Wave Equations: EM Waves

Electromagnetic waves for E field for B field

In general, electromagnetic waves Where  represents E or B or their components

# A plane wave satisfies wave equation in Cartesian coordinates # A spherical wave satisfies wave equation in spherical polar coordinates # A cylindrical wave satisfies wave equation in cylindrical coordinates

Solution of 3D wave equation In Cartesian coordinates Separation of variables

Substituting for  we obtain Variables are separated out Each variable-term independent And must be a constant

So we may write where we use

Solutions are then Total Solution is plane wave

Traveling 3D plane wave

spherical coordinates

spherical waves

Alternatively The wave equation becomes

Put Then  Hence

Therefore Wave equation transforms to 

Which follows that Separation of variables Solutions are Total solution is

outgoingwaves incomingwaves Final form of solution General solution spherical wave

Cylindrical waves

with angular and azimuthal symmetry, the Laplacian simplifies and the wave equation

The solutions are Bessel functions. r For large r, they are approximated as

A plane wave satisfies one-dimensional wave equation in Cartesian coordinates The position vector must remain perpendicular to the given plane

The wave then satisfies the generalization of the one-dimensional wave equation

Plane EM waves in vacuum

Wave vector k is perpendicular to E Wave vector k is perpendicular to B

B is perpendicular to E

B, k and E make a right handed Cartesian co-ordinate system

Plane EM waves in vacuum