Wave Equation

WAVE EQUATION 

Plane Electromagnetic Wave Equation in Vacuum

Maxwell's equations in general form are

Now, for the free space (vacuum) the permittivity and permeability are denoted by ε0 and μ0, respectively. Therefore,

Also, the conductivity σ = 0, that is the medium is a perfect insulator. Therefore there is no conduction current in the medium which implies

Also there is no charge present in the vacuum therefore ρ = 0 and as a result eqn.(1) reduces to

Wave equation for electric field vector

Taking the curl on both sides of equation (3), we get

Now from vector calculus identity, we have

But from eqn. (5), and substituting this in equation (7) we get

substituting eqn (8) in eqn (6)

This is general electromagnetic wave equation in terms of electric field vector for free space.

Wave equation for magnetic field vector

Taking curl on both sides of the equation (4), we have

Now from vector calculus identity, we have

But from eqn (2), we have

Substituting eqn (13) in eqn (12), we get

on substituting eqn (14) in eqn (11), we have Using eqn (14) and eqn (11)

Substituting the eqn (3) in eqn (15)

This general electromagnetic wave equation in terms of for free space.

Discussion 

1. The electromagnetic wave equation for and is written as

In one dimension say along x-axis, the wave equations are given by the x-components of the above expression. That is