# The mathematical definition of "wave"?

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## 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!

• Delta2

Staff Emeritus
2019 Award
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."

• Dale, DaveE, etotheipi and 2 others
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?

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 realised 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.

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• etotheipi
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
Gold Member
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
Gold Member
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

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• Delta2, etotheipi and PeroK
etotheipi
Gold Member
2019 Award
• jasonRF and SSequence
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.

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"?

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

• PeroK and etotheipi
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.

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.

• Dale
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".

• PeroK
DaveE
Gold Member
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.

• Dale
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.

• Math_QED
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?

• Dale
DaveE
Gold Member
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
Homework Helper
Gold Member
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.

• Math_QED, jasonRF and etotheipi
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.

• Dale, jasonRF and DaveE
jasonRF
Gold Member
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
for a quick and dirty derivation of group velocity in 3D.

jason

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• Dale
atyy
• ChinleShale and vanhees71
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
Gold Member
2019 Award
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'.

• • vanhees71, Math_QED and pbuk