aeik.r. (This is a footnote in Section 1.3.3, page 18 of the 1999 version. In section 1.3.4 (p. 20), they give the equation

${\displaystyle V(z,t)=a[e^{-i(\omega t-kz)}+...}$

and then say "The symbol R (it is not a proper R but a squiggly one which they used earlier to mean 'the real part of) is omtted here with the convention explained earlier."

I think what they are saying is that the phase part of the initial amplitude can be left out, as it is not relevant in any optical situation, since only relative phase is ever measureble. This is not the case with other sorts of waves.

Longhurst (Geometric and Physical Optics) uses a real amplitude in his expressions for waves, as equation 10-18

${\displaystyle \phi _{0}=\int \int {\frac {a}{2\lambda r}}\cos blahe^{kt(ct+blah}}$

Heavens & Ditchburn (Insight into Optics) discuss the representation of waves by complex quantities, and say

"the expression E0 e is referred to as the complex amplitude"

I think this shows that in optics, it is normal to describe light waves using a real amplitude. I would settle for |A| and Arg(A) if that is the consensus

PS. I have corrected the equations -whoops.
PPS Does anyone know why the Longhurst equation displays in small font?Epzcaw (talk) 17:56, 21 April 2011 (UTC)

Amended and corrected equations for consideration

 ${\displaystyle u(\mathbf {r} ,t)}$ ${\displaystyle =Ae^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}=|A|e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t+\phi )}}$ ${\displaystyle =Ue^{i2\pi (\mathbf {n} \cdot \mathbf {r} /\lambda -ft)}=|A|e^{{i[2\pi (\mathbf {n} \cdot \mathbf {r} /\lambda -ft)}+\phi ]}}$ ${\displaystyle =Ue^{i(2\pi /\lambda )(\mathbf {n} \cdot \mathbf {r} -ct)}=|A|e^{{i[(2\pi /\lambda )(\mathbf {n} \cdot \mathbf {r} -ct)}+\phi ]}}$

where

• ${\displaystyle A\,\!}$ is the complex amplitude of the wave.
• ${\displaystyle |A|\,\!}$ is the real value of ${\displaystyle A\,\!}$
• ${\displaystyle \phi \,\!}$ =${\displaystyle Arg(A)}$ and is the phase of the wave at t =0
• ${\displaystyle \mathbf {k} }$ is the wave vector, ${\displaystyle \mathbf {k} =(2\pi /\lambda )\mathbf {n} }$
• ${\displaystyle \omega \,\!}$ is the angular frequency, ${\displaystyle \omega =2\pi f\,\!}$
• ${\displaystyle \mathbf {n} }$ is a unit vector in the direction of travel
• ${\displaystyle \lambda \,\!}$ is the wavelength
• ${\displaystyle f\,\!}$ is the frequency
• ${\displaystyle c\,\!}$ is the velocity of the wave.

Epzcaw (talk) 18:07, 21 April 2011 (UTC)

If you don't understand complex numbers, you shouldn't be making major edits to this page. |A| is the magnitude, not the "real value". — Steven G. Johnson (talk) 18:43, 21 April 2011 (UTC)

My mistake - corrected below

where

• ${\displaystyle A\,\!}$ is the complex amplitude of the wave.
• ${\displaystyle |A|\,\!}$ is the amplitude of ${\displaystyle A\,\!}$
• ${\displaystyle \phi \,\!}$ =${\displaystyle Arg(A)}$ and is the phase of the wave at t =0
• ${\displaystyle \mathbf {k} }$ is the wave vector, ${\displaystyle \mathbf {k} =(2\pi /\lambda )\mathbf {n} }$
• ${\displaystyle \omega \,\!}$ is the angular frequency, ${\displaystyle \omega =2\pi f\,\!}$
• ${\displaystyle \mathbf {n} }$ is a unit vector in the direction of travel
• ${\displaystyle \lambda \,\!}$ is the wavelength
• ${\displaystyle f\,\!}$ is the frequency
• ${\displaystyle c\,\!}$ is the velocity of the wave.
Steven G, you raise an interesting point. I myself having just made a similar gaff, I take this position. If it wasn’t for guys like Epzcaw (and me) pages like this would be in a sorry state indeed. Guys like Epzcaw and myself who clearly don’t understand complex numbers as well as you, are never the less willing to take time out of our lives to improve this page. And this page is definitely in need of improvement. What is ultimately important is that what is on the wikipage is true and accurate. I am very thankful that there are guys like you who are looking over our shoulders because the last thing I would want to do is mislead the Wikipedia readers. I don’t see what is wrong with us doing the ‘grunt” work while you guys make sure that it “get done right”. I agree that some people can be “more trouble than they’re worth” but Epzcaw, and me are not two of them. The fact is that there doesn’t seem to be enough people like you willing to take the time to make pages like this more accessible to the general reader. As it happens I am presently working on doing just that. I suggest that we compromise and that means having errors temporarily existing on pages like this. You are an important part of those errors being temporary and its unfortunate that you begrudge your “role”. I’d be interested in your view and the view of others on this as I suspect my position may have some weak points. Dave3457 (talk) 20:36, 21 April 2011 (UTC) PS. Note that his particular error was on a talk page.

Given this discussion about symbols I should point out something that I'm working on. I plan to be including the real formalism of a plane wave before this complex formalism. My work in progress is here User:Dave3457/Sandbox/Talk_space_referred.(Note:The images came from the wave vector page.)
Below are the first few lines.
xxxxxxxxxxxxxxxxxxxxxxxxxxxx
One of the simplest formalisms of a plane wave involves defining it along the direction of the x-axis.

${\displaystyle A(x,t)=A_{o}\sin(kx-\omega t+\varphi )}$

In the above equation…
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
The related image is SVG so I can quickly create our own version to suit the equations. I'm personally partial to using ${\displaystyle \varphi \ rather\ than\ \phi }$ for the phase shift. (Note the LaTeX page seems to suggest we not use the ${\displaystyle \phi \,\!}$ version for angles.)
I'm not to familiar with conventions but I think things would be clearest if we used...

${\displaystyle A(x,t)=A_{o}\sin(kx-\omega t+\varphi )}$

for the real, and ...

${\displaystyle U(x,t)=U_{o}e^{i(kr-\omega t)}\,\!}$

for the complex. One could discuss how ${\displaystyle U_{o}}$ can be broken down into magnitude and phase in the discussion of the meanings of each symbol. I also like Constant314's idea of also showing the equation is this form

${\displaystyle U(r,t)=A_{o}e^{{i[(2\pi /\lambda )(\mathbf {n} \cdot \mathbf {r} -ct)}+\phi ]}}$

as it would provide continuity. It just dawned on me that I should probably use cos rather than sin as cos is the real part of the above equation. Dave3457 (talk) 20:42, 21 April 2011 (UTC)

Dave3457, I've looked at your sand box, and I think your approach has much to commend it. Your first equation is undoubtedly the simplest way to model a monochromatic wave mathematically (should the first varaible in the defition list be A, not phi?) and will be more comprehensible to those who don't want to get involved in what will look like 'hard sums' to them.

I agree with your comments to Stevenj - sarcastic comments like his are not within the spirit of Wikipedia where people give their time and effort freely to spread knowledge. It is possible to be critical and point out errors in a pleasant and friendly way. It is inevitable that errors arise whenever one is trying to explain a topic, and Wikipedia provides an ideal environment for ironing these out.

Also to Stevenj - you say: "saying that A is "generally" chosen to be real is hogwash in my experience, and would need a reference even if it were true.". I have provided references above to several very respectable optics books, all of whom chose A to be real.

Epzcaw (talk) 21:10, 21 April 2011 (UTC)

Epzcaw, I fixed the phi thing as well as a couple other variable issues. By the way, I was thinking of posting a "ready for main-space" version in my user-space in the same manner that I just posted that working version. That way you or anyone else, who has the time, could first give it a quick look over to find similar issues. We could even use the associated talk page to comment back and forth before posting it on main-space. This would probably suit Steven G :)

Dave3457 (talk) 23:11, 21 April 2011 (UTC)

Excellent. Will watch out. Epzcaw (talk) 23:27, 21 April 2011 (UTC)

I think what Stevenj meant was that you guys probably need to be more careful; I expect you understand complex numbers well enough, and need to check your work carefully, because those of us who spend a whole lot of time cleaning up after sloppy edits do get grumpy about it sometimes. I can see that you're smart and well intentioned. I was a bit frustrated with Epzcaw about being relatively unresponsive to stuff on his talk page, and repeatedly removing stuff without so much as a word of edit summary, but I assume he's learning. Thanks for helping, both of you. But keep in mind, that when you work to make it better, in whatever way, that you may also be dissing the work of others, so feel your way along, respect the pushback you get, and try not to make stuff that others need to clean up. You might want to review the version from before you started, where I think the magnitude and angle of A were correctly handled in the real version. It may also be just great to start with a real version instead of complex; but find a source that does that, and cite it. Stuff that follows and cites a source is much more likely to survive the next well-intentioned newby that comes along. Dicklyon (talk) 00:39, 22 April 2011 (UTC)

The Born and Heavens passages you cite simply write the complex amplitude in polar form (e.g. ${\displaystyle E_{0}e^{i\delta }}$). This is not the same thing as choosing it to be real. And just because one typically measures magnitude (e.g. time-average power), not phase, does not mean that phase is not important, because interference effects between multiple waves are extremely common and important in optics, and to express the relative phase of different planewaves you need at least some of their amplitudes to be complex. (Nor is absolute phase not measurable in electromagnetism, especially at longer wavelengths.) (And planewaves are not limited to electromagnetism; they appear in any wave equation.) The second part of the Born that you cite doesn't say anything about the amplitude typically being real, either; you impose your own odd interpretation on them mentioning taking the real part—there is a big difference between taking the real part of a complex planewave expression and assuming the amplitude to be real!! — Steven G. Johnson (talk) 04:48, 22 April 2011 (UTC)

Steven, I'm not sure what you're getting at with "to express the relative phase of different planewaves you need at least some of their amplitudes to be complex." In real waves (E&M, water, or otherwise) the waves are real-valued, but phase is still relevant. The only reason complex values are used is because those are true eigenfunctions of linear systems, whereas the cosine with phase expressions are not. Dicklyon (talk) 06:14, 22 April 2011 (UTC)
First, we are talking about the complex-exponential form (and the disputed claim that in this form the amplitude is usually chosen real). Having nonzero phase in the complex-exponential form is equivalent to having a complex amplitude.
Second, the cosine expressions are also eigenfunctions of linear systems (cosine and sine are eigenfunctions of the second-derivative operations that appear in the usual wave equation), although exponentials are more general in diagonalizing any linear operator with translational symmetry (for deep reasons involving the representation of the translation group). There are several reasons why one uses complex exponentials. e.g. linearity allows one to superimpose the real and imaginary parts and take the real part at the end, and complex exponentials are often algebraically easier to work with than trigonometry because the angle-addition rules are simpler for exponentials. Also, real wave equations typically have material dispersion (e.g. the index of refraction is a function of frequency), which is most simply expressed in the Fourier domain where the wave equation coefficients (in a lossy dispersive medium) become complex numbers—you can still take the real part at the end to get a real solution, but writing the intermediate steps in terms of purely real numbers becomes nasty.
(PS. There is one physical wave equation where the solutions are actually complex: the Schrödinger equation.)
— Steven G. Johnson (talk) 14:40, 22 April 2011 (UTC)
Yes, I agee with all that, though with reservations on how "physical" a QM wave function is. So by "to express the relative phase of different planewaves you need at least some of their amplitudes to be complex," you meant for complex-valued wave; for real waves that's not the case. I thought you were saying at least one wave had to be complex-valued; it's the overload amplitude term confusing me again. Dicklyon (talk) 17:11, 22 April 2011 (UTC)
Yes, I am talking about the complex exponential wave; that is the context in which the amplitude must generally be complex. (While the QM wavefunction is only indirectly observable, it is definitely not true that taking the real part of it makes it more physical or more observable.) — Steven G. Johnson (talk) 18:06, 22 April 2011 (UTC)
I think sines, cosines and exponentials are eigenfunctions of linear time invarient dynamical systems and that the complex amplitudes are the eigenvalues.Constant314 (talk) 17:34, 22 April 2011 (UTC)
No, this is wrong, in two ways. Consider the linear time-invariant operator d/dt (first derivative). exp(iωt) is a (generalized) eigenfunction of this operator, and the eigenvalue is iω – not the amplitude. (Multiplying an eigenfunction by a scalar amplitude never changes the eigenvalue; you are confused about the definition of eigenvalue.) However, sin(ωt) is not an eigenfunction, because (d/dt)sin(ωt) = ωcos(ωt) — not just a multiple of sin(ωt). (Interestingly, Gil Strang once pointed out to me that sine and cosine could be thought of as the analogue of singular vectors of such operators, which is a generalization of the eigenvector concept.) — Steven G. Johnson (talk) 18:01, 22 April 2011 (UTC)
You put a sinusiodal wavefrom into the input of a linear time invarient system and the output is a sinusoidal waveform of exactly the same frequency of possibly a different amplitude and possibly phase shift relative to the input. You let the complex eigenvalue take care of the amplitude and the phase shift.Constant314 (talk) 22:55, 30 April 2011 (UTC)
I second what Steve said. Because of the phase change, sines and cosines are not eigenfunctions; that's why we need to go to complex exponentials. And amplitudes are not eigenvalues; input amplitudes get multiplied by the eigenvalues to make output amplitudes, all of which are generally complex. Dicklyon (talk) 23:11, 30 April 2011 (UTC)
I should be more precise: the eigen-values of a linear time invariate system comprise the (complex) frequency response.Constant314 (talk) 04:56, 1 May 2011 (UTC)
I knew that's what you meant to say; but I wouldn't use comprise in that backwards sense; the complex frequency response comprises the eigenvalues. And the corresponding eigenfunctions are complex exponentials. Dicklyon (talk) 05:51, 1 May 2011 (UTC)
Then let me say the eigenvalues compose the frequency response.Constant314 (talk) 06:21, 1 May 2011 (UTC)
You still aren't using the words properly.
First, "systems" don't have eigenvalues, "operators" do. Whether the eigenvalues of an linear time-invariant operator O are "frequencies" however, depend on what the operator is—they are only frequencies per se if O is the operator that yields the time derivative of the state ψ at a given time, i.e. if the operator O appears in the system i∂ψ/∂t = Oψ, or if you are talking about the ∂/∂t operator itself. (Yes, I know this looks like a Schrodinger equation, but you can actually write Maxwell's equation in this form if you let ψ be a 6-component vector of the E and H fields.) There are lots of other operators whose eigenvalues are not frequencies. For example, the ∇2 operator that appears in the scalar wave equation is related to the second time derivative, and so its eigenvalues are -1 times the frequency squared, not frequency. And an operator that doesn't describe time evolution will probably have eigenvalues that have nothing to do with temporal frequencies at all. (e.g. the eigenvalues of the ∇2 operator in Poisson's equation are not "frequencies" in that context.)
Second, to the extent that the frequencies are eigenvalues, they aren't the "frequency response". They are the frequency spectrum. The Frequency response usually refers to some kind of transfer function: it describes the amplitude as a function of frequency in response to inputs at the given frequencies.
(Aside: To be clear, sines and cosines can be eigenfunctions, Dicklyon. It depends on the operator. The are not eigenfunctions of first derivatives, but they are eigenfunctions of second derivatives. Because most (not all) wave equations involve second derivatives, it is not unreasonable to use sines and cosines in this context, although complex exponentials are more flexible.)
— Steven G. Johnson (talk) 06:40, 1 May 2011 (UTC)
Let's say I have physical object like an AC coupled bandwidth limited amplifier and I make a mathematical model of the object. The equations in my model are all ordinary linear equations or linear differential equations and all the coefficients are time invariant. There is a variable in my model called Vin that is associated with the input node of the object. There is also a variable in my model called Vout that is associated with the output. And I define an operator on my model that maps functions applied to Vin to the functions observed at Vout. Although I did not consciously choose to do it, the way I setup the equations to model the object are such that the operator is a mapping of the Laplace transform (LT) of the input to the LT of the output. I'll use my oscilloscope to search for Eigen-functions of the actual object. If I find one, I will know it because the output function is the same as the input function except multiplied by a scalar and maybe phase shifted. I apply a high frequency square wave function. The edges of the output function are rounded off. This function is not an Eigen-function. I apply a low frequency square-wave and I see overshoot at the output. This function also is not an Eigen-function. I try impulses, triangle waves, ramps, clarinet sounds. Always the output is different from the input. All those functions are not Eigen-functions. But, if I put in a sine, cosine, exponential, exponentially damped sine or exponentially damped cosine, I get back exactly what I put in except the amplitude is different and may be phase shifted. By happy circumstance I find that when I apply my operator to the LT of the input function, the output is equal to the LT of the input times a complex scalar. That scalar is the Eigen-value of the operator associated with the LT of the input function. The LT of the input function is an Eigen-function of the operator. But, we become so comfortable working with LT's that we start thinking that the LT of the function is the function.Constant314 (talk) 02:41, 2 May 2011 (UTC)
If your sine wave is phase shifted (other than by multiples of 180 degrees as Steven points out), it's not an eigenfunction; the output is not the input times a factor in that case. Dicklyon (talk) 03:30, 2 May 2011 (UTC)
It is an eigen function if I can define a scalar multiplication that accounts for the amplitude change and the phase shift. And a question about editing a this talk page. Can we make a new section on this page and move all this eigen stuff there?Constant314 (talk) 20:48, 2 May 2011 (UTC)
Steven, I'm not sure why you're being so picky about the language. Can't we say a "system" that has an input and an output has eigenfunctions and eigenvalues just like what you call an "operator" has? And how did we get off on the "frequencies" thing? You're interpreting something we said as meaning eigenvalues proportional to frequency? The "frequency response" is what you're calling the "frequency spectrum", but I think it's odd that you call it that. What field are you coming from? I'm an EE. Dicklyon (talk) 03:30, 2 May 2011 (UTC)

Contrary to what I said earlier I’ve decided to post my changes directly to the article as I don’t believe anything in it is misleading. I plan to make a couple more additions later.
Epzcaw, I hope you don’t mind but several of your recent changes have had to go by the wayside because of the change in order. Never the less most of it is still there but in a different form.
While the symbols, A,Ao and U, Uo, used for the formulas suit this page I'm not sure to what extent they conform to the general literature. Let me know if it is a problem and I'll be happy to look after it as it will involve changing the image. In the paragraph following my contribution I changed the references to variable A to Ao. I'm not certain that Uo wouldn't have been more appropriate.
The first paragraph in the lede has not been changed nor anything below the comment about quantum mechanics,(except the variable A).

Note to Dicklyon: I read your comment about using the math inline template and would have been happy to do so but I could not get it to produce a PNG image. I can't help but feel that it reads better and is clearer when the variable in a sentence visually matches the variable in the equation and this requires converting it to a PNG image. My feeling is that using the text version of a variable in a sentence makes the reader work harder to "piece things together". The reader is forced to do a "mental" conversion rather than make quick visual associations. I appreciate that this is not the norm but I also don't believe that it is "hard core" MOS. The relevant MOS section seems to be here Wikipedia:Manual_of_Style_(mathematics)#Using_LaTeX_markup. Admittedly here is a quote from it..."try to avoid in-line PNG images". That page lists the cons but I can't help but feel that the pros out weight the cons. I looked for some pages that use the inline PNG image method and found these do, if not to a much lesser extent,Ordered_pair,Exponential_growth I would be interested in your views. I'm prepared to begrudging make the changes but I would do so with a heavy heart because it would result in a less pleasant and straight forward experience for the reader. I'm a person that tends to put functionality ahead of "prettiness".
Of course I would welcome anyone else chiming in.
Dave3457 (talk) 02:48, 30 April 2011 (UTC)

The point of the template is to match sentence style when the math is in a sentence, rather than making a png to match display math. This is also very useful for blind readers, whose screen readers have a harder time with pngs. Dicklyon (talk) 21:40, 30 April 2011 (UTC)

Well done Dave3457 (no problem about omitting anything I suggested). I believe it is much better to start with a simple system and move on to a more complex one. Many Wiki readers will be totally intimidated by complex variables, while they will find trig functions comprehensible. Longhurst (Geometrical and Physical Optics) discusses much of wave optics using trig functions only.

As far as the math inline template vs. LaTex, I have to say I agree with Dicklyon - I think it looks better and is more readable. This may, of course, be purely a matter of taste - we need an ergonomist!! Bur presumably, Wikipedia style guidelines are based on some expertise about this? Will maybe try converting some of it, and putting it here to gauge opinion.

Is there any way of avoiding the large white space at the start of the section? Epzcaw (talk) 19:48, 30 April 2011 (UTC)

I don't see a large white space, are you sure you're looking at the actual main-space page and not my user-space sandbox.
Concerning you statement "... much better to start with a simple system and move on to a more complex one." I agree, for a page like this the "target audience" should be the high school student trying to make sense of his physics class.
As far as text versus LaTex thing...( I'm about to push back, don't read in negative emotions)...you say in part.."I think it looks better" but looks shouldn't be the issue, in my mind it should be functionality. The question should be "is it easier to digest the information?". My position is that a person can, with less mental effort, visually connect a variable in a sentence with a variable in an equation. I'm the first to admit that it isn't as "pretty".
As far as placing them side by side, thats not a bad idea, Below are the two methods side by side.

xxxxxxxxx Below is inline PNG images

${\displaystyle A(x,t)=A_{o}\cos(kx-\omega t+\varphi )}$

In the above equation…

• ${\displaystyle A(x,t)\,\!}$ is the magnitude or disturbance of the wave at a given point in space and time. An example would be to let ${\displaystyle A(x,t)\,\!}$ represent the variation of air pressure relative to the norm in the case of a sound wave.
• ${\displaystyle A_{o}\,\!}$ is the amplitude of the wave which is the peak magnitude of the oscillation.
• ${\displaystyle k\,\!}$ is the wave’s wave number or more specifically the angular wave number and equals ${\displaystyle 2\pi /\lambda \,\!}$, where ${\displaystyle \lambda \,\!}$ is the wavelength of the wave. ${\displaystyle k\,\!}$ has the units of radians per unit distance and is a measure of how rapidly the disturbance changes over a given distance at a particular point in time.
• ${\displaystyle x\,\!}$ is a point along the x-axis. ${\displaystyle y\,\!}$ and ${\displaystyle z\,\!}$ are not part of the equation because the wave's magnitude and phase are the same at every point on any given y-z plane. This equation defines what that magnitude and phase are.

xxxxxxx Below is inline Text

${\displaystyle A(x,t)=A_{o}\cos(kx-\omega t+\varphi )}$

In the above equation…

• ${\displaystyle A(x,t)}$ is the magnitude or disturbance of the wave at a given point in space and time. An example would be to let ${\displaystyle A(x,t)}$ represent the variation of air pressure relative to the norm in the case of a sound wave.
• ${\displaystyle A_{o}}$ is the amplitude of the wave which is the peak magnitude of the oscillation.
• ${\displaystyle k}$ is the wave’s wave number or more specifically the angular wave number and equals ${\displaystyle 2\pi /\lambda }$, where ${\displaystyle \lambda }$ is the wavelength of the wave. ${\displaystyle k}$ has the units of radians per unit distance and is a measure of how rapidly the disturbance changes over a given distance at a particular point in time.
• ${\displaystyle x}$ is a point along the x-axis. ${\displaystyle y}$ and ${\displaystyle z}$ are not part of the equation because the wave's magnitude and phase are the same at every point on any given y-z plane. This equation defines what that magnitude and phase are.

xxxxxxxxx
Again I can’t help but feel the first version is easier to digest and make sense of because of the visual similarity of the inline variables and the variables in the equation.

Things don’t seem to be looking good for my position so I propose the following compromise where only the variables in the text that are referencing variables in the equation are PNG images.

xxxxxxxxx Version where only the variables being referenced in the equation are PNG images

${\displaystyle A(x,t)=A_{o}\cos(kx-\omega t+\varphi )}$

In the above equation…

• ${\displaystyle A(x,t)\,\!}$ is the magnitude or disturbance of the wave at a given point in space and time. An example would be to let ${\displaystyle A(x,t)\,\!}$ represent the variation of air pressure relative to the norm in the case of a sound wave.
• ${\displaystyle A_{o}\,\!}$ is the amplitude of the wave which is the peak magnitude of the oscillation.
• ${\displaystyle k\,\!}$ is the wave’s wave number or more specifically the angular wave number and equals ${\displaystyle 2\pi /\lambda }$, where ${\displaystyle \lambda }$ is the wavelength of the wave. ${\displaystyle k\,\!}$ has the units of radians per unit distance and is a measure of how rapidly the disturbance changes over a given distance at a particular point in time.
• ${\displaystyle x\,\!}$ is a point along the x-axis. ${\displaystyle y\,\!}$ and ${\displaystyle z\,\!}$ are not part of the equation because the wave's magnitude and phase are the same at every point on any given y-z plane. This equation defines what that magnitude and phase are.

xxxxxxxxx
Again, I can't help but feel the above is much clearer.

As a favor could I get you to directly respond to my view that putting PNG variable images in the sentence makes it easier to connect referenced variables in the text with the variables in the equation. Am I the only one that believes this?
Dave3457 (talk) 21:32, 30 April 2011 (UTC)

If the matching is a big deal for you, try going the other way and doing it all with the math template instead of PNGs. Dicklyon (talk) 21:45, 30 April 2011 (UTC)
Concerning the blind, LaTeK generates a default Alt text for the blind when creating images, refer here Alt_text. I do appreciate that the math template makes things easier to edit, but again, no PNG feature.
I don't think doing it all with the math template is an option because then the equations wouldn't stand out. Also, no one else does. I'm leaning toward the compromise because it would also reduce the chances of someone in the future reverting it all to template. Dave3457 (talk) 23:15, 30 April 2011 (UTC)
OK, I won't worry about the blind problem then. I still don't see why you want a PNG; the point of the template is to make suitably styled html. Dicklyon (talk) 04:24, 1 May 2011 (UTC)

The large white space occurs after the introduction - at the start of the Mathematical formalism (definitely on the main page, not your user page). Probably to do with my laptop screen resolution, but gives an initial impression that there is nothing there , and I have to scroll down to see the text. Obviously to do with the size and location of the figures.

As far as math template is concerned, I think it looks better because it is easier to read than the larger font symbols created by LaTeX whihc I find slightly irritating but as I said that is my opinion. Maybe need to search Wikipedia for an articel on 'Readability of maths within text'!!! Epzcaw (talk) 10:13, 1 May 2011 (UTC)

xxx

### More inline PNG images talk

Dicklyon and Epzcaw: Well, first I did the comprise version thing, now only variables that are directly referring to variables in the equations are PNG images. I think that should help your (Epzcaw) “slightly irritating” concern because now the vast majority of PNG images occur at the beginning of the sentences.

By the way I found several pages that do what I did by going 1/4 of the way down the List of equations page.

Equations_of_motion#Equations_of_uniformly_accelerated_linear_motion.
Equation_of_state#Major_equations_of_state
Bernoulli's_equation#Incompressible_flow_equation
Doppler_equations#General
Quartic_equation#Solving_a_quartic_equation I think this guy thinks along the same lines as myself.

Personally I suspect that the reason it is not done more often is because some don’t know how to force the text to display as a PNG image and so they “leave it to chance”. Gibbs–Helmholtz_equation might be an example and Relativistic_Euler_equations is definitely an example.

Dicklyson, I’m not sure I know what you mean by “suitably styled html”. If you mean for the benefit of the editors I have to say I think the readers should come first.

Given that you ”still don't see why I want PNG” I’ve collected my points from above and reproduced them below in bullet form.

• I feel that it reads better and is clearer when the variable in a sentence visually matches the variable in the equation... My feeling is that using the text version of a variable in a sentence makes the reader work harder to "piece things together". The reader is forced to do a "mental" conversion rather than make quick visual associations.
• The question should be "is it easier to digest the information?". My position is that a person can, with less mental effort, visually connect a variable in a sentence with a variable in an equation.
• (I) feel it is easier to digest and make sense of things because of the visual similarity of the inline variables and the variables in the equation.

Another point that is worth making is that while the extra effort may not make much difference to individuals such as you two who are dealing with this stuff a lot and have grown lots of brain synapses to deal with variables and equations, the typical person reading a page such as this may be new to even the cosine function and how its implemented. Some may have never even seen the ${\displaystyle \varphi \,}$ symbol before. They haven’t got the familiarity with these symbols that you guys have. I feel that we should do everything we can to make things as straight forward as possible for them, particularly with a page such as this.
Dave3457 (talk) 05:43, 2 May 2011 (UTC)

See Template:math, which says The Math template formats HTML- or wikimarkup generated mathematical formulas in the same manner as HTML based TeX formulas, which uses a serif-based font. The generated formula is displayed using the same-size font as the adjoining text. The template will prevent line-wrapping. ... Use this template for non-complex formulas as an alternative to using the "math" tag format.
And see HELP:Math. These have been that way since late 2008, but I never saw the use of the math template until recently, which means most editors weren't using it. It's so much better than the old scheme of just putting math variables in italics, keeping the ugly sans serif font. Anyway, that's what I meant by "suitably styled html". Knowing this is available means that there's less reason to resort to images for equations. Dicklyon (talk) 06:07, 2 May 2011 (UTC)

I still feel the in-line PNG images look untidy, and are therefore distracting - for example, the 'y' on the line where x is defined is sitting well above the rest of the text. Many of them look as if they were literally 'cut and pasted' by someonee rather untidy. I assume there is a mechanism for altering their position, but this will make the job much longer and fiddlier.

Preusmably it is not acceptable to reduce the size of the text in the LaTeX equations to mathc the text size in the article, as occurs in books. If this were done, then the Math template in-line symbols would match exactly the symbols in the equations, and we would all be in agreement.Epzcaw (talk) 10:08, 2 May 2011 (UTC)

Dicklyon and Epzcaw, you're not addressing my concerns or reasons. Do you think they are unfounded?
I agree Dicklyon, the math template is much better than italics and such. Concerning the quote "Use this template for non-complex formulas as an alternative to using the math tag format." . This isn't a MOS page and was presumably written by the same people who created the tag and they are just encouraging people to use it when they can.
I would point out that with only a few exceptions the PNG images are not "in-line" PNG images so much as "before-line" images :)
Epzcaw, The "y" doesn't stand up in my Firefox browser and only stands up a bit in my Explorer. Here is a screen shot [1] It doesn't look bad if you ask me. I would also point out that the help pages talk about how they are attempting to refine the placement of the images. Here is a quote from the Math page.."Vertical alignment with the surrounding text can also be a problem. The css selector of the images is img.tex. It should be pointed out that solutions to most of these shortcomings have been proposed by Maynard Handley, but have not been implemented yet."
In the end, these are my points.
• There are clear reason's for doing it.
• Other people are doing it.
• In the end the MOS leaves it optional. Call it a loop-hole if you want :)
• In my view your cons don't out weight the pros.
• I'm not a "drop in" editor of this page as I created the associated text and in the past created the plane wave animation and the "Polarized electromagnetic plane waves" section, including its illustrations. If you're prepared to leave the decision to me, thank you :) Dave3457 (talk) 06:34, 3 May 2011 (UTC)

Just my opinion - certainly not down me to say how it should be. So up to you!! Epzcaw (talk) 11:26, 3 May 2011 (UTC)

## Blank space at the start of Mathematical Formalism section

Sorry to keep banging on about this, but I do think it does distract from the quality of the article. I have put two screen dumps into a JPG files so you can see what I mean.

I realise that different screen resolutions always pose a problem fro web page display. My screen is 1280x800 so not particularly low.

Is there a Wikipedia fix for this?Epzcaw (talk) 10:25, 2 May 2011 (UTC)

Well it not the browser, I use Firefox (1600:900) but tried Explorer with no problem. That being said if it is happening to you it is probably happening to others. As it happens I tried a couple of different things when I was working on it, even putting the first two images side by side below the lede but it didn't look good. What I often do is insert the following tag when I start a new section and want the images directly to the right. The below causes the text to clear the images before beginning. {{-}}. I didn't use it because it brought down the headings as well, however I just discovered that you can use __TOC__ to force placement of the Heading list. Relevant link...Help:Section#Positioning_the_TOC. I made the changes to the page and added a comment about the situation. I personally like it better this way anyway. Dave3457 (talk) 20:39, 2 May 2011 (UTC)
Most likely it is the browser. What versions of IE did you guys try? This was pretty standard wiki format, and the new version has a lot of extra trapped while space, so I think we should go back to the normal way and let Epzcaw debug his setup. Dicklyon (talk) 22:12, 2 May 2011 (UTC)

Again, even if Epzcaw wasn't having problems, I personally like it this way. It makes sense to have the illustrations related to the section Mathematical formalisms distinctly in the proper section. This way the image related to the equation is directly across from the equation. Things are more straight forward for the reader this way. One could shrink the wavefronts image to make the size of the blank space smaller. I guess Epzcaw will be the tie breaker unless someone else chimes in. Dave3457 (talk) 03:30, 3 May 2011 (UTC)

This works for me, and I think it is better to have diagrams adjacent to the text to which they relate.

I am using IE 7.0.6002. I haven't upgraded to 8 in spite of constant messages to do so, because I found the last upgrade extremely annoying and I am sure there are many other people out there using older versionns.

I am not complaining about this on my own account. I am looking at it from the point of view of someone new to to the section who is using Wikipedia to learn about the subject, and I think that at least some would assume when they saw the blank space that the article was incomplete, and leave it. Epzcaw (talk) 09:28, 3 May 2011 (UTC)

First, thanks for the hidden comment about the white space. It saved me time & effort and directed me here. Also, to give you background on the pros and cons of taking my advice, I was in technical communications for about 10 years until I got too sick to work, in part from daily fever spikes. I have one of those right now, so I ask your patience and I hope this makes sense. :)
I am in favor of moving the second image to the next section. The way the layout is right now requires more pre-frontal processing to comprehend. The gigantic white space registers as, "Oh, is this an error or a stub?" This interrupts the comprehension process. We're not talking about moving the image to the bottom of the article, just down a little bit.
Unless the image's meaning is somehow tied only to one sentence in all senses, ways, denotations, connotations, etc., there's no reason to lock it that tightly to a paragraph that's not on a printed page (in which case you would just need to make sure the reader doesn't have to flip pages to comprehend the text and the image together). The image will still be in a de facto "left-hand column" in the same relative order, which novices to the topic can easily skim. Experts and novices can both relate the image to the most relative text as well as to the page as a whole.
Sorry that all I can do is post my input & that I might not make it back here to answer any comments. I'm leaving the decision of what action to take up to other people, because due to this chronic illness I can only spend unknown & limited amounts of time & energy on any one edit. Actually, I'm typing with a feverish deficit already today. Ah, and I really need to get around to reading my talk page again sometime! Oh, well, nap time. :) Thanks!
--Geekdiva (talk) 02:57, 23 November 2011 (UTC)

## The first picture has bad perspective

The first picture (the one with the caption "The wavefronts of a plane wave traveling in 3-space") shows a series of rectangles in three-dimensional space. The problem is that the artist used perspective on each rectangle individually (the far edge of a rectangle is shorter than the near edge), but did not apply perspective to the plates as a whole (the back rectangle is not smaller than the frontmost rectangle, and the top edges of all the rectangles to do converge to one perspective point). The result is a an eyesore to anyone used to dealing with 3D pictures on a detailed level. Either the whole picture should have correct perspective (which would require more than copy and pasting a tapered rectangle in a draw program - it would require modeling it in a 3D renderer, or have it drawn by an artist that understands perspective), or the whole picture should have no perspective at all. 129.63.129.196 (talk) 19:05, 26 February 2013 (UTC)

## A simple plane wave animation

I appreciate the effort, but I'm removing this animation because there is already an animation on the page that is better. My main criticism is that this animation does not illustrate the planes of constant phase and amplitude.Constant314 (talk) 17:03, 28 September 2013 (UTC)

Agreed, the one removed was only a 2D wave. Dave3457 (talk) 03:42, 13 October 2013 (UTC)

## Momemtum

In the complex form, k is proportional to the momemtum of the point? Jackzhp (talk) 01:18, 19 October 2013 (UTC)

## Regarding Section: Polarized electromagnetic plane waves (paragraph 6)

For convenience, here is a paste of paragraph 6.... "Not indicated in either illustration is the electric field’s corresponding magnetic field which is proportional in strength to the electric field at each point in space but is at a right angle to it. Illustrations of the magnetic field vectors would be virtually identical to these except all the vectors would be rotated 90 degrees perpendicular to the direction of propagation.".... The last 6 words of this paragraph "perpendicular to the direction of propagation." are either wrong, misleading, or just very unclear. If the magnetic field is perpendicular to the electric field at each point in space as stated earlier in this paragraph, then for both linear and circular polarization it appears to be a contradiction to state "all the vectors would be ... perpendicular to the direction of propagation." — Preceding unsigned comment added by 97.125.86.96 (talk) 19:41, 30 October 2014 (UTC)

Yes. there was something vague or wrong with that sentence. It has been fixed.Constant314 (talk) 21:14, 30 October 2014 (UTC)
Thank you and much improved. Your edit makes it clear at each point in space the magnetic field vectors occupy the same plane as the electric field vectors only rotated 90 degrees, which might be worth stating for clarity. Is there a particular consistent description which defines this 90 degree rotation? For ex: The rotation is always clockwise facing the direction of propagation, or is the rotation context dependent, inconsistent, or unknown? Also, the magnetic field is proportional in strength to the electric field. Are we using the term "proportional" in a math sense in which the two fields have a constant but unspecified ratio, or are we saying more specifically that the two field strengths are the same and the ratio is 1 to 1? If we know the proportionality let's state it explicitly. How these two fields relate is central and very interesting to readers. Thanks for your time and expertise. — Preceding unsigned comment added by 97.125.86.96 (talk) 17:50, 31 October 2014 (UTC)
Circularly polarized light can rotate CW or CCW. Often, circularly polarized light is in a mix with ordinary polarized light. The proportionality constant between the electric field and the magnetic field in free space is the impedance of free space which is about 377 ohms in MKS units. It depends on the units. In some systems of units, the proportionality constant is unity.Constant314 (talk) 22:30, 31 October 2014 (UTC)

## "harmonic and homogeneous"

The article currently takes plane waves to mean any constant-velocity wave with planar wavefronts, however in my experience the term "plane wave" only ever refers to waves with a well defined frequency and well defined wavevector, i.e., "harmonic and homogeneous" as is now stated in the article. Is there a source for this more general, expanded notion of "plane wave"? If not, what is the simplest term I can use to refer to a wave of the form ${\displaystyle e^{i{\vec {k}}\cdot {\vec {x}}-i\omega t}}$? --Nanite (talk) 10:00, 23 November 2016 (UTC)