How to Solve TE Waveguide with Different Materials?

[dalarev]

Homework Statement

I'm attempting to find the General solution and boundary conditions for a TE$$_{m0}$$ waveguide split up into two different materials: a (any) dielectric on the bottom, and a metallic conductor on top. Image attached.

Homework Equations

General (homogenous) Helmholt's equation (not sure how to use latex),
E$$_{z}$$(x,y) = 0, because it's TE.
E$$_{x}$$(x,b) = 0, tangential to top surface.

The Attempt at a Solution

The problem here is the curved line, which I have called as
l = $$\stackrel{b}{a}$$*x. I'm thinking I can cancel out the partial derivatives with respect to y if I make the length, l, a function of x only.

Also, I can easily plug in n=0, but I have to derive the particular solutions before.

 

[LightHero]

I'm thinking that the particular solution will be linear, since there is no source. Any suggestions? Thanks.

 

[Mene]

I would first commend you on your efforts in attempting to find a solution for this problem. Your approach of using the general Helmholt's equation and considering the boundary conditions for TE waves is a good starting point. However, there are a few points that I would like to address and provide some guidance on.

Firstly, the use of LaTeX is a helpful tool in writing mathematical equations, but it is not necessary for understanding your solution. It is more important to clearly explain your thought process and steps in solving the problem.

Secondly, when dealing with a waveguide that is split into two different materials, it is important to consider the material properties of each section. This includes the dielectric constant and conductivity of the materials, as they will affect the propagation of the wave.

Thirdly, the curved line that you have labeled as "l" is not clear in its purpose. I would suggest providing a more detailed explanation of what this line represents and how it relates to the problem.

Furthermore, I would recommend considering the use of separation of variables to solve the Helmholt's equation for this problem. This involves assuming a solution of the form E(x,y) = X(x)Y(y) and then substituting it into the equation to obtain two separate ordinary differential equations.

Lastly, it is important to verify your solution by checking if it satisfies the boundary conditions and if it is physically meaningful. This includes checking for continuity of the electric field and ensuring that it satisfies the wave equation.

In conclusion, I would suggest revisiting your approach and incorporating the suggestions mentioned above. Keep in mind the material properties and the use of separation of variables to find a solution for this problem. Good luck!