The mathematical definition of "wave"?

[whoohm]

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.

 

[martinbn]

Wave map is standrd terminolgy.

 

[Stephen Tashi]

Wave map is standrd terminolgy.

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

 

[jasonRF]

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]

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.

 

[Dale]

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.

 

[Stephen Tashi]

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.

 

[Dale]

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.

 

[Baluncore]

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.

 

[martinbn]

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.

 

[ChinleShale]

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. ..."

 

[ChinleShale]

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)

and this should read

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.