19.1 Wave Equation

An electromagnetic wave is governed by three important features: the spatial curvature of the electric field ∇²E, the temporal curvature of the electric field Ë, and the temporal slope of electric current density J, which may also be considered the acceleration of electric current density. These parameters are expressed in the wave equation as follows:

∇E = μ(ε₀ Ë, + J),

where μ is the magnetic permeability and ε₀ is the electric permittivity.

The ∇ operator defines the gradient as

∇ = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k

where i, j, and k are unit vectors. The ∇² operator defines the Laplacian as

∇² = (∂²/∂x²)i + (∂²/∂y²)j + (∂²/∂z²)k.

The temporal versions of the spatial operators are

Á = (∂/∂t)A

and

Ä = (∂/∂t²)A.