section_path: Mathematical description/Single waves
section_name: Single waves
paragraph_text: Single waves
  
  A wave can be described just like a field, namely as a function F (x, t) {\displaystyle F(x,t)} where x {\displaystyle x} is a position and t {\displaystyle t} is a time.
  
  The value of x {\displaystyle x} is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a vector in the Cartesian three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}}. However, in many cases one can ignore one dimension, and let x {\displaystyle x} be a point of the Cartesian plane R 2 {\displaystyle \mathbb {R} ^{2}}. This is the case, for example, when studying vibrations of a drum skin. One may even restrict x {\displaystyle x} to a point of the Cartesian line R {\displaystyle \mathbb {R} } — that is, the set of real numbers. This is the case, for example, when studying vibrations in a violin string or recorder. The time t {\displaystyle t}, on the other hand, is always assumed to be a scalar; that is, a real number.
  
  The value of F (x, t) {\displaystyle F(x,t)} can be any physical quantity of interest assigned to the point x {\displaystyle x} that may vary with time. For example, if F {\displaystyle F} represents the vibrations inside an elastic solid, the value of F (x, t) {\displaystyle F(x,t)} is usually a vector that gives the current displacement from x {\displaystyle x} of the material particles that would be at the point x {\displaystyle x} in the absence of vibration. For an electromagnetic wave, the value of F {\displaystyle F} can be the electric field vector E {\displaystyle E}, or the magnetic field vector H {\displaystyle H}, or any related quantity, such as the Poynting vector E × H {\displaystyle E\times H}. In fluid dynamics, the value of F (x, t) {\displaystyle F(x,t)} could be the velocity vector of the fluid at the point x {\displaystyle x}, or any scalar property like pressure, temperature, or density. In a chemical reaction, F (x, t) {\displaystyle F(x,t)} could be the concentration of some substance in the neighborhood of point x {\displaystyle x} of the reaction medium.
  
  For any dimension d {\displaystyle d} (1, 2, or 3), the wave's domain is then a subset D {\displaystyle D} of R d {\displaystyle \mathbb {R} ^{d}}, such that the function value F (x, t) {\displaystyle F(x,t)} is defined for any point x {\displaystyle x} in D {\displaystyle D}. For example, when describing the motion of a drum skin, one can consider D {\displaystyle D} to be a disk (circle) on the plane R 2 {\displaystyle \mathbb {R} ^{2}} with center at the origin (0, 0) {\displaystyle (0,0)}, and let F (x, t) {\displaystyle F(x,t)} be the vertical displacement of the skin at the point x {\displaystyle x} of D {\displaystyle D} and at time t {\displaystyle t}.