Talk:Wave equation

Removed

The basic equation is:

$v=f\lambda$

I'm not sure it should be removed. Although "perfectly correct", it's the same as listing the differential-only forms of Newtonian mechanics -- correct, but not useful.

Perhaps you (or I) should add it back in a section describing specific solutions, such a standing wave patters, or in this case, singletons.

Fair enough. as it stood it was confusing and seemed unrealed to the differential equation this article is about -- Tarquin 13:11 Jan 6, 2003 (UTC)

Not sure if this is the right way to suggest this,(bit of a beginner with wikipedia) -- surly the above formula should be in the article as it's the basic formula and very widely used, i was looking for it when i searched "wave equation". Sorry again if this is the wrong way of suggesting.

scalar or vector?

the function in search, u, is scalar or vector-valued? i.e. when is it what?

Unless we have departed from classical mechanics, space only has three dimensions.

The introduction contains: "and one or more spatial variables x₁, x₂, ..., x_n (variables representing a position in a space under discussion)"

It is obscure why the author has chosen to deal with n-dimensional space in the article whose preamble clearly addresses classical physics: "waves or standing wave fields – as they occur in classical physics ..." (my emphasis)

The introduction of n-dimensions is clearly misleading and should be removed.

Gpsanimator (talk) 21:50, 18 December 2023 (UTC)

Sorry I only partly agree. We should depart from classical mechanics: the article is named "wave equation" not "classical wave equation". However we don't want to go into quantum here, just any aspect directly related to the differential equation per the short description.

It also seems suitable under the topic to discuss n-dimensional wave equation. However I agree 100% that this aspect need not be in the introduction.

Overall the article seems to be focused on physics in 3 or less real dimensions, a good choice. A section on n-dimensions and on complex valued wave equations seems appropriate especially as summaries of other Wikipedia articles. Johnjbarton (talk) 00:30, 19 December 2023 (UTC)

I performed an 'undo' on the recent changes from Gpsanimator due to overloading of the word Amplitude, and with it removed the earlier edit changing n dimensions to just 3. My mistake - sorry about that. A change to 3 dimensions seems reasonable to me, and (pending outcome here) it probably makes sense for that change to be restored. Chumpih ^t 02:06, 19 December 2023 (UTC)

I am pleased that we have partial agreement on the edits, that's great!

You say:"We should depart from classical mechanics:", but I have to draw your attention to the opening sentence of the abstract: "The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields – as they occur in classical physics" (my emphasis)

The article is indeed about waves in classical physics.

You " performed an 'undo' on the recent changes from Gpsanimator due to overloading of the word Amplitude", but all I did was to selectively replace the word "displacement" with "amplitude" since the former clearly refers to an amplitude measured in a distance property, while "amplitude" more appropriately can be applied to properties of distance, pressure, magnetic field, gravitational field or light intensity - all equally well defined by the wave equation.

I note you have also removed my edits restricting the space dimensions to 3 from n.

Is it possible to restore my edits? Gpsanimator (talk) 02:25, 19 December 2023 (UTC)

Regarding "classical physics", my point is that this phrase can be deleted without altering the title of the article. A "wave equation" is a particular second order differential equation. In fact I think the lede sentence is simply incorrect. Johnjbarton (talk) 02:51, 19 December 2023 (UTC)

The wave equation arises as the solution to real-world phenomena. The second order differential equation is proposed to describe the propagation of waves in acoustics, light, E-M radiation, gravitational waves etc. These phenomena only exist in the 3-dimensions of classical space.

Sure, as an exercise in mathematics, or theoretical physics, you might chose to experiment with the equation in more dimensions, but in the "real world" they only exist in 3 dimensions.

Let's not over-complicate the issue, it's difficult enough in 3 dimensions. Gpsanimator (talk) 06:23, 19 December 2023 (UTC)

@Chumpih Please open a separate topic next time. Johnjbarton (talk) 02:30, 19 December 2023 (UTC)

I understand your request was directed to @Chumpih, but I'm not quite sure I follow. Can you give me an example of what you're looking for? Gpsanimator (talk) 02:38, 19 December 2023 (UTC)

Your topic was Unless...three dimensions. @Chumpih's topic was "I performed an undo". Now we have two topics. So we have to preface every reply with "regarding dimensions" or "regarding undos". That is why the Talk page has an "Add topic" feature. Johnjbarton (talk) 02:42, 19 December 2023 (UTC)

Thank you for the clarification. I see your point. Gpsanimator (talk) 03:40, 19 December 2023 (UTC)

I was merely giving context for my removal of the 'n to 3' edit, which is in scope of this section. The discussion here of displacement versus amplitude is indeed out of scope, and per WP:BRD (if you're in to that) it would be for someone else to create the section. Chumpih ^t 03:13, 19 December 2023 (UTC)

I'll create a new talk topic "displacement vs amplitude" where we can arm-wrestle that topic. Gpsanimator (talk) 03:46, 19 December 2023 (UTC)

I wonder how you would react to a suggestion that we replace all references to x₁, x₂, x₃ with x, y, z respectively? Gpsanimator (talk) 01:27, 1 January 2024 (UTC)

Not unreasonable to me, but I don't have a strong opinion here. May I suggest you create another section on the matter to see if consensus can be found? Then again, that sounds like a lot of typing for something that might prove totally uncontroversial, and acceptable to the watchers. So perhaps just go in with an edit per WP:BRD and be later forgiven (or not)? Chumpih ^t 01:41, 1 January 2024 (UTC)

OK, thanks, will do. Gpsanimator (talk) 00:09, 2 January 2024 (UTC)

Regarding dimensions, do you have a reference that shows classical wave equations only exist in 3 or less dimensions? I don't think that is true. I think you can have wave equations in any number of dimensions. Johnjbarton (talk) 02:45, 19 December 2023 (UTC)

re:"I think you can have wave equations in any number of dimensions" Can you offer a reference in the context of classical physics?

I don't think it's necessary for me to offer a reference since in classical physics space only, and always, has three dimensions. However wave theory is often presented in one or two dimensions in order to simplify the maths. Gpsanimator (talk) 03:53, 19 December 2023 (UTC)

Classical mechanics, eg Hamiltonian mechanics, Hamilton-Jacobi equation and the like all use multi-dimensional spaces, eg configuration space (physics) or phase space. That part is entirely the same as in quantum mechanics. In QM the equations just have operators, and QM certain works with multidimensional wave equations.

In addition, surfaces of constant action form "wavefronts", see Hamilton's optico-mechanical analogy. In optics these surfaces are EM wavefronts so they should obey the wave equation.

Having said that, I am surprised to admit that I can't find a ref for >3 dimensional non-QM wave equation. Given all the obscure corners of physics I would expect this one to be covered ;-) I concede the point about dimensions and raise a separate topic for the lede "classical". Johnjbarton (talk) 06:24, 19 December 2023 (UTC)

To quote Feynman :

Mathematicians, or people who have very mathematical minds, are often led astray when “studying” physics because they lose sight of the physics. They say: “Look, these differential equations—the Maxwell equations—are all there is to electrodynamics; it is admitted by the physicists that there is nothing which is not contained in the equations. The equations are complicated, but after all they are only mathematical equations and if I understand them mathematically inside out, I will understand the physics inside out.” Only it doesn’t work that way. Mathematicians who study physics with that point of view—and there have been many of them—usually make little contribution to physics and, in fact, little to mathematics. They fail because the actual physical situations in the real world are so complicated that it is necessary to have a much broader understanding of the equations.

Space has three dimensions of distance. Let's not lose sight of the physics. Gpsanimator (talk) 01:38, 23 December 2023 (UTC)

Some physics not in your sight:

• Pedagogical Introduction to Extra Dimensions Thomas G. Rizzo
• Spacetime
• Kaluza–Klein theory
• Large extra dimensions
• Universal extra dimensions

But let's also not lose sight of our basic agreement that this article has the right content for the intended audience. Johnjbarton (talk) 01:59, 23 December 2023 (UTC)

So, we'll stay with 3 dimensions, right? Gpsanimator (talk) 09:08, 23 December 2023 (UTC)

I wonder how you would react to a suggestion that we replace all references to x₁, x₂, x₃ with x, y, z respectively? Gpsanimator (talk) 01:26, 1 January 2024 (UTC)

Green's function - Lack of proper references.

A new section has been introduced that has some issues regarding relevancy, context (WP:PCR) and lack of references (WP:V) especially regarding claims (WP:NOR) at the beginning of the subsection, e.g.:

• [...there are two impulse responses: an acceleration impulse and a velocity impulse. The effect of inflicting an acceleration impulse is to suddenly change the wave velocity $\partial _{t}u$.]
• [For velocity impulse, $s(t,x)=\partial _{t}\delta ^{D+1}(t,x)$, so if we solve the Green function $G$, the solution for this case is just $\partial _{t}G$.]

After addressing the issues with the author directly some improvements have been made regarding adding context to some of the other subsections, but when it comes to providing proper references we unfortunately reached a stalemate. The only reference is a lecture note on the homogeneous wave equation that in no way directly supports the claims regarding the inhomogeneous wave equation.

Given that the author clearly put a lot of effort in writing the subsection, I'd like to ask for opinions of others on this matter. For now I'll just add a {{citation needed}}. Roffaduft (talk) 05:58, 21 August 2024 (UTC)

Ok, the claims regarding the source terms $s(t,x)=\delta ^{D+1}(t,x)$ and $s(t,x)=\partial _{t}\delta ^{D+1}(t,x)$ follow by substituting them in the "additional term" given in the "Duhamel's principle" subsection, i.e., they are only part of the solution of the wave equation.

The following statement just introduces a lot of ambiguity IMHO:

• Since the wave equation [...] has order 2 in time, there are two impulse responses: an acceleration impulse and a velocity impulse. The effect of inflicting an acceleration impulse is to suddenly change the wave velocity $\partial _{t}u$. The effect of inflicting a velocity impulse is to suddenly change the wave displacement $u$.

First, because it fully ignores the initial conditions of $u$ and $\partial _{t}u$ and, second, because either choice of source term $s(x,t)$ can (indirectly) affect both the displacement and velocity of the wave. Roffaduft (talk) 08:55, 22 August 2024 (UTC)

Explanation of wave equation is not right

Some of the explanation of the wave equation is not correct, or at least is written from a much more casual tone than the rest of the article. For example it actually uses the word “pointy” in reference to potentials u with very high second derivatives. Unless anyone objects, I’ll clean this up in a second, and I would appreciate it if someone could review what I replace the current explanation with. OverzealousAutocorrect (talk) 02:58, 26 August 2024 (UTC)

WP:BOLD, just assume you did a good job when nobody responds to your edits.

I think the whole article could use a good cleanup/restructuring. For example:

• There is an "introduction" subsection after the introduction
• The article goes from 1D to 3D to 2D to multi-D
• Too much emphasis on scalar/vector in the subsection titles IMHO
• The sections on the inhomogeneous wave equation are a bit blehh.
• The referencing is pretty horrible throughout the article.

If you're willing to work on the "introduction" subsection, I'd love to help out. Maybe we can move some of the general introductions to the actual intro of the article, e.g, introduce the distinction between vector and scalar over there to allow reorganising the rest of the article and changing some of the subsection titles. Roffaduft (talk) 07:35, 26 August 2024 (UTC)

Vote to remove "Investigation by numerical methods" subsection

I'd like to remove the subsection Investigation by numerical methods from the article. It's just a crude example of a numerical method applied to the wave equation. It does not provide insight or unique information regarding the wave equation nor does the wave equation leads to some unique application of a numerical method.

If there are no objections, I'll remove the subsection in due time. Roffaduft (talk) 10:27, 26 August 2024 (UTC)