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)

Indiscriminate use of "displacement" for the general solution of the wave equation.

• Displacement typically refers to the physical movement of a point or particle from its original position. In an acoustic wave, the pressure changes are not caused by particles physically moving to new positions (although the particles do oscillate slightly around their equilibrium positions). Instead, they are caused by the compression and expansion of the medium (air, water, etc.) as the wave propagates.
• Pressure is a scalar quantity, meaning it has magnitude but no direction. Displacement, on the other hand, is a vector quantity, meaning it has both magnitude and direction. Thus, calling the pressure changes a displacement implies a directional aspect that isn't quite accurate.

Whilst "displacement" is appropriate int the case of the transverse wave of vibrating string, or a ship's wake, it is not appropriate as the solution of a wave equation for other longitudinal wave phenomena such as the pressure in an acoustic wave, the electric field in an electromagnetic wave, the intensity of light in a light wave or the intensity of the gravitational field in a gravitational wave.

I have suggested that the term "amplitude" would better describe the solution of the wave function, but am open to suggestions as to a better word. I certainly think that displacement, used 18 times in the article, is in indiscriminate use of a narrow term that detracts from the scalar aspect of the solution of a wave equation. Gpsanimator (talk) 06:01, 19 December 2023 (UTC)

For sure, Amplitude is not an improvement. It already has a definition in this context.

When describing the phenomenon in several contexts, displacement an appropriate and commonly used term.

1. Pond water wave: In a pond wave, 'displacement' refers to the vertical distance that a point on the water's surface moves from its equilibrium position. This term captures both the upward and downward movement of water particles as the wave passes. In deeper waves, the motion is more circular, but again 'displacement' isn't inappropriate.
2. Sound wave: In acoustics, 'displacement' can refer to the small changes in position of air molecules from an equilibrium due to the pressure variations inherent in sound. While the term "pressure deviation" is also pertinent, 'displacement' specifically refers to the physical movement of the particles, which is what creates the sound wave.
3. Plucked string: In the case of a string instrument, 'displacement' describes the distance a point on the string moves from its rest position. When a string is plucked, each point of the string oscillates, creating waves that travel along the string, and 'displacement' aptly names the momentary aspects of this movement.

Shift and deviation are also applicable in various contexts but are less specific than displacement:

• Shift is a more general term that can imply a change or movement from an original position, but it doesn't explicitly convey the oscillatory nature of wave motion.
• Deviation suggests a divergence from a norm or standard (like ambient pressure in the case of sound waves), but it doesn't inherently imply a movement, and it also carries other meanings. It's not so widely used in this context, as far as I see.

So it would appear that displacement is an adequate term for describing the movement.

You state that the use of the word displacement is "indiscriminate" in the article. Can you please provide an example of such indiscriminate use? Chumpih ^t +ai 12:59, 19 December 2023 (UTC)

I used "indiscriminate" in the sense of not showing careful thought or planning: This implies doing something without considering the consequences.

Your three examples above clearly demonstrate my point. All three refer to distance. You have overlooked the application of wave mechanics to the wider applications where the displacement is not distance:

I am not skilled in Wiki editing, but check the line containing "is a term for how the displacement accelerates,"

• acoustics where the displacement is pressure,
• light where the displacement is intensity,
• EM radiation where the displacement is magnetic or electric field.

Referring to the second differential of distance w.r.t time (as an "acceleration" is perfectly valid, but referring to the second differential of pressure, intensity or electric field as "acceleration" as is done in the article is clearly misleading. Gpsanimator (talk) 20:33, 20 December 2023 (UTC)

P.S. I would challenge your explanation of a sound wave. Displacement in a sound wave does not refer to "small changes in position of air molecules" at all.

It refers to changes in the pressure in the transmission medium (air, water, rock etc.) Gpsanimator (talk) 21:48, 20 December 2023 (UTC)

There's a purpose here to be educational. And while it's probably fair that 'displacement' isn't ideal for every application of the wave equation, it's arguably good enough for many of the circumstances, and gets the idea across even for the media where it's not as accurate. Shift and deviation aren't as good, per arguments above. And distance begs the question "distance from what?". We could have some compound phrase like "magnitude of change from a resting state", or some alternative, but that seems clumsy. And in many circumstances such a phrase would be synonymous with the word displacement. We already have a symbol throughout the article, u, and that makes plenty of appearances. But if the goal is to educate, there should be verbal descriptions, too, so we probably shouldn't delete the descriptions for less equation-loving readers . The question remains: what word would be an improvement? Chumpih ^t 22:26, 20 December 2023 (UTC)

I still don't see what's wrong with amplitude. I think the linked description of Amplitude as "Measure of change in a periodic variable" is misleadingly limited. A quick search for a definition of amplitude turns up:

In Physics and Waves:

• Displacement: For waves and vibrating objects, amplitude is the maximum displacement of a point from its rest position. This can be the distance a wave peak is above the equilibrium point or the distance a pendulum swings away from its central position.
• Magnitude: More generally, amplitude refers to the magnitude of a periodic variable, like the change in air pressure for a sound wave or the voltage swing of an alternating current.

Which, I think quite nicely covers all bases. Gpsanimator (talk) 00:45, 23 December 2023 (UTC)

Britannica Dictionary definition of AMPLITUDE

: a measurement that indicates the movement or vibration of something (such as a sound wave or a radio wave) Gpsanimator (talk) 01:34, 23 December 2023 (UTC)

Per your other definitions amplitude is the maximum displacement. So using amplitude instead of displacement might muddle the concepts - amplitude is 'peak-to-peak' or similar, whereas the wave equation considers the instantaneous and local. Chumpih ^t 08:23, 23 December 2023 (UTC)

Amplitude may have those overtones, but displacement is quite specifically a property of space: not pressure, volume, field strength or loudness!.

And the second differential of pressure, volume, field strength or loudness is not acceleration! Gpsanimator (talk) 09:12, 23 December 2023 (UTC)

Is there any evidence to suggest displacement is not acceptable terminology in this context? Are there reliable sources that show another word or phrase being used in preference to displacement?

Here's a page: Wien's displacement law. The title itself illustrates a wider applicability of the word displacement. Chumpih ^t 05:08, 27 December 2023 (UTC)

The Shorter Oxford Dictionary defines displacement "2. Physics. The amount by which anything is displaced; the difference between the initial position of a body an a subsequent position"

And, position: "the place occupied by a thing"

All of which clearly and indisputably involve the idea of position as a location in space.

The explanation in the Article, of the second differential of the wave function as "a term for how the displacement accelerates," also reinforces this narrow interpretation of the wave equation defining a displacement in a spatial dimension, whereas it has a much wider applicability as the intensity, amplitude or strength of pressure, electrical charge, light intensity, sound etc. being modelled in the wave equation.

These are the reasons that displacement is far too limiting to be used generally as the interpretation of the solution of the wave equation.

I have suggested amplitude as being more generally applicable and the abandonment of the term "acceleration" to generally refer to the second differential w.r.t. time. Gpsanimator (talk) 23:04, 30 December 2023 (UTC)

Thanks for that. So there doesn't appear to be any reliable sources that show another word or phrase being used in preference to displacement in this context.

Is there an issue with the Wien's displacement law use of the word displacement? Its presence solidly refutes the notion that displacement is quite specifically a property of space.

If it helps: consider the case of plotting the pressure in a tube as a line on a chart - the higher the line at some point, the higher the pressure. (X is position along the tube.) As the pressure raises or lowers due to some wave, the line goes up or down - the value is displaced. Chumpih ^t 23:31 + 23:45, 30 December 2023 (UTC)

Can we at least agree that to call the second differential w.r.t. time of pressure in acoustics, or of an electric field in an EM wave, an acceleration is misleading? Gpsanimator (talk) 21:11, 31 December 2023 (UTC)

Yeah, that's a fair point. An edit has been made to the page - do with it what you will. Chumpih ^t 00:56, 1 January 2024 (UTC)

The "wave equation" is not classical

The current lede starts:

• 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 – such as...

This is a definition of the topic of the article, but the subject "wave equation" would be correctly divided in to "classical wave equation" and "quantum wave equation". I will write a new first sentence that omits the classical and add a sentence explaining the limits placed in this article. Johnjbarton (talk) 06:29, 19 December 2023 (UTC)

Please don't. I think the quantum wave equation is covered in Wave Function. Gpsanimator (talk) 19:55, 20 December 2023 (UTC)

Wave function is not an equation, it is a solution to an equation. Johnjbarton (talk) 23:02, 20 December 2023 (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)