Editing Wave (section)

=== Single waves ===
{{See also|Solitary wave (disambiguation){{!}}Solitary wave}}
A wave can be described just like a field, namely as a [[function (mathematics)|function]] <math>F(x,t)</math> where <math>x</math> is a position and <math>t</math> is a time.

The value of <math>x</math> is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a [[vector (mathematics)|vector]] in the [[analytic geometry|Cartesian three-dimensional space]] <math>\mathbb{R}^3</math>. However, in many cases one can ignore one dimension, and let <math>x</math> be a point of the Cartesian plane <math>\mathbb{R}^2</math>. This is the case, for example, when studying vibrations of a drum skin. One may even restrict <math>x</math> to a point of the Cartesian line <math>\R</math> – that is, the set of [[real number]]s. This is the case, for example, when studying vibrations in a [[string (music)|violin string]] or [[recorder (musical instrument)|recorder]]. The time <math>t</math>, on the other hand, is always assumed to be a [[scalar (physics)|scalar]]; that is, a real number.

The value of <math>F(x,t)</math> can be any physical quantity of interest assigned to the point <math>x</math> that may vary with time. For example, if <math>F</math> represents the vibrations inside an elastic solid, the value of <math>F(x,t)</math> is usually a vector that gives the current displacement from <math>x</math> of the material particles that would be at the point <math>x</math> in the absence of vibration. For an electromagnetic wave, the value of <math>F</math> can be the [[electric field]] vector <math>E</math>, or the [[magnetic field]] vector <math>H</math>, or any related quantity, such as the [[Poynting vector]] <math>E\times H</math>. In [[fluid dynamics]], the value of <math>F(x,t)</math> could be the velocity vector of the fluid at the point <math>x</math>, or any scalar property like [[pressure]], [[temperature]], or [[density]]. In a chemical reaction, <math>F(x,t)</math> could be the concentration of some substance in the neighborhood of point <math>x</math> of the reaction medium.

For any dimension <math>d</math> (1, 2, or 3), the wave's domain is then a [[subset]] <math>D</math> of <math>\mathbb{R}^d</math>, such that the function value <math>F(x,t)</math> is defined for any point <math>x</math> in <math>D</math>. For example, when describing the motion of a [[drumhead|drum skin]], one can consider <math>D</math> to be a [[disk (mathematics)|disk]] (circle) on the plane <math>\mathbb{R}^2</math> with center at the origin <math>(0,0)</math>, and let <math>F(x,t)</math> be the vertical displacement of the skin at the point <math>x</math> of <math>D</math> and at time <math>t</math>.