- #1

Stephen Tashi

Science Advisor

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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:

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 "

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:

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!

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, awaveis 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!