Wave equation

The wave equation is a partial differential equation describing disturbances that propagate through space and time at a finite speed, such as sound or vibrations.

D'Alembert formula: A formula expressing solutions of the one-dimensional wave equation as left- and right-moving waves. It gives the general solution for the equation on an unbounded one-dimensional line.
Partial differential equation: An equation relating a function of multiple variables to its partial derivatives. The wave equation belongs to this broad class of equations.
Acoustic wave equation: A wave equation governing small pressure or density disturbances in a fluid. It predicts the propagation of sound through air, water, and other fluids.
Jean le Rond d'Alembert: An eighteenth-century French mathematician and physicist who studied the vibrating-string equation. His 1747 analysis produced a general solution for the one-dimensional string equation.
Heat equation: A parabolic partial differential equation modeling diffusion of heat or other quantities. It smooths disturbances over time, unlike the wave equation's traveling fronts.
Initial value problem: A differential equation problem specified by initial data at a particular time. Initial displacement and velocity determine which wave solution develops.
Second derivative: The derivative of a function's first derivative, measuring its local curvature or acceleration. Spatial and temporal curvature are equated in the standard wave equation.
Elastic wave equation: A system of equations describing how elastic displacements propagate through a solid. It extends wave propagation to coupled compressional and shear motion in materials.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work shaped analysis, mechanics, and mathematical physics. Euler developed early mathematical treatments of vibrating strings and their physical interpretation.
Laplace equation: An elliptic partial differential equation stating that a function's Laplacian is zero. It describes static equilibrium rather than time-dependent propagation.

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