Wave equation - Knowlpedia

Definition

On a Lorentzian manifold $(M,g)$ , the scalar wave equation for a smooth real- or complex-valued field $\phi$ is

\Box_g\phi=0,

where $\Box_g$ is the d’Alembert operator. The inhomogeneous equation is $\Box_g\phi=f$ for a prescribed source $f$ .

Flat-spacetime form

On Minkowski spacetime, this convention gives

\partial_t^2\phi-\sum_{j=1}^{n-1}\partial_{x_j}^2\phi=0.

Plane waves $e^{-i\omega t+i k\cdot x}$ solve the equation precisely when $\omega^2=|k|^2$ , so their frequency covector is null.

Initial-value formulation

The wave equation is an initial-value problem rather than an elliptic boundary-value problem. The precise global existence, uniqueness, and finite-propagation statement is the Cauchy theorem for normally hyperbolic operators.

Variants

For a field valued in a vector bundle, the scalar operator is replaced by a normally hyperbolic operator. Adding a mass term produces the Klein–Gordon equation. Nonlinear wave equations add terms depending nonlinearly on $\phi$ or its derivatives and require separate well-posedness analysis.

References

Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: Chapter 3.
Lars Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Springer, 1997. Publisher record. Relevant: Chapters 1–2.