- (0 pts) Study the derivation of the TE modes in a rectangular waveguide in Jackson’s section 8.4.
- (4 pts) Work out the TM modes for the same waveguide. [You should end up with formulas that parallel (8.44-8.46).]
- (2 pts) Which of the TE and TM modes has the lowest cutoff frequency in a rectangular waveguide with ? Is there only one mode with this cutoff frequency?
- (3 pts) Find the second-lowest, third-lowest, and fourth-lowest cutoff frequencies in a rectangular waveguide with . How many modes have each of these cutoff frequencies?
- (6 pts) Calculate the attenuation constant for the mode with the lowest cutoff frequency in a rectangular waveguide with , using the method of Jackson’s section 8.5.]
- (0 pts) Note that the power flow tends to zero near the cutoff, and the attenuation constant found in step #5 diverges. This result is unphysical, because the wavevector should always be finite. Therefore, the method of section 8.5 fails near the cutoff. [A more general method is described in section 8.6, which you should review.]
- (2 pts) Consider a waveguide with a circular cross-section. Write the eigenvalue equation in cylindrical coordinates.
- (2 pts) Use variable separation of show that the solutions can be written as .
- (2 pts) Show that and explain why must be integer.
- (2 pts) Use the results of steps #7 and #8 to obtain the ordinary differential equation for . Identify this equation by name.
- (2 pts) Recall which special functions represent the solutions of the resulting equation. Identify those that are singular at the origin and explain why they should be discarded.
- (3 pts) Using the results of steps #9 and #10, establish the procedure for finding the cutoff frequencies of all TE and TM modes in a circular waveguide. Introduce the notation and , where is the integer number from step #8 and is the sequential number of the solution.
- (4 pts) Find 5 lowest cutoff frequencies and label the corresponding modes using the notation from step #11. Note the degeneracies.
- (2 pts) In step #13 you should have found that some cutoff frequencies are doubly degenerate. Identify the relation between the relevant special functions that is responsible for this degeneracy.
- (0 pts) Consider a waveguide whose cross-section is an isosceles right triangle. Note that the variables can not be separated in this case.
- (1 pts) Note that the isosceles right triangle can be obtained by slicing a square along its diagonal. Using the fact that the mirror reflection along the diagonal is a symmetry operation for the square, argue that the TE and TM modes for the square can be chosen to be even or odd under this reflection.
- (2 pts) The solutions for the TE and TM modes of a rectangular waveguide were found in section 8.4 and in step #2. Specialize these solutions to the case of a square cross-section and list the field configurations for all TE modes and the field configurations for all TM modes.
- (2 pts) Explain why degenerate modes are only defined up to an arbitrary linear combination.
- (3 pts) Form linear combinations of the modes found in step #17 that relabel the TE and TM modes in terms of their parity (even or odd) under mirror reflection along one the square diagonal where . Which combinations of labels are possible?
- (3 pts) Use the modes from step #19 to find the TE and TM modes for the triangular waveguide.
- (2 pts) Identify the TE and TM modes with the lowest cutoff frequency for the triangular waveguide.
Total: 47 points.