- #1

fluidistic

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## Homework Statement

1)Find the EM fields that go through a coaxial waveguide (inner radius equals a, outer radius equals b) filled with a dielectric material where both TE and TM modes propagates.

2)Find the cutoff frequency.

3)Find the cutoff frequency if we close both ends of the waveguide with a conducting material, separated by a distance L.

## Homework Equations

Helmholtz equation for both TE and TM modes. Each one of these have a different boundary conditions.

## The Attempt at a Solution

1)First I want to get the explicit form for both TE and TM modes, so that I can write both ##\vec E _{\text{TM}}## and both ##\vec H _{\text{TM}}## as well as both ##\vec E _{\text{TE}}## and both ##\vec H_{\text{TE}}##, the sum of this all is the total EM field they ask for, I suppose.

So I tackled the problem by trying to find the TE modes, in which case I must solve ##(\nabla_\perp ^2 + \gamma ^2)H_z(\rho, \theta)=0## where I use cylindrical coordinates. The boundary conditions are ##\frac{\partial H_z(\rho, \theta)}{\partial \rho} \big |_S=0##. Notice that there are 2 surfaces, one for when ##\rho=a## and one for when ##\rho=b##. So I solve this Helmholtz equation via separation of variables and I got to solve the Bessel equation for the radial part/function.

Now my "problem" appears when solving this Bessel equation and applying the boundary conditions above. Namely the radial part is ##R(\rho)=A_{mn}J_m(\rho n \gamma)+B_{mn}Y_m(\rho n \gamma)##. Unfortunately when applying the boundary conditions none of ##A_{mn}## nor ##B_{mn}## must be 0... so this complicates things a lot.

Applying the boundary conditions indeed yield 2 equations from which (after eliminating both ##A_{mn}## and ##B_{mn}##), ##J'_m(an\gamma) Y'_m(bn\gamma)-Y'_m(an\gamma ) J'_m(bn\gamma )=0##. I hoped to obtain an expression for the eigenvalues of the modes, ##\gamma _{mn}## but how can I get an explicit form for them? So far I can only give an implicit form for ##\gamma_{mn}##.

Had I reached an explicit form, I would have obtained the cutoff frequency using the relation ##\omega _c=\frac{\gamma _c}{\sqrt{\mu \varepsilon}}## where this gamma_c is the smallest possible I believe.

So I'd like any comment, especially if I'm missing something or if I'm on the right track. Thanks a lot!