Other meanings of Wave equation
Mathematics & physics
The wave equation is a partial differential equation describing the propagation of waves through space and time. In its simplest form, it relates the second time derivative of a disturbance to its spatial curvature, with the constant of proportionality setting the wave speed.
The wave equation expresses how a changing disturbance propagates rather than remaining fixed in one place. For a scalar field u, the standard homogeneous form is ∂²u/∂t² = c²∇²u, where c is the propagation speed and ∇² is the Laplacian, measuring spatial curvature.1 A local peak therefore accelerates according to the surrounding shape of the field. The equation models small-amplitude disturbances in strings, air, fluids, elastic solids, and electromagnetic fields, although each physical system may require a different interpretation of u and c.
Its solutions can be traveling, standing, or transient waves. The equation is hyperbolic: disturbances generally propagate at finite speed, so initial data influence only a limited region bounded by a characteristic cone. This contrasts with diffusion equations, whose idealized disturbances spread instantaneously.
Solutions are determined by both the differential equation and the initial and boundary conditions. In one dimension, the general free-space solution can be written as the sum of a right-moving and a left-moving profile, a result associated with d'Alembert.2 The initial displacement and initial velocity select the particular profiles.
Boundaries change the solution fundamentally. Fixed endpoints favor standing waves whose wavelengths fit an integer number of half-wavelengths into the domain; these discrete patterns are normal modes. Separation of variables and a Fourier series represent more complicated initial shapes as sums of such modes.3 Damping, forcing, and nonreflecting boundaries add terms or conditions that describe energy loss, external excitation, or radiation away from the modeled region.
The basic equation is an idealization, and its coefficients encode the medium. A variable wave speed produces a variable-coefficient equation; anisotropic materials can replace the scalar Laplacian with coupled spatial operators. Electromagnetic waves are described by vector fields derived from Maxwell's equations, while elastic waves involve coupled longitudinal and transverse displacements.4
Inhomogeneous equations include a source term, such as an applied force or a radiating charge. A Green's function gives the response to an idealized point source and can build solutions for distributed sources. In numerical work, finite-difference, spectral, and finite element method schemes approximate the equation; stability restrictions, including the Courant condition for many explicit schemes, limit the allowed time step.5
The wave equation appears in fields that do not look like ordinary waves. In general relativity, linearized gravitational disturbances satisfy wave equations on spacetime, while in quantum mechanics the time-dependent Schrödinger equation is wave-like but is not the classical wave equation: it is first order in time and its complex field is a probability amplitude.6
Geometry can create subtle effects. Waves in a bounded cavity have a spectrum governed by the domain's shape, and irregular boundaries can produce complicated interference or sensitive ray patterns. On curved surfaces, propagation may differ from flat-space intuition because geodesics focus or spread. Even the one-dimensional idealization has edge cases: discontinuous initial data can produce weak solutions, and nonlinear corrections can cause steepening or shock formation, phenomena outside the linear equation's assumptions.
The notation ∇²u is appropriate for a scalar field in Euclidean space; vector, anisotropic, curved-space, and nonlinear wave models require corresponding generalizations.
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