Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in physics—such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.
Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.
The wave equation is a hyperbolic partial differential equation. It typically concerns a time variable t, one or more spatial variables x1, x2, …, xn, and a scalar function u = u (x1, x2, …, xn; t), whose values could model, for example, the mechanical displacement of a wave. The wave equation for u is
where ∇2 is the (spatial) Laplacian and c is a fixed constant.
Solutions of this equation describe propagation of disturbances out from the region at a fixed speed in one or in all spatial directions, as do physical waves from plane or localized sources; the constant c is identified with the propagation speed of the wave. This equation is linear. Therefore, the sum of any two solutions is again a solution: in physics this property is called the superposition principle.
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