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## 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.