# The mathematical definition of "wave"?

• Stephen Tashi
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In summary, there is no single, precise mathematical definition for "wave" in the context of physics and mathematics. The informal definition states that a wave is a propagating dynamic disturbance of one or more quantities, sometimes described by a wave equation. However, even the wave equation may not be the only equation considered a wave equation. Therefore, it is safer to declare a wave to be something that satisfies a certain system of equations rather than attempting to explicitly state all classes of functions that qualify as waves. However, this then poses the problem of defining "wave equation" and determining which equations fall under this category.
Heuristic definition:
Waves can be described as the set of any (quasi-)periodic functions within some domain (or function space), where these functions are themselves the solution to some partial or ordinary differential equation or difference equation on some unknown but sufficiently nice manifold, i.e. the solution to some dynamical system; when the dynamical system is nonlinear, this definition can capture solitons as well. In any case, when these dynamical systems are sufficiently linearized and the correct manifold identified, then we automatically recover the canonical special functions from the classical theory of analysis.

Background:
Waves as mathematical/physical objects have seemed to begin with Huygens in the 17th century, i.e. during the era of classical mathematics where there wasn't a clear distinction between the fields physics and mathematics. Seeing that waves have been generalized and abstracted to the extreme, today we would probably classify the subject as belonging to mathematics, but it would be more accurate to classify it as belonging to mathematical physics proper, being among other things the basis of perturbation theory.

The informal name of the mathematical study of the dynamics of waves is synchronization theory. Over the centuries however, this field has not really been intensely studied from within the context of being a unified subject, but instead bits and pieces of the study of waves have occurred as secondary projects within countless subfields and disciplines, scattered across all the sciences. It isn't an exaggeration to say that to a large extent, synchronization theory is one of the most unifying subjects within all of mathematics; e.g. Euler's identity is one of the key results of synchronization theory.

A surface level treatment of synchronization theory is taught almost universally in secondary education, but under another moniker: trigonometry. Classical trigonometry (and all generalizations thereof) is de facto concerned with the study of waves; the pedagogical problem however is that this does not seem to be so directly because the focus is more on right triangles within the unit circle, i.e. the focus in trigonometry is on the geometry of the state space representation of the waves instead of on the waves themselves.

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There are many physical phenomena for which the wave equations and their solutions are only approximations. Water waves, sound waves, and blast waves are three examples that come to mind.

A "wave" is more properly a physical phenomena than a mathematical one, since the equations and their solutions are only approximations in many cases.

The challenge with a single definition doesn't trouble me. Biologists have a similar challenge defining "species" and at the higher levels talk about the species concept. And physicists have at least three definitions for mass.

There is always some awkwardness with multiple definitions running around for a single word in its scientific uses. But once that word has been so widely used in the literature with different definitions, it's hard to put the cat back in the bag with a "single" definition that excludes a significant subset of uses in the historical scientific literature.

jasonRF and PeroK
But I think the requirement for a single definition of waves that applies broadly in science is misplaced. The primary role of science is to describe nature in a way that provides testable predictions - it is not to "define." Yes, for clarity, terms need to be defined in each specific context, but one errs in demanding that a given tern needs the same definition in contexts where the testable predictions are not related.

A couple decades ago, a definition of "ballistic pressure wave" was needed in the context of whether projectiles could injure tissue at a distance - without direct contact. Skeptics of the emerging theory often demanded a definition - was it the "shock wave", the "sonic wave", or the "shear wave" of the passing projectile? After some thought and discussion, colleagues and I defined it as "a pressure transient that can be measured with a high speed pressure transducer" for the purpose of our published papers on the subject. Our definition has worked well for over 20 years in that context.

But even that definition will fail at some point. As a wave travels outward from the source, eventually many waves become too small to detect with a given sensitivity. Do they cease to be waves at that point? Will improving the detectors make them waves again? This is the silliness one arrives at when defining is more important than describing.

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Participants are reminded that posts on PF should be consistent with the professional scientific literature.

atyy said:
Hyperbolic equations are often said to represent waves.
I have seen that in the literature.

Auto-Didact said:
Waves can be described as the set of any (quasi-)periodic functions within some domain
This one I have not seen and it is a bad definition since many solutions to the wave equation are completely non-periodic.

Dr. Courtney said:
After some thought and discussion, colleagues and I defined it as "a pressure transient that can be measured with a high speed pressure transducer" for the purpose of our published papers on the subject.
Excellent! This clearly shows the variety in the literature.

Stephen Tashi said:
Perhaps a "wave" can be viewed (abstractly) as a generalization of a cellular automata.
I have never seen anything close to this in the literature, but I have not read much on cellular automata.

Due to the rampant personal speculation in this thread I am asking in advance for people suggesting alternative definitions to proactively provide literature references.

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vanhees71
SSequence said:
@jasonRF
I have a few basic questions out of curiosity (if you don't mind).

I am trying to understand whether (informally) my "guess" about the wave equation is right (or whether there are some inaccuracies or deficiencies in it).

After looking at this topic, my guess is that wave equation (1D) represents solutions to the following:
----- a traveling wave with speed v
----- two traveling waves with speed v (moving in opposite direction). It seems that this would also help create a standing wave.
----- Any finite number of traveling waves. Possibly with different shapes(?) but with speed v and any of left-right direction.
The simple wave equation has no dispersion, so all waves travel with the same speed (group velocity = phase velocity). in 1-D they can propagate in either direction. The d'Almbert solution to the wave equation makes that pretty clear (google it if you haven't seen it already)

I see no reason why we must constrain ourselves to a finite number of traveling waves. Indeed, we can represent a solution to the wave equation using Fourier series or the Fourier transform, both of which include an infinite number of sinusoids.

SSequence said:
----- A wave "group" of sort [some kind of more complicated solution to equation] where an "infinite" number of waves (moving with the "phase velocity") give us a single wave moving with the "group velocity" (which is something like the speed of the peak ... or something similar). Obviously this is informal, I expect the math to be involved.
Common terms to use are "wave packet" or "pulse"; in principle a wave packet does not need to include an infinite number of sinusoids, but it usually does via a Fourier transform. Each sinusoid propagates at the phase velocity. The sum of the sinusoids yields a waveform that has an envelope that travels at the group velocity (at least to a good approximation in many cases).

SSequence said:
(1) Is it correct to say that two traveling waves moving with unequal speeds v1 and v2 won't represent a solution to a single wave equation?
By definition in dispersive media you definitely can have two waves with unequal speeds that satisfy the same equation (or set of equations). edit: just to be clear, phase velocity and group velocity are both frequency dependent in general.

It can be even more complicated. A single wave equation (or system of equations) might support different modes with significantly different characteristics and dramatically different speeds. An unmagnetized plasma is a good example. It can support waves with longitudinal electric fields propagating near the thermal speeds of the particles (ion-acoustic waves and Langmuir waves), as well as transverse electromagnetic waves with phase velocities greater than the speed of light but group velocities less than the speed of light.

SSequence said:
(2) In real-life, we would often place some boundary conditions (e.g. a standing wave created between two end-points). My question is that when we talk about "phase velocity"/"group velocity" in case of a "wave group", are there some conditions placed sometimes to rule out solutions which might be mathematically correct (but not physically correct)?
Yes. One example are conditions to enforce causality - if you have a source then the waves propagate away from the source, not towards it. If you google "sommerfeld radiation condition" you should find some examples.

SSequence said:
I am trying to understand the informal picture. Finally, can you describe a few references which describes the formation of "wave group" (mathematics) in a detailed way.
What kind of reference you are looking for? All of the references I know are in the context of the physics, and while derivations about group velocity are pretty much always agnostic to any specific model, you don't seem interested in that approach.

jason

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SSequence
My questions (including the one below):
SSequence said:
(1) Is it correct to say that two traveling waves moving with unequal speeds v1 and v2 won't represent a solution to a single wave equation?
were specifically with regards the "simple wave equation" (one finds in introductory texts etc.).

So regarding this:
jasonRF said:
By definition in dispersive media you definitely can have two waves with unequal speeds that satisfy the same equation (or set of equations). edit: just to be clear, phase velocity and group velocity are both frequency dependent in general.

It can be even more complicated. A single wave equation (or system of equations) might support different modes with significantly different characteristics and dramatically different speeds. An unmagnetized plasma is a good example. It can support waves with longitudinal electric fields propagating near the thermal speeds of the particles (ion-acoustic waves and Langmuir waves), as well as transverse electromagnetic waves with phase velocities greater than the speed of light but group velocities less than the speed of light.
I am not getting this particular point. I mean I get the basic idea that a more sophisticated equation will definitely allow traveling waves with different speeds. But I am not fully clear on what equation are you specifically talking about.

But it seems that you are also saying(?) that [by combining "infinite" number of sinusoids via Fourier transform formalism etc.], the "simple wave equation" [the one finds in introductory undergrad. texts etc.] alone can actually also be used to represent two traveling wave moving with different speeds.

Now the last paragraph seems to make some sense. I didn't think of this possibility when writing post#32. But it would be good if you could confirm (or deny) it.

Dale said:
Due to the rampant personal speculation in this thread I am asking in advance for people suggesting alternative definitions to proactively provide literature references.

In regards to general idea that a wave describes the behavior of a field where what happens next at a location depends only on the current state of things at that location and nearby locations, the generalization of cellular automata to continuous spatial automata described in http://web.eecs.utk.edu/~bmaclenn/papers/CSA-tr.pdf uses an example ("continuous game of life") that has these properties, but I don't understand whether the definition of "continuous spatial automata" proposed in that paper is required to have these properties.

A paper by Terrance Tao https://www.math.ucla.edu/~tao/preprints/wavemaps.pdf uses the terminology "wave map". I don't know whether "wave map" is standard terminology and I don't know enough math to understand the paper. Perhaps someone can comment on whether the concept of a membrance vibrating on an abstract manifold captures the general idea mentioned above.

Dale
Hi SSequence,

When I replied to you for some reason I was thinking I was replying to the OP. That is also why my reply regarding references was not generous - please accept my apology.

Since your questions are a little different than the main thread, I wonder if it makes sense to break these off into a separate thread?

In any case, when you asked about solutions to a wave equation, I interpreted that as not referring to any specific wave equation. And since some of the questions were about group velocity , I was assuming you were asking about solutions to equations that are more complicated than the wave equation, ##v^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}## . For the simple wave equation, all disturbance travel at the same velocity ##v##, whether or not they are single sinusoids our an infinite number of sinusoids arranged as a wave packet. Sorry for any confusion I caused you!

Reasonable basic references at the sophomore level are

https://www.people.fas.harvard.edu/~hgeorgi/new.htm
https://scholar.harvard.edu/david-morin/waves
http://farside.ph.utexas.edu/teaching/315/315.html

All of these have sections that discuss group velocity.

jason

jasonRF said:
Since your questions are a little different than the main thread, I wonder if it makes sense to break these off into a separate thread?
I do not have many more questions except the ones below, so it should be fine I think (unless mods decide for a separate thread). The main thread can continue its course.

I just have two final questions:
jasonRF said:
For the simple wave equation, all disturbance travel at the same velocity , whether or not they are single sinusoids our an infinite number of sinusoids arranged as a wave packet
Yes my question in post#45 was regarding this. What I was asking was (very roughly) that is it possible that one could use an "infinite" number of traveling disturbances [such as sinusoids] to make two different wave packets with different group velocity, while also satisfying the simple wave equation?

Or is it impossible?

=================

Also, I don't know that if, say, a single "wave packet" is a solution to the simple wave equation whether it can/will change its "shape" [e.g. in the sense of spreading out etc.] or not (in the simple wave equation). As I re-call, a simple traveling wave (of any shape) is also a solution to wave equation but keeps its shape intact.

So it would be good to be sure about its comparison to wave packets.

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atyy said:
https://web.math.princeton.edu/~seri/homepage/papers/gowers-Aug4-2006.pdf
"Another important, characteristic, property of hyperbolic equations is finite speed of propagation."

That paragraph of the paper gives a hint about how to define "propagation speed" and how to define a "disturbance". I don't find the definitions in the paper completely rigorous. The general idea is to define a disturbance as a change in the initial conditions of a PDE and to measure propagation speed by measuring (according to some metric) the distance between where the solutions are the same and where they differ. ( I don't know how one converts that distance to an idea of speed ).

The paper says:
Another important, characteristic property of hyperbolic equations is finite speed of propagation. This property can be best understood in terms of domains of dependence. Given a point ##p \in R^{1+d}##, outside the initial hypersurface ##\mathbb{H}##, we define ##\mathbb{D}(p) \subset \mathbb{H}## as the complement of the set of points ##q \in \mathbb{H}## with the property that any change of the initial conditions made in a small neighborhood ##V## of ##q## does not influence the value of solutions at ##p##. More precisely if ##u,v## are two solutions of the equation whose initial data differ only in ##V##, [they] must also coincide at ##p##. The property of finite speed of propagation simply means that, for any point ##p##, ##\mathbb{D}(p)## is compact in ##\mathbb{H}##.

SSequence said:
I do not have many more questions except the ones below, so it should be fine I think (unless mods decide for a separate thread). The main thread can continue its course.

I just have two final questions:

Yes my question in post#45 was regarding this. What I was asking was (very roughly) that is it possible that one could use an "infinite" number of traveling disturbances [such as sinusoids] to make two different wave packets with different group velocity, while also satisfying the simple wave equation?

Or is it impossible?
It is impossible. All waves described by the simple wave equation travel at the same velocity, so phase velocity = group velocity = constant independent of frequency. Basic solutions are of the form ##f(x - v t)## for waves moving in the ##+x## direction and ##g(x+vt)## for waves moving in the ##-x## direction, where ##f## and ##g## are arbitrary functions of a single variable (that should be twice differentiable for classical solutions). Basically the shapes stay intact as they propagate.
SSequence said:
Also, I don't know that if, say, a single "wave packet" is a solution to the simple wave equation whether it can/will change its "shape" [e.g. in the sense of spreading out etc.] or not (in the simple wave equation). As I re-call, a simple traveling wave (of any shape) is also a solution to wave equation but keeps its shape intact.

So it would be good to be sure about its comparison to wave packets.
You can have a wave packet that is a solution to the simple wave equation, but like I wrote above it will not change shape at all. It is only when you have more complicated wave equations that model waves in dispersive media that you get wave packets that can change shape, and where phase velocity and group velocity are no longer equal.

jason

SSequence
Dr. Courtney said:
If we allow the more fundamental definition of wave to be mathematical (class of functions or wave equations), then we must also define how well a physical phenomena is represented by the math before we determine whether or not it is really a "wave."

Does "we" refer to physicists? I'm sure they won't be limited by what pure mathematics says.

Relevant to the topic of this thread would be a precise description of what criteria physicists use to determine whether a phenomenon is a wave or whether it isn't. I'm sure anyone who has taught physics has used the term "wave" hundreds of times in lectures - as if it means something specific.

Auto-Didact
Stephen Tashi said:
Relevant to the topic of this thread would be a precise description of what criteria physicists use to determine whether a phenomenon is a wave or whether it isn't.
No. The term is not used precisely by physicists. That is (I believe) @Dr. Courtney ’s basic point. Not only is it used imprecisely by physicists, but they are always free to define what they mean in a specific context, as he did, and such redefinitions are commonly accepted in the professional scientific literature.

If you want a precise mathematical definition (eg solutions to the wave equation) then it will not cover all of the use cases.

This is why I emphasized the distinction between the mathematical definition and the colloquial usage in my first post here.

Stephen Tashi said:
I'm sure anyone who has taught physics has used the term "wave" hundreds of times in lectures - as if it means something specific.
Whenever I mean something specific by the term “wave”, I specifically mean a solution to the wave equation.

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jasonRF
An off topic post and several responses have been deleted. Please stay on topic. This is not the foundations forum.

weirdoguy
Stephen Tashi said:
In regards to general idea that a wave describes the behavior of a field where what happens next at a location depends only on the current state of things at that location and nearby locations, the generalization of cellular automata to continuous spatial automata described in http://web.eecs.utk.edu/~bmaclenn/papers/CSA-tr.pdf
This reference is not particularly relevant to the thread. There is no suggestion in that paper that continuous spatial automata could be used in any specific way to provide a rigorous mathematical definition of a wave. In fact, the paper contains no mention of waves at all.

Stephen Tashi said:
A paper by Terrance Tao https://www.math.ucla.edu/~tao/preprints/wavemaps.pdf uses the terminology "wave map". I don't know whether "wave map" is standard terminology and I don't know enough math to understand the paper. Perhaps someone can comment on whether the concept of a membrance vibrating on an abstract manifold captures the general idea mentioned above.
This one seems good and relevant.

Instead of the singular form, "What is a wave?" I find it more enlightening as ask, what are wave(s) in physics? For example non-dispersive waves can be defined as any 'disturbance' of the medium from its equilibrium form that may be propagated in a direction at a speed v. The shape propagated is called the waveform. This propagation is typically unique to the non-dispersive wave equation. Of course, a person could also segment their study into dispersive waves, electromagnetic waves, water waves, acoustics wave, etc., anything that interests them. Understanding/recognizing several of these different wave phenomena and the associated wave equations/solutions is key a fundamental physics education/understanding. A single definition of the word "wave" is of very little use, in my opinion.

A good resource for the study of vibrations and waves in physics is "Vibrations and Waves in Physics" by Iain G. Main.

etotheipi
Wave map is standrd terminolgy.

martinbn said:
Wave map is standrd terminolgy.

Which examples of waves (in physics) are covered by the concept of "wave map" ?

Stephen Tashi said:
Which examples of waves (in physics) are covered by the concept of "wave map" ?
The document you linked seemed to indicate that wave maps aren't that well understood yet. However, it does explicitly say that they do not include dissipation. For linear waves in causal media, the Kramers-Kronig relations tell us that dissipationless media are non-dispersive. So at best wave maps can model waves in the limit where dissipation is weak enough to ignore, but they certainly leave out a lot of phenomena. For example, wave maps cannot help you if you want to understand why you receive more AM stations at night than in the day, since the primary issue is dissipation in lower regions of the ionosphere during the daytime.

jason

Stephen Tashi
Dale said:
This reference is not particularly relevant to the thread. There is no suggestion in that paper that continuous spatial automata could be used in any specific way to provide a rigorous mathematical definition of a wave.

As to http://web.eecs.utk.edu/~bmaclenn/papers/CSA-tr.pdf , I disagree. The example of the "continuous game of life" in that article illustrates (for the case of discrete times) a situation where a disturbance is propagated with a finite speed. That property has been mentioned, by other posters, as characterizing waves.

The general concept suggested by that example (but not necessarily by the definition of continuous spatial automaton given in the article) is as follows.

Define a real valued function ##u(x,t)## of location ##x## and discrete time ##t## by defining at each location ##x_1## how to compute ##u(x_1,t+1)## as a function ##f_{x_1}## whose domain consists only of the set of values of ##u(x,t)## for ##x## within a finite distance ##r_{x_1}## of ##x_1##.

For two locations ##x_1, x_2## it may be that ##u(x_2,t+1)## is not a function of a set that contains the argument ##u(x_1,t)## due to ##x_1## being farther from ##x_1## than distance ##r_{x_2}##. However, ##u(x_2,t+2)## is (implicitly) a function of a set of values of ##u## that are farther from ##x_2## than ##r_{x_2}##. The value of ##u(x_2,t+3)## is implicitly a function of a set of values of ##u## that are even farther away.

If there is a smallest integer ##k## such that ##u(x_2,t+k)## is implicitly a function of a set of values that contains ##u(x_1,t)##, we can define ##k## to be the speed of propagation of ##u## from ##x_1## to ##x_2##.

To formulate a definition that encompasses prominent examples of physical waves, the following are needed.

Generalize to the case where ##x## and ##u## can be any sort of mathematical objects that contain location information for a point in a metric space. For example, ##x## might be a vector that include spatial coordinates of a point and also contains additional information.

Generalize to the case of continuous time.

In the case where partial derivatives of ##u## exist, generalize to the case of continuous time in a way that the condition of the dependence of ##u(x,t)## only on values of ##u## within radius ##r_{x}## of ##x## is replaced by the condition that ##u(x)## depends only on partial derivatives of ##u## evaluated at x.

Stephen Tashi said:
That property has been mentioned, by other posters, as characterizing waves.
Please stick to what is in the scientific literature, not merely comments by other posters.

If you have a reference that provides a definition of waves in terms of cellular automata then post it. Otherwise that claim is personal speculation which is already problematic in this thread.

I personally am highly skeptical that cellular automata can be used to provide a rigorous definition of waves. And your reference does not even investigate that claim, let alone support it.

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hutchphd
Dale said:
Please stick to what is in the scientific literature, not merely comments by other posters.

Please clarify whether you agree that the description of a wave as a "disturbance" that "propagates" through a medium is used in scientific literature.

Stephen Tashi said:
Please clarify whether you agree that the description of a wave as a "disturbance" that "propagates" through a medium is used in scientific literature.
In your own words you are looking for a “standard mathematical definition for ‘wave’”. I don’t believe that any professional scientific source would attempt to pretend that such a colloquial description is sufficient for a mathematical definition.

The only “standard mathematical definition for ‘wave’” that I have previously seen is a solution to the wave equation. Although I did like the wave map reference earlier.

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hutchphd
The term “wave” is used for too many different travelling, (and standing), phenomena.
All “waves” are conceptually merged in the thoughts and language of humans.

Physics has classified those phenomena, in part according to the mathematics involved.

I do not see how merging those diverse classes again, into a single mathematical definition, can be constructive.

jasonRF and Dale
Dale said:
The only “standard mathematical definition for ‘wave’” that I have previously seen is a solution to the wave equation. Although I did like the wave map reference earlier.
Which syas the same. Wave maps are solutions to the wave map equation. So it will not satisfy the OP.

Dale and weirdoguy
Stephen Tashi said:
As I said in the original post, it is obvious that the defect in defining a wave to be a solution to "the" wave equation is that there more equations than "the" wave equation which are also called wave equations. Do I need to explain that further?

Here is an excerpt from the introduction to G.B. Whitham's textbook, Linear and Nonlinear Waves.(pages 2-3)
"
1.1 The Two Main Classes of Wave Motion

There appears to be no single precise definition of what exactly
constitutes a wave. Various restrictive definitions can be given, but to
cover the whole range of wave phenomena it seems preferable to be guided
by the intuitive view that a wave is any recognizable signal that is
transferred from one part of the medium to another with a recognizable
velocity of propagation. The signal may be any feature of the disturbance,
such as a maximum or an abrupt change in some quantity, provided that it
can be clearly recognized and its location at any time can be determined.
The signal may distort, change its magnitude, and change its velocity
provided it is still recognizable. This may seem a little vague, but it turns
out to be perfectly adequate and any attempt to be more precise appears to
be too restrictive; different features are important in different types of
wave.
Sec 1.1 THE TWO MAIN CLASSES OF WAVE MOTION 3

Nevertheless, one can distinguish two main classes. The first is formu-
lated mathematically in terms of hyperbolic partial differential equations,
and such waves will be referred to as hyperbolic. The second class cannot
be characterized as easily, but since it starts from the simplest cases of
dispersive waves in linear problems, we shall refer to the whole class as
dispersive and slowly build up a more complete picture. The classes are not
exclusive. There is some overlap in that certain wave motions exhibit both
types of behavior, and there are certain exceptions that fit neither.

The prototype for hyperbolic waves is often taken to be the wave
equation
...

although the equation

φ_t + φ_x = 0 (1.2)

is, in fact, the simplest of all. As will be seen, there is a precise definition
for hyperbolic equations which depends only on the form of the equations
and is independent of whether explicit solutions can be obtained or not.
On the other hand, the prototype for dispersive waves is based on a type of
solution rather than a type of equation. A linear dispersive system is any
system which admits solutions of the form

φ = acos(κx — ωit), (1.3)

where the frequency to is a definite real function of the wave number k and ω
the function ω(k) is determined by the particular system. The phase speed
is then ω(k)/k and the waves are usually said to be "dispersive" if this
phase speed is not a constant but depends on κ. The term refers to the fact
that a more general solution will consist of the superposition of several
modes like (1.3) with different k. [In the most general case a Fourier
integral is developed from (1.3).] If the phase speed ω/k is not the same for
all k, that is, ω≠c_0 k where c_0 is some constant, the modes with different k
will propagate at different speeds; they will disperse. ..."

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atyy, jasonRF, martinbn and 3 others
There were a couple of typos when I copied the section for the text. I omitted the constant c_0 and equation (1.2) reads

φ_t + c_0φ_x = 0 (1.2)

where the frequency ω is a definite real function of the wave number k and
the function ω(k) is determined by the particular system.

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