Finding the Dominant Mode in a Rectangular Waveguide

[georgeh]

Homework Statement

For an air-filled waveguide rectangular wave guide with the top and bottom made of PEC and
the left wall made of PMC and the right wall of PEC. The dimensions are a=5cm b=3m. Find the dominant mode propagating in this wave guide
(a is length and b is the height)

Homework Equations

The Attempt at a Solution

I am trying to attempt to understand the boundary conditions.
I know that for a PMC the tangential magnetic field and the normal electric field must be equal to zero. For a PEC the tangential electric field is zero, the magnetic field normal to the surface is 0.
I am not sure how to attempt to find the field. I am looking at a diagram where the box is facing the y-x axis and the direction of propagation is in the z. I am suppose to be able to reason how the function by looking at the boundary conditions.
the solution they have is
for TE:
H_z = sin(2m+1/(2a)*pi*x)*cos(pi*n/b*y) e^-jbetaz
and for TM Mode:
E_z = cos(2m+1/a* x)*sin(n*pi/b *y)*e^-jbetaz
I don't understand how they figuered this out by applying the boundary conditions. I am
getting confused as to how to apply the boundary conditions. If someone could clarify it, i would greatly appreciate it.

 

[gabbagabbahey]

PEC (perfect electric conductor?) and PMC (perfect magnetic conductor?) are not terms I am familiar with. They must be used mainly in engineering and perhaps materials science courses, as opposed to electrodynamics courses.

If you can provide a decent definition of those terms (a quick Google search found next to nothing), I'm sure I can help you with this one...

 

[georgeh]

Yup. PEC - Perfect Electrical conductor. This is implies that the tangential electric field is zero and the tangential magnetic field is maximum. The Normal electric field is maximum and the normal magnetic field is zero.
PMC - Perfect Magnetic Conductor - Tangential magnetic field is zero, tangential electric field is maximum. Normal magnetic field is maximum and the normal electric field is zero.

 

[gabbagabbahey]

In that case, start with the general solution to Maxwell's Equation in a rectangular waveguide: have you derived that yet?

 

[Phrak]

A picture is worth a thousand words and a page of equations. This looks like the usual rectangular wave guide, split down the middle.

Split where? Find a drawing of the fields for the usual reactangular wave guide. Locate the plane through which the magnetic fields are always perpendicular.