Math 2.0

Partial differential equations Definition

Wave equation

Builds on Hyperbolic partial differential equationLaplacianPartial derivative

Definition

Let n≥1n \ge 1, let c>0c > 0, and let Ω⊆ℝn\Omega \subseteq \R^n be open. Let I⊆ℝI \subseteq \R be an interval. The wave equation with speed cc is the hyperbolic partial differential equation for u ⁣:Ω×I→ℝu \colon \Omega \times I \to \R: ∂2u∂t2=c2Δu,\frac{\partial^2 u}{\partial t^2} = c^2 \Delta u, where Δ=∑i=1n∂2/∂xi2\Delta = \sum_{i=1}^{n} \partial^2/\partial x_i^2 is the Laplacian in the space variables. The inhomogeneous wave equation is utt−c2Δu=fu_{tt} - c^2 \Delta u = f for a given function ff.

Remarks

Example: for n=1n = 1 and any F∈C2(ℝ)F \in C^2(\R), the function u(x,t)=F(x−ct)u(x, t) = F(x - ct) is a solution, since utt=c2F′′(x−ct)u_{tt} = c^2 F''(x - ct) and uxx=F′′(x−ct)u_{xx} = F''(x - ct). Non-example: u(x,t)=x2u(x, t) = x^2 fails for n=1n = 1, since utt=0u_{tt} = 0 but c2uxx=2c2≠0c^2 u_{xx} = 2c^2 \ne 0. The operator ∂t2−c2Δ\partial_t^2 - c^2\Delta is called the d'Alembertian or wave operator. The Cauchy problem prescribes u(⋅,0)=gu(\cdot, 0) = g and ut(⋅,0)=hu_t(\cdot, 0) = h.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHyperbolic partial differential equation, concept, no evidence levelHyperbolic partial di…conceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptWave equation, definition, no evidence levelWave equationdefinition
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  1. Euclidean space object
  2. Function concept
  3. Hyperbolic partial differential equation concept
  4. Total derivative concept
  5. Function of several variables concept
  6. Vector-valued function concept
  7. Differentiable function of several variables concept
  8. Partial derivative concept
  9. Vector field concept
  10. Gradient concept
  11. Divergence concept
  12. Laplacian concept
  13. Wave equation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Wave_equation
About this page
TagS977PT8S3S
URIm2://S977PT8S3S
Kinddefinition
WritingW0 stub
MSC35L05, 35L15
DefinesWave equation
UsesHyperbolic partial differential equation, Laplacian, Partial derivative
Changed2 November 2026

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