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The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields - as they occur in classical physics - such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation, which is much easier to solve and also valid for inhomogeneous media. The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions u = u (x1, x2, ..., xn; t) of a time variable t (a variable representing time) and one or more spatial variables x1, x2, ..., xn (variables representing a position in a space under discussion), while there are vector wave equations describing waves in vectors such as waves for an electrical field, magnetic field, and magnetic vector potential and 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., a space and time). For example, in the Cartesian coordinate system, for as the representation of an electric vector field wave in the absence of wave sources, each coordinate axis component (i = x, y, z) must satisfy the scalar wave equation. Other scalar wave equation solutions u are for physical quantities in scalars such as pressure in a liquid or gas, or the displacement along some specific direction of particles of a vibrating solid away from their resting (equilibrium) positions.

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