Editing Wave equation (section)

== Introduction ==
The wave equation is a [[hyperbolic partial differential equation]] describing waves, including traveling and [[standing waves]]; the latter can be considered as [[Superposition principle|linear superpositions]] of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in [[Scalar field|scalars]] by scalar functions {{math|1=''u'' = ''u'' (x, y, z, t)}} of a time variable {{mvar|t}} (a variable representing time) and one or more spatial variables {{math|x, y, z}} (variables representing a position in a space under discussion). At the same time, there are vector wave equations describing waves in [[Vector field|vectors]] such as [[Inhomogeneous electromagnetic wave equation|waves for an electrical field, magnetic field, and magnetic vector potential]] and [[#Elastic waves|elastic waves]]. By comparison with vector wave equations, the scalar wave equation can be seen as a special case of the vector wave equations; in the [[Cartesian coordinate system]], the scalar wave equation is the equation to be satisfied by each component (for each coordinate axis, such as the ''x'' component for the ''x'' axis) of a vector wave without sources of waves in the considered domain (i.e., space and time). For example, in the Cartesian coordinate system, for <math>(E_x, E_y, E_z)</math> as the representation of an electric vector field wave <math>\vec{E}</math> in the absence of wave sources, each coordinate axis component <math>E_i</math> (''i'' = ''x'', ''y'', ''z'') must satisfy the scalar wave equation. Other scalar wave equation solutions {{mvar|u}} are for [[Physical quantity|physical quantities]] in [[Scalar field|scalars]] such as [[Acoustic_wave_equation#Equation|pressure]] in a liquid or gas, or the [[displacement (vector)|displacement]] along some specific direction of particles of a vibrating solid away from their resting (equilibrium) positions.

The scalar wave equation is
{{Equation box 1
|indent=:
|equation=<math>\frac{\partial^2 u}{\partial t^2} = c^2 \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right)</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA
}}where
* {{mvar|c}} is a fixed non-negative [[real number|real]] [[coefficient]] representing the [[Wave#Wave_velocity|propagation speed]] of the wave
* {{mvar|u}} is a [[scalar field]] representing the displacement or, more generally, the conserved quantity (e.g. [[pressure]] or [[density]])
* {{mvar|x}}, {{mvar|y}} and {{mvar|z}} are the three spatial coordinates and {{mvar|t}} being the time coordinate.

The equation states that, at any given point, the second derivative of <math>u</math> with respect to time is proportional to the sum of the second derivatives of <math>u</math> with respect to space, with the constant of proportionality being the square of the speed of the wave.

Using notations from [[vector calculus]], the wave equation can be written compactly as
<math display="block">u_{tt} = c^2 \Delta u,</math>
or
<math display="block">\Box u = 0,</math> 
where the double subscript denotes the second-order [[partial derivative]] with respect to time, <math>\Delta</math> is the [[Laplace operator]] and <math>\Box</math> the [[d'Alembert operator]], defined as:
<math display="block"> u_{tt} = \frac{\partial^2 u}{\partial t^2}, \qquad \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} +  \frac{\partial^2}{\partial z^2}, \qquad \Box = \frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \Delta.</math>

A solution to this (two-way) wave equation can be quite complicated. Still, it can be analyzed as a linear combination of simple solutions that are [[sinusoidal]] [[plane wave]]s with various directions of propagation and wavelengths but all with the same propagation speed {{mvar|c}}. This analysis is possible because the wave equation is [[linear differential equation|linear]] and homogeneous, so that any multiple of a solution is also a solution, and the sum of any two solutions is again a solution. This property is called the [[superposition principle]] in physics.

The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as [[initial conditions]], which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by [[boundary conditions]], for which the solutions represent [[standing waves]], or [[harmonics]], analogous to the harmonics of musical instruments.