The mathematical definition of "wave"?

[Stephen Tashi]

TL;DR Summary

What is the mathematical definition of a "wave" and the quantities associated with a wave, such as group velocity, phase velocity etc.?

Is there a standard mathematical definition for "wave"? What is the definition? Assuming that there is a definition, what are the mathematical definitions of the properties of waves? For example, how is the "group" of a wave defined? ( as in the "group" that has a "group velocity").

I'm not asking for examples of functions that are waves or informal explanations of waves. I'm asking if there is a precise mathematical definition for "wave" and its associated properties.

The term "wave function" is used in quantum mechanics with a specialized meaning. That's not what I'm asking about. Assuming "wave" denotes a general type of function in mathematics, I'm asking for the definition of that type of function.

We could define a "wave" to mean the same thing as "periodic function". That wouldn't work out well in applications of math. For example, there is the phenomena of sound "waves" decaying in amplitude as we get farther from the source of sound.

One might say that a wave is any function that is a solution to a "wave equation". That would move the task of defining "wave" to the task of defining "wave equation". If we take that approach, what's the definition of a "wave equation"?

It's interesting to read Wikipedia's current attempt to define "Wave". https://en.wikipedia.org/wiki/Wave

The informal definition is:

In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities, sometimes as described by a wave equation.

To formulate a mathematical definition of "wave" based on that idea, we have to define "disturbance" and set up a scenario where "equilibrium" is precisely defined. It's interesting that the article says "sometimes as described by a wave equation" , implying that there can be waves that are not described by a wave equation.

Usually if a standard mathematical definition exists for something, a Wikipedia article will quote it in a straightforward manner. However, under the heading "Single wave" we find only:

A wave can be described just like a field, namely as a function $F(x,t)$ where $x$ is a position and $t$ is a time.

That definition departs from pure mathematics by introducing the physical concepts of position and time.

The article then goes on to give numerous examples and possibilities for waves without saying there are any restrictions on $F(x,t)$. Based on that, a "wave" is any function of 2 or more variables!

 

[Vanadium 50]

What don't you like about "a wave is something that satisfied the wave equation"? The wave equation is certainly something conventionally understood: $\frac{\partial^2 x}{\partial t^2} = v^2 \nabla^2 x$ . If that is too circular for you, sometimes the answer to "What's a giraffe?" is to point and say "That. That is a giraffe."

 

[Stephen Tashi]

What don't you like about "a wave is something that satisfied the wave equation"? The wave equation is certainly something conventionally understood: $\frac{\partial^2 x}{\partial t^2} = v^2 \nabla^2 x$

I like that except that "the" wave equation you state may not be the only equation called a wave equation. Are we to define a "wave equation" to be one specific equation?

 

[nuuskur]

That definition departs from pure mathematics by introducing the physical concepts of position and time.

I think you are hung up on semantics, here. One is not forced to think of $x$ and $t$ as position and time respectively, but it's likely helpful to the intuition.

We could define a "wave" to mean the same thing as "periodic function". That wouldn't work out well in applications of math. For example, there is the phenomena of sound "waves" decaying in amplitude as we get farther from the source of sound.

That could be remedied by multiplying a periodic function with a parameter that handles the decay e.g $p=p(\|x-x_0\|)$.

There's a problem with such an approach, though. It's 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. Otherwise, how would we know we got all of them covered?

One could then, of course, prove some necessary condition for a solution, thus likely identifying the classes of solutions.

I would make an analogy here, because I was having a similar problem, myself. Namely, what is a set? When I asked what a set was I was referred to the ZF axioms and I was like "no, I wanted to know what a set is.." and I realized some time later that I was just worrying about semantics. "Set" is just a label. It would be much better to know how to identify sets and which operations with sets produce sets. ZF answers all of that and more.

Similarly here, it's not so interesting to me what a "wave" is. We can call them "Phenomenon X" for all good it does us. But knowing that these objects appear as solutions to systems of eqs tells us how to construct them.

 

[Stephen Tashi]

I think you are hung up on semantics, here. One is not forced to think of $x$ and $t$ as position and time respectively,

Right - but that makes the problem worse! Now we need a mathematical definition that works in an even broader class of applications.

but it's likely helpful to the intuition.

I agree.

There's a problem with such an approach, though. It's 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. Otherwise, how would we know we got all of them covered?

I think you mean that declaring an equation to be a wave equation implicitly defines the properties of all its solutions. I agree that this is a safer approach in the sense that if we attempt to make an explicit list of properties, we might inadvertently omit some of them.

However, this shifts the difficulty from defining "wave" to the problem of defining "wave equation". Which equations are wave equations and which are not? Instead of the properties of "wave", we must define the properties of a "wave equation".

 

[DaveC426913]

I'd say it is not surprising that coming up with the definition of a wave is difficult to do.

As you note, we use the word to describe all sorts of things, whether they are related to each other or not. Words are sloppy and ham-fisted when it comes to defining real-world phenomena.

If you want the definition of something in physics, look to the math.

 

[jasonRF]

Summary:: What is the mathematical definition of a "wave" and the quantities associated with a wave, such as group velocity, phase velocity etc.?

I'm not sure if there is a universally accepted definition for wave. I'm not convinced it is useful, either. Where have you already looked to find fine one?

I just pulled out Physics (4th edition) by Resnick, Halliday and Krane. Chapter 19 is on wave motion gives a reasonable notion of waves as disturbances that transport energy and momentum without the transport of material particles (edit: this is reasonable for classical physics). Please tell me you have already looked at basic sources such as this. If you haven't, you need to do some work on your own before asking more questions.

Likewise, three are many many books at different levels that provide the precise definition of phase velocity and group velocity. Have you looked at any of those? If not, then you need to do more work on your own. Here is one example at the sophomore level:
https://www.people.fas.harvard.edu/~hgeorgi/new.htm

By the way, what is your background? How much math and physics do you already know?

jason

 

[etotheipi]

Likewise, three are many many books at different levels that provide the precise definition of phase velocity and group velocity. Have you looked at any of those? If not, then you need to do more work on your own. Here is one example at the sophomore level:
https://www.people.fas.harvard.edu/~hgeorgi/new.htm

Morin has also published a draft of a textbook on waves at the undergrad level, which is freely available here:
https://scholar.harvard.edu/david-morin/waves

 

[Stephen Tashi]

Chapter 19 is on wave motion gives a reasonable notion of waves as disturbances that transport energy and momentum without the transport of material particles.

As you recall, in a recent thread in the physics section, I was heavily criticized for employing the vague concept of "disturbances". Apparently textbook writers can get away with it!

Please tell me you have already looked at basic sources such as this. If you haven't, you need to do some work on your own before asking more questions.

I know you mean well, but those remarks make no sense vis-a-vis my question. This is a question in the mathematics section. I'm asking about a mathematical definition, not for an intuitive description or examples. If I ask for a mathematical definition for something (e.g. "tangent bundle") then the definition may involve sophisticated concepts. If so, then let the definition be presented and it may require "work on my own" to understand the definition. But there is no reason to withhold the definition.

Likewise, three are many many books at different levels that provide the precise definition of phase velocity and group velocity.

I disagree, as far as a mathematical definition goes. Many books provide specific examples of what they choose to call waves and precisely define a phase velocity for each specific example. A collection of examples in not a general mathematical definition.

Here is one example at the sophomore level:
https://www.people.fas.harvard.edu/~hgeorgi/new.htm

It looks like a collection of examples. Where in that PDF is there a mathematical definition of "wave"? The same can be asked of a link you gave in another thread http://farside.ph.utexas.edu/teaching/jk1/jk1.html

I'm not sure if there is a universally accepted definition for wave.

Mathematically, I'm starting to think there isn't. There might be not-well-known articles where people have proposed a sophisticated definition. I haven't found any.

 

[Stephen Tashi]

Morin has also published a draft of a textbook on waves at the undergrad level, which is freely available here:
https://scholar.harvard.edu/david-morin/waves

Which says:

A wave is a correlated collection of oscillations.

So how shall we define "a correlated collection of oscillations"?

 

[weirdoguy]

What's the definition of "mathematical definition"? And why physical phenomenon has to have one?

 

[Stephen Tashi]

What's the definition of "mathematical definition"?

On a sophisticated level, that's a good question, but a question that should be asked in a different thread!

And why physical phenomenon has to have one?

I'm not saying it must - and that's also a topic for a different thread.

 

[weirdoguy]

but a question that should be asked in a different thread!

But if we don't know what "mathematical definition" is in general, how can we answer your question asked in this thread? We can give you a lot of answers, but you can always say "no, that's not it". So what's "it"? For me @Vanadium 50 definition is the best so far.

 

[Stephen Tashi]

One of the reasons is that linearity applies to its solutions.

I've thought about such an approach in the following way. We might define "elementary wave functions" to be specific functions (e.g. sine waves, ramp waves, etc. and define their specific velocities on a case by case basis.). Then we could define "wave" in general to be any superposition of them. I don't know how we'd work out a general definition for "wave velocity" that relates it to the velocities of a the components, but at least we get as far as defining "wave"

However, not all equations that are called wave equations are linear differential equations. So a component of a solution need not be a solution. In such a case, we are at odds with the previously mentioned idea of defining a "wave" to be a solution to (a single given) wave equation. We'd have to generalized that approach to saying that a "wave" is a solution to a differential equation and a solution that can be expressed as a superposition of functions, each of which is a solution to some "elementary wave equation".

 

[DaveE]

If you don't like the "solution to the wave equation" definition, then I'm afraid you'll need to explicitly define what you do mean. There is some value in the subtle vagueness of human communication. Either deal with it or define what you are talking about. I'm thinking something like the beginning of a math proof, ∀ a ∈ N if a ≠ 0, then ∃ a⁻¹ ∈ N... , for example.

"Wave" is a useful but perhaps imprecise way technical people talk to each other about "wave-like" things.

Why I (an engineer and ex-math guy) can't really deal with Mathematicians, even though they are correct. Absolutely, annoyingly, stupifyingly correct.

 

[Stephen Tashi]

But if we don't know what "mathematical definition" is in general, how can we answer your question asked in this thread?

And how can we answer anybody's questions about any other mathematical definitions in any other threads? Discussions in the math section commonly answer questions about definitions. I see no reason why asking for the definition of "wave" is a special case.

 

[SSequence]

@Stephen Tashi
Sorry I deleted my previous post.

 

[Stephen Tashi]

If you don't like the "solution to the wave equation" definition, then I'm afraid you'll need to explicitly define what you do mean.

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?

 

[DaveE]

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?

That's my point. Yes, you might need to, depending on who you are talking to or what you are talking about. However, you are correct, it doesn't always mean the same thing to everyone.

 

[PeroK]

What's the definition of "mathematical definition"? And why physical phenomenon has to have one?

Exactly:

Only mathematical objects have a mathematical definition. A Hilbert space, for example. A wave is not a mathematical object, which is why it has no mathematical definition. A wave could be described as a physical concept that is useful in describing certain physical phenomena; and, has a relationship to various mathematical objects such as the wave equation - which is a mathemetical object.

 

[hutchphd]

What don't you like about "a wave is something that satisfied the wave equation"? The wave equation is certainly something conventionally understood: ∂2x∂t2=v2∇2x . If that is too circular for you, sometimes the answer to "What's a giraffe?" is to point and say "That. That is a giraffe."

My problem is that it this is a very narrow definition. For instance it would eliminate consideration of various "solitary wave" solutions to nonlinear equations which are pretty impressively "wavelike". The solution to diffusion equations can give pulses. How about Schrodinger's with potentials.

That being said, rather than querying the cohort l ask @Stephen Tashi to make the attempt. Otherwise this is an endless game of "no, that's not it". Having played one inning of that game I choose to not play another.

 

[jasonRF]

As you recall, in a recent thread in the physics section, I was heavily criticized for employing the vague concept of "disturbances". Apparently textbook writers can get away with it!I know you mean well, but those remarks make no sense vis-a-vis my question. This is a question in the mathematics section. I'm asking about a mathematical definition, not for an intuitive description or examples. If I ask for a mathematical definition for something (e.g. "tangent bundle") then the definition may involve sophisticated concepts. If so, then let the definition be presented and it may require "work on my own" to understand the definition. But there is no reason to withhold the definition.

Fair enough. I will stay out of your quest for a mathematical definition of a wave. I would be surprised if you find one that doesn't leave out some phenomenon commonly accepted to be a wave, but I have been wrong on many occasions in the past.

EDIT: I was assuming your goal was to find a definition that includes most (or perhaps all?) phenomena that are commonly called waves. This may be a false assumption on my part.

I disagree, as far as a mathematical definition goes. Many books provide specific examples of what they choose to call waves and precisely define a phase velocity for each specific example. A collection of examples in not a general mathematical definition.

The concepts of phase velocity and group velocity in general only apply to linear waves and naturally come out of a Fourier analysis of a wave packet. The book I linked does provide the correct mathematical definitions in 1D (equations 10.32 and 10.36), and the derivation is not at all in the context of a specific example. Did you read any portion of chapter 10 in that book, especially the section 10.2.1 that has the title "group velocity"?

By the way, the generalization to 3D is pretty straightforward. See my post here
https://www.physicsforums.com/threads/group-velocity-definition.842980/post-5294777
for a quick and dirty derivation of group velocity in 3D.

jason

 

[atyy]

I like that except that "the" wave equation you state may not be the only equation called a wave equation. Are we to define a "wave equation" to be one specific equation?

Hyperbolic differential equations are often said to have wave-like solutions.
https://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation

 

[Stephen Tashi]

EDIT: I was assuming your goal was to find a definition that includes most (or perhaps all?) phenomena that are commonly called waves. This may be a false assumption on my part.

Ok, I understand what you're saying now and I apologize for saying your recommendations weren't relevant. You are saying that if I myself want to create an appropriate mathematical definition of "wave" then I should study in detail a variety of examples of waves in physics. In the OP, I did not expect to be the one proposing a definition of "wave".

In the OP, I expected that asking for a mathematical definition of "wave" would be similar to asking for a mathematical definition of "velocity". For "velocity", we have the mathematical definitions of derivative and gradient that cover the topic in physics. I expected a similar situation for "wave" since there are many discussions where experts confidently use the term "wave" as if it means something specific. The scholarpedia article is pessimistic about finding a general definition of "wave", but I thought there would be a complicated mathematical structure that filled the bill and that most textbooks would choose not to mention it because it would be an unnecessary digession from the specific cases they treat.

I agree that your recommendations are relevant to the topic - if it turns out that it's me who is to propose a definition.

The concepts of phase velocity and group velocity in general only apply to linear waves and naturally come out of a Fourier analysis of a wave packet.

I'm glad to hear this! Posts about the group velocity of a wave in other threads left me wondering "How in the world are they defining 'group velocity' for a general type of disturbance propagating through a medium?"When people are taught how to classify things of type X by examples, the usual procedure is present examples of type X and examples not of type X. The non-X examples are chosen to emphasize the essential properties of type X. It would be helpful if the copious examples of waves can be supplemented by examples of things that share many common properties with a wave, but are not a wave.

However, I don't know if the "wave" terminology in physics is formal enough that examples of nearly-a-wave-but-not-a-wave would be uncontroversial. If Alice says "This is a wave" and Bob says "No, it isn't", what criteria are used to settle the dispute? Perhaps no one makes a big deal about it, one way or the other.

 

[etotheipi]

A Mexican wave is a propagating disturbance of people from their 'equilibrium positions'. If you were so inclined, you could model that physical phenomenon with mathematical functions (e.g. you could take the continuum limit) and make a pretty, purely mathematical, simulation. Maybe you can find a wave equation for it, based on reaction times, or something. But surely you could lump both ideas (the phenomenon, and the model) under the umbrella term of a 'wave'.

 

[Stephen Tashi]

That being said, rather than querying the cohort l ask @Stephen Tashi to make the attempt. Otherwise this is an endless game of "no, that's not it".

Fair enough.

My original post isn't intended as the challenge "Bet you can't find a mathematical definition of 'wave'!". I thought there would be a standard definition. Before posting, I tried to guess a definition and rejected several ideas for reasons that have been stated.

I don't claim to have the appropriate definition. I'm willing to offer specific ideas.

Perhaps a "wave" can be viewed (abstractly) as a generalization of a cellular automata.

Cellular automata agree with the intuitive idea of phenomena where what happens next at a location depends only on what is happening now at the location and what is happening near it. Cellular automata do not agree with intuitive notion that space and time are continua and that things in space-time can take on a continuum of values. So it is necessary to generalize the notion of cellular automata.

(Disclaimer(!): In looking into the topic of cellular automata, I encountered references to Stephen Wolfram's book "A New Kind of Science". As yet, I haven't read the book. I don't know if Wolfram has already worked all this out or done something different. I don't know what the grand ideas of the book are, so I have no opinion about them.)

According to Wikipedia
https://en.wikipedia.org/wiki/Continuous_automaton , cellular automata have been genearlized to "continuous cellular automata". The wikipedia article is restricted to a function that is defined on discrete locations in space and time and takes on values in [0,1]. One goal is to further generalize the concept so the function can take on values in any set of mathematical objects - in particular so the function can be vector valued.

The Wikipedia article https://en.wikipedia.org/wiki/Continuous_spatial_automaton describes a generalization of cellular automaton to the case of a funtion defined on spatial continuum while keeping the requrement that the values of the function are discrete. However, the article doesn't give a specific definition.

A common way to define continuous mathematical objects is as limits (in some sense) of sequences of discrete mathematical objects. This suggests generalizing the concept of "cellular automaton" to the concept of things that are limits of sequences of cellular automata. This is only a vague idea until the concept of such a limit is precisely defined. I have ideas about how to do this, but perhaps it has already been done. If so, I'd like to hear about it before posting any attempt.

 

[DaveE]

I don't claim to have the appropriate definition. I'm willing to offer specific ideas.

But are you willing to accept the PF consensus that there isn't a good (generally accepted) definition? I don't think you're going to get an answer here.

 

[Stephen Tashi]

But are you willing to accept the PF consensus that there isn't a good (generally accepted) definition?

Talking about a "PF consensus" on anything could be controversial! Yes, I do accept that , so far, nothing has been presented in this thread that indicates there is a generally accepted definition of "wave". Among contributors, this is the majority opinion. I can't rule out the possibility that some expert in a little known field will pop-in and say "Didn't you know that a wave is just a special case of a javyhonkus?". (Maybe an expert on generalized cellular automata will do something like that.)

 

[jasonRF]

I would still like an answer to my question

The book I linked does provide the correct mathematical definitions in 1D (equations 10.32 and 10.36), and the derivation is not at all in the context of a specific example. Did you read any portion of chapter 10 in that book, especially the section 10.2.1 that has the title "group velocity?

From your replies so far it isn’t clear that you are investing any of your own time to actually learn about the topics you are asking about.

Jason

 

[Stephen Tashi]

I would still like an answer to my questionFrom your replies so far it isn’t clear that you are investing any of your own time to actually learn about the topics you are asking about.

Jason

If the suggested material answers the question in the OP then I am interested in studying it. Are you saying that it does?

I am investing my own time in thinking about how to generalize cellular automata - and other projects such as remodeling a house. I have not made a detailed study to the text leading up to equations 10.32 and 10.36. Am I capable of doing that eventually? Probably. I did take a course in partial differential equations over 40 years ago and a graduate course in complex analysis.

My outlook on this: If equations 10.32 and 10.36 say something important about a general definition of wave, the let someone who has reviewed this material say what that something is.

When a question is asked in the math section of PF, is the focus to be on answering the question or is the focus to be on giving personal guidance to the person who asks it in order that they answer the question themselves?

If the question has as a known answer, then I think either course of action is reasonable. One can give the answer or take the other approach (similar to the methods in the homework section) and try to guide the questioner to discover the answer.

If the question asked does not have a known answer (to PF members), then offering suggestions for the questioners further study may be helpful. But I don't think it is appropriate for a person who offers such suggestions to imply that the questioner is being negligent or disruptive by not following them. If the suggestions for further study are not known to lead to an answer, then I see no reason that following them should be mandatory.

From my point of view, if a person asks a question that is mathematically interesting and gets no answer, I focus on answering the question myself. I don't focus on the person asking the question. I don't worry about their character or how many hours of the day they are devoting to the question. If I worry about the extent of their knowledge, I only do it because I want to discuss something specific and need to understand what terminology I can use.

(It's curious that asking for a mathematical definition of "wave" can provoke indignation and frustration. I've had similar experiences as a teacher. I'd be lecturing away about some topic using customary terminology like "wave". Then some student would ask a simple question like "What is a wave?" Frustration at having the flow of the lecture interrupted, self-doubt when no precise answer came to mind, give a bunch of examples and hope the issue is dropped, How dare you ask such a question when you haven't studied partial differential equations - I've felt all those emotions.)

 

[jasonRF]

Hi Steven Tashi,

thanks for answering. A simple “no” would have been fine - I wasn’t trying to be that inflammatory. For some reason I thought you were interested in seeing the precise mathematical definitions of objects such as phase and group velocities, especially after you stated you couldn’t find books that defined them outside of specific examples. Clearly I was wrong.

I wish you luck in your quest to develop a wave theory based on cellular automata.

Jason

 

[SSequence]

@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.

----- 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.What are the deficiencies and/or mistakes in the above descriptions of solutions. Also, are there are some "other type" of solutions too, in addition to the ones above?

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

And two further questions:
(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?

(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)?

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

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.

 

[Stephen Tashi]

Can we define a wave to be a special type of transformation group? I think the abstract approach to dynamics establishes a context for doing this. For those unfamiliar with that approach, I'll describe my understanding of it.

Consider the case of a real valued function $u(x,t)$ whose arguments are coordinates of a spatial location $x = (x_1,x_2,x_3)$ and a real number $t$, which I will call "time".

For a fixed time $t$, we can view $u$ as defining a real valued function of position, which will be denoted by $u_t(x)$ (For example, $u_{5.2}(x)$ is the function defined by $u_{5.2}(x) = u(x,5.2)$ )

On the set of functions $\{u_t(x) | -\infty < t < \infty \}$ we can define a set of relations $\mathbb{T}$ by defining the relation $T_{\Delta t} \in \mathbb{T}$ to be the relation that associates the function $u_t(x)$ with the function $u_{t + \Delta t}(x)$.

The above scenario is a standard set-up in texts on continuous 1-parameter transformation groups, except that in those texts $T_{\Delta t}$ is defined as the function that maps $u_t(x)$ to $u_{t + \Delta t} (x)$.

The scenario stated above does not rule out possibilities like this:
$u_{5.2}(x) = u_{7.3}(x)$ considered as functions of $x$
and $u_{6.2}(x) \ne u_{8.3}(x)$ as functions of $x$. The relation $T_{1.0}$ is not a function since it relates the same function $u_{5.2}(x) = u_{7.3}(x)$ to two different functions $u_{6.2}(x), u_{8.3}(x)$.

If we add an assumption that $u(x,t)$ has a property that ensures that $T_{\Delta t}$ is a function then we have imposed a significant constraint on $u(x,t)$.

One possible constraint is make $u(x,t)$ behave "by definition".

Constraint 1: Assume $u(x,t)$ has the property that for each $-\infty < \Delta t < \infty$, the relation $T_{\Delta t}$ is a function.

( This constraint allows us to say that $\mathbb{T}$ is a group, in the algebraic sense of the concept "group". It is an additive group under "+" using the definition that the sum of transformations $T_{\Delta_1 t} + T_{\Delta_2 t}$ is the transformation $T_{\Delta_1t + \Delta_2t}$ )

Constraint 1 fails to describe solutions to differential equations. In general, a function $u(x,t)$ being a solution to a partial differential does not prohibit things like $u_{5.2}(x)$ and $u_{7.3}(x)$ being the same function In a manner of speaking, as time passes the same shape $u_{5.2}(x) = u_{7.3}(x)$ might evolve to different shapes because how they change depends not only on the shape, but also the shapes leading up to it.

The usual approach to making Constraint 1 useful for physics is to expand our definition of "space". Instead of saying the variable $x$ represents a location, we define it to include more information. For example, the variable $x$ could be defined to be a vector including location coordinates $(x_1,x_2,x_3,t)$ and also coordinates representing partial derivatives like $\frac{\partial u(x,t)}{\partial x_2}$ or $\frac{\partial u(x,t)}{\partial t}$ or higher order derivatives.

By expanding the definition of $x$ in this manner we can make $u_{5.2}(x)$ and $u_{7.3}(x)$ different functions, even though they are they describe the same shape in 3-D space.

From a strictly mathematical point of view, expanding the definition of $x$ in this manner introduces the assumption that the various derivatives of $u(x,t)$ exist. However, physics treats non-differentiable functions such as step functions.

I don't know how to precisely describe a mathematical scenario where the definition of $x$ is expanded by dropping the assumptions of (global) differentiability and introducing conditions that make it possible to work with step functions. Perhaps there is a generalized definition of derivatives that works?

Glossing over that difficulty, expanding the definition of $x$ and using Constraint 1 goes too far! It allows $u(x,t)$ to be a solution of a partial differential equation, but this just says that $u(x,t)$ describes the dynamics of some system. It doesn't say that $u(x,t)$ must describe a wave-like phenomena.

What additional constraints could be placed on $u(x,t)$ to force it to be a wave? That's a subject for a different post.

 

[Dale]

Is there a standard mathematical definition for "wave"? What is the definition?

Yes. Mathematically a wave is a function which is a solution to the wave equation (I assume you know that equation). Colloquially we may call things waves which do not satisfy that definition, but that is the precise mathematical definition. It would be acceptable to use generalizations as long as the usage is clear.

For example, how is the "group" of a wave defined? ( as in the "group" that has a "group velocity").

The group velocity is $v_g=\partial \omega/\partial k$. The thing that travels at the group velocity for a narrow-band signal is:

$$\int_{-\infty}^{\infty} A(k) e^{i (k-k_0)(x-\omega’_0 t)}dk$$

So I would call that function the group.

See: https://en.m.wikipedia.org/wiki/Group_velocity

To formulate a mathematical definition of "wave" based on that idea, we have to define "disturbance" and set up a scenario where "equilibrium" is precisely defined.

Do not go down this route again! We had that discussion already and it was locked for cause.

Can we define a wave to be a special type of transformation group?

If you can provide a professional reference that does so, then yes.

 

[atyy]

Hyperbolic equations are often said to represent waves. The wave equation is a hyperbolic equation. A feature of hyperbolic equations is that they represent disturbances that travel.

This idea that a wave is something that travels is used to distinguish waves from "almost waves". For example, one can define a "velocity" in the passive cable theory used to model neurons in which voltage-dependent channels are not yet activated (https://warwick.ac.uk/fac/sci/systemsbiology/staff/richardson/teaching/ma4g4/ITN_LN3.pdf). However, it is recognized that this velocity is not a true velocity (https://www.amazon.com/dp/0195181999) as the equation is not hyperbolic.

In contrast, the action potential, which obeys a different equation, due to voltage-dependent channels being activated is modeled by a hyperbolic equation, and has a true velocity.

https://web.math.princeton.edu/~seri/homepage/papers/gowers-Aug4-2006.pdf
"Another important, characteristic, property of hyperbolic equations is finite speed of propagation."
"Elliptic, parabolic and dispersive equations do not have this property."
"Elliptic and hyperbolic equations are the most robust, useful, classes of PDE. Other important classes, such as parabolic and dispersive, can be interpreted as lying at the boundaries of these two classes. A neat classification of all linear equations into, elliptic, hyperbolic, parabolic and dispersive is unfortunately not possible, even for second order equations in two space dimensions."