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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.

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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