Waveguide Guide: 
and V
Wave Equations:
Lorentz Condition:
and 
:
Boundary Conditions:
Dispersion Relation in Rectangular Coordinates:
It is the same for traveling, standing, or combined standing/traveling (waveguide) waves:
Rectangular Waveguide:
The wave guide is infinitely long in the x-direction and goes from 0 to a in the y-direction and from 0 to b in the z-direction.
Waveguide Waveform:
Two Mode Types
The rather complicated vector eigenvalue problem involved in
waveguides turns out to be simpler than might have
been thought.
There are two distinct types of solutions.
One type has 
, i.e., the mode electric field
only has components perpendicular (or transverse)
to the long axis of the wave guide.
These are the TE modes.
The second type has 
, i.e., the mode magnetic field
only has components perpendicular (or transverse)
to the long axis of the wave guide.
These are the TM modes.
If you work with and , this reduces
the number of vector components you need to find from
6 to 5, but if you work with and V, as
we are doing here, it reduces the number from 4 to 2.
TE
Modes (
and V=0):
The condition that turns out to require
that
and V=0.
Hence, we only have to find 
and 
.
Applying both the boundary conditions and the
Lorentz gauge condition leads to
and
with
This condition relates the two amplitudes, but does not determine the overall magnitude. This is determined by the person who shoots the energy into the waveguide.
TE
and TE
Special Cases:
If you set either m=0 or n=0 in the amplitude relation above you will see that only one of the vector field components survives.
TM Modes (
and 
):
The condition that turns out to require
that
and .
Hence, we only have to find 
and V(y,z).
Applying both the boundary conditions and the
Lorentz gauge condition leads to
TM and TM Special Cases:
Setting m=0 or n=0 in the formula for 
makes the fields vanish,
so 
and 
for TM modes.
TEM Mode:
In a completely hollow guide waves with both 
and 
parallel to the axis of the guide are impossible.
But with a conductor along the axis these waves are possible.
Their dispersion relation is simply
with 
parallel to the axis of the guide.
The electric field points outward from the central conductor
and terminates on the outer surface of the guide while
the magnetic field circulates around the central conductor
and runs parallel to the outer conductor.
Hence, they are just about like free space waves with ,
, and mutually perpendicular.
In a cylindrical guide (a coaxial cable) of radius a
these waves are
described in cylindrical coordinates by
where 
is the electric field amplitude at r=a.