1. The problem statement, all variables and given/known data 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) 2. Relevant equations 3. 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.