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Wave equation: overview, mathematics, history and applications

A concise encyclopedia article on the wave equation: its form, properties, classical solutions, historical origins, common applications, and important variants in physics and engineering.

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The wave equation is the class of mathematical equations that model how disturbances propagate through space and time. In physical settings these disturbances appear as mechanical waves on rivers, waves on lakes and oceans, as well as variations of pressure or displacement that constitute sound and many aspects of light. The same basic wave formalism arises in diverse fields such as acoustics, electromagnetics, and fluid dynamics, where it provides a first model for how a localized change moves and spreads.

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Mathematical form and properties

In its simplest linear and homogeneous form for a scalar field u(x,t), the wave equation relates the second time derivative to a spatial Laplacian multiplied by the square of a speed parameter c. This structure makes the equation hyperbolic: disturbances travel with finite speed, information propagates along characteristic surfaces (light cones in relativistic contexts), and superposition holds for linear problems. Typical consequences include reflection at boundaries, refraction in media with variable coefficients, and the existence of standing waves when boundary conditions force interference of counterpropagating modes.

Classical solutions and methods

Exact solutions and constructive representations depend on dimension and geometry. In one spatial dimension a classical decomposition expresses solutions as a sum of a right‑travelling and a left‑travelling wave (the d'Alembert formula). In higher dimensions, solution formulas often use spherical means, Fourier transforms, or Green's functions to produce the fundamental solution and to represent forced or inhomogeneous problems. Energy methods yield conserved or nearly conserved quantities for many linear models and provide basic a priori estimates for existence and uniqueness theorems.

Historical context

Interest in the wave equation goes back to eighteenth‑century studies of musical strings and vibrating membranes. Questions about the motion of a string on a musical instrument motivated work by prominent mathematicians and physicists such as Jean le Rond d'Alembert, Daniel Bernoulli, Leonhard Euler, and Joseph‑Louis Lagrange. In 1746 d'Alembert wrote the one‑dimensional form of the wave equation; subsequent work by Euler and others extended the analysis to higher dimensions and to more general mechanical systems. These early investigations helped establish methods still used in modern analysis of partial differential equations.

Applications and examples

  • Acoustics: modeling sound in open air and enclosed spaces, including reverberation and waveguides; see acoustics.
  • Electromagnetism: Maxwell's equations in free space reduce to wave equations for electric and magnetic fields, a core topic in electromagnetics.
  • Water waves: surface gravity waves and shallow‑water approximations are related to wave‑type models in fluid dynamics, though free‑surface flows often require nonlinear treatments.
  • Seismology and engineering: wave models describe seismic waves, structural vibrations, and the design of systems to control or exploit wave propagation.

Variants and modern topics

Real systems frequently deviate from the simplest linear, homogeneous form. Dispersive media introduce frequency‑dependent speeds and lead to wave packets that spread; nonlinearities produce phenomena such as solitons and shocks (examples include the Korteweg–de Vries or nonlinear Schrödinger type models in specialised regimes). Dissipation and attenuation require damping terms. Mathematically, one studies inhomogeneous wave equations with forcing, variable coefficients to model nonuniform materials, and coupled systems that reflect multi‑component fields.

Computation and analysis

When closed‑form solutions are unavailable, numerical methods are essential. Finite difference, finite element, and spectral methods are commonly used; preserving correct wave speeds, stability, and appropriate conservation properties is crucial. Analytical topics of ongoing interest include propagation of singularities, influence of boundary geometry on solutions, scattering theory for obstacles, and stability of nonlinear waves. For introductory background one finds classical expositions in texts on mathematical physics and partial differential equations and in historical discussions of the work by d'Alembert, Bernoulli, Euler and Lagrange.

The wave equation in one spatial dimension

The D'Alembert operator in one spatial dimension

{\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}}

decays due to Schwarz's theorem as in the binomial formula {\displaystyle (a^{2}-b^{2})=(a-b)(a+b)}into the product

{\displaystyle \Box =\left({\frac {1}{c}}{\frac {\partial }{\partial t}}-{\frac {\partial }{\partial x}}\right)\left({\frac {1}{c}}{\frac {\partial }{\partial t}}+{\frac {\partial }{\partial x}}\right)}.

Therefore, the wave equation in one spatial dimension has the general solution

u\left(t, x\right) = f(x + ct) + g(x - ct)

with any twofold differentiable functions f(x)and g(x). The first summand {\displaystyle f(x+ct)}is a wave running to the left and the second summand {\displaystyle g(x-ct)}a wave running to the right with unchanged shape. The straight lines {\displaystyle x\pm ct={\text{konstant}}}are the characteristics of the wave equation.

Be

{\displaystyle \phi (x)=u(0,x)=f(x)+g(x)}

the initial value and

{\displaystyle \psi (x)={\frac {1}{c}}{\frac {\partial u}{\partial t}}(0,x)=f'(x)-g'(x)}

the initial time derivative of the wave. These functions of space are collectively called initial values of the wave.

The integration of the last equation gives

{\displaystyle f(x)-g(x)=\int _{x_{0}}^{x}\psi (\xi )\,\mathrm {d} \xi \ .}

By dissolving one obtains

{\displaystyle f(x)={\frac {1}{2}}\left(\phi (x)+\int _{x_{0}}^{x}\psi (\xi )\,\mathrm {d} \xi \right)\ ,}

{\displaystyle g(x)={\frac {1}{2}}\left(\phi (x)-\int _{x_{0}}^{x}\psi (\xi )\,\mathrm {d} \xi \right)\ .}

Expressed by their initial values, the solution of the wave equation is therefore

{\displaystyle u(t,x)={\frac {1}{2}}\left(\phi (x+ct)+\phi (x-ct)+\int _{x-ct}^{x+ct}\psi (\xi )\,\mathrm {d} \xi \right)\ .}

This is also known as the D'Alembert solution of the wave equation (d'Alembert, 1740s).

The wave equation in three spatial dimensions

The general solution of the wave equation can be expressed as a linear combination of plane waves

{\displaystyle u({\vec {x}},t)=\int \mathrm {d} \omega \int \mathrm {d} ^{3}{\vec {k}}\,A(\omega ,k)e^{\mathrm {i} ({\vec {k}}\cdot {\vec {x}}-\omega t)}\delta (\omega -c|{\vec {k}}|)}

write. The delta distribution ensures that the dispersion relation ω {\displaystyle \omega =c|{\vec {k}}|}maintained. Such a plane wave moves in the direction of {\vec {k}}. In the superposition of such solutions, however, it is not obvious how their initial values are related to the later solution.

In three spatial dimensions, the general solution of the homogeneous wave equation can be represented by mean values of the initial values. Let the function {\displaystyle u(t,{\vec {x}})}and its time derivative at the initial time t=0be \psi given by functions ϕ \phi and ψ

{\displaystyle u(0,{\vec {x}})=\phi ({\vec {x}}),\quad {\frac {1}{c}}{\frac {\partial }{\partial t}}u(0,{\vec {x}})=\psi ({\vec {x}})\,,}

then the linear combination of mean values is

{\displaystyle u(t,{\vec {x}})=ct\,M_{t,{\vec {x}}}[\psi ]+{\frac {1}{c}}{\frac {\partial }{\partial t}}(ct\,M_{t,{\vec {x}}}[\phi ])}

is the corresponding solution of the homogeneous wave equation. Here denotes

{\displaystyle M_{t,{\vec {x}}}[\chi ]={\frac {1}{4\,\pi }}\int _{-1}^{1}\mathrm {d} \cos \theta \int _{0}^{2\pi }\mathrm {d} \varphi \,\chi ({\vec {x}}+ct{\vec {n}}(\theta ,\varphi ))\quad {\text{mit}}\quad {\vec {n}}(\theta ,\varphi )={\begin{pmatrix}\sin \theta \cos \varphi \\\sin \theta \sin \varphi \\\cos \theta \end{pmatrix}}}

the mean value of the function χ \chi\,,averaged over a spherical shell around the point {\vec {x}}with radius {\displaystyle c|t|.}In particular, {\displaystyle M_{0,{\vec {x}}}[\chi ]=\chi ({\vec {x}}).}

As this representation of the solution by the initial values shows, the solution depends continuously on the initial values and depends at time tat location {\vec {x}}only on the initial values at locations {\vec {y}}from which one ccan reach {\vec {x}}in the running time |t|with velocity It thus satisfies Huygens' principle.

For one-dimensional systems and in even spatial dimensions, this principle does not apply. There, the solutions at time talso depend on initial values at closer points {\vec {y}}from which one reaches {\vec {x}}with less speed.

The solution of the inhomogeneous wave equation in three spatial dimensions

{\displaystyle u(t,{\vec {x}})=ct\,M_{t,{\vec {x}}}[\psi ]+{\frac {1}{c}}{\frac {\partial }{\partial t}}(ct\,M_{t,{\vec {x}}}[\phi ])+{\frac {1}{4\pi }}\int _{|{\vec {z}}|\leq c|t|}\mathrm {d} ^{3}{\vec {z}}\,{\frac {v(ct-\operatorname {sign} (t)|{\vec {z}}|,{\vec {x}}+{\vec {z}})}{|{\vec {z}}|}}}

depends at the location {\vec {x}}at time t>0on the inhomogeneity on the backward light cone of x → {\vec {x}}negative times only on the inhomogeneity on the forward light cone. The inhomogeneity and the initial values affect the solution at the speed of light.

Retarded potential

The retarded potential

{\displaystyle u_{\text{retardiert}}(t,{\vec {x}})={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}\mathrm {d} ^{3}{\vec {z}}\,{\frac {v(ct-|{\vec {z}}|,\,{\vec {x}}+{\vec {z}})}{|{\vec {z}}|}}}

is a solution of the inhomogeneous wave equation which requires that the inhomogeneity von all backward light cones decays faster than 1/r^{2}It is the wave that is completely generated by the medium without a passing wave.

In electrodynamics, the continuity equation restricts inhomogeneity. Thus, the charge density of a non-vanishing total charge cannot vanish everywhere at any time. In perturbation theory, inhomogeneities occur that do not decay spatially fast enough. Then the associated retarded integral diverges and has a so-called infrared divergence.

The somewhat more elaborate representation of the solution by its initial values at finite time and by integrals over finite sections of the light cone is free from such infrared divergences.

Lorentz invariance of the D'Alembert operator

The D'Alembert operator \Box is invariant under translations and Lorentz transformations Λ \Lambda sense that applied to Lorentz concatenated functions {\displaystyle f\circ \Lambda ^{-1}}yields the same as the Lorentz concatenated derivative function

{\displaystyle (\Box f)\circ \Lambda ^{-1}=\Box \,(f\circ \Lambda ^{-1})\ .}

Accordingly, the Laplace operator is invariant under translations and rotations.

The homogeneous wave equation is invariant even under conformal transformations, especially under stretching.

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