Wave equation

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light 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 partial differential equation that may constrain some scalar function u = u (x1, x2, …, xn; t) of a time variable t and one or more spatial variables x1, x2, … xn. The quantity u may be, for example, the pressure in a liquid or gas, or the displacement, along some specific direction, of the particles of a vibrating solid away from their resting positions. The equation is where c is a fixed non-negative real coefficient. Using the notations of Newtonian mechanics and vector calculus, the wave equation can be written more compactly as where ◻ ¨ {displaystyle {ddot {scriptstyle square }}} denotes double time derivative, ∇ is the nabla operator, and ∇2 = ∇ · ∇ is the (spatial) Laplacian operator: u ¨ = ∂ 2 u ∂ t 2 ∇ = ( ∂ ∂ x 1 , ∂ ∂ x 2 , … , ∂ ∂ x n ) ∇ 2 = ∂ 2 ∂ x 1 2 + ∂ 2 ∂ x 2 2 + ⋯ + ∂ 2 ∂ x n 2 {displaystyle {ddot {u}}={frac {partial ^{2}u}{partial t^{2}}}quad quad abla ={igl (}{frac {partial }{partial x_{1}}},{frac {partial }{partial x_{2}}},ldots ,{frac {partial }{partial x_{n}}}{igr )}quad quad abla ^{2}={frac {partial ^{2}}{partial x_{1}^{2}}}+{frac {partial ^{2}}{partial x_{2}^{2}}}+cdots +{frac {partial ^{2}}{partial x_{n}^{2}}}} A solution of this equation can be quite complicated, but it can be analyzed as a linear combination of simple solutions that are sinusoidal plane waves with various directions of propagation and wavelengths but all with the same propagation speed c. This analysis is possible because the wave equation is linear; 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.