Can someone prove that the $\tilde{E}$ and $\tilde{H}$ fields in a waveguide looks as pictured? Hi,

I'm trying to use the solution to the wave's equation in a rectangular waveguide for $\tilde{E}$ and $\tilde{H}$ to show how I can get the above picture. For example, why is the magnetic field looping around? Why is the electric field diverging towards the walls?

Can someone show me how I can get the above picture based on the following set of equations? (taken from here) Thank you!

• "Can someone prove [...]" As usual you use the proposed structure of the fields to make a observable prediction (say what the cut-off frequency for the mode would be) and then subject it to experimental test under conditions where different modes give different observables. – dmckee Nov 28 '14 at 16:21

For example, why is the magnetic field looping around?

Yours is nothing more (nor less) than the intuitive statement of $\nabla\cdot B=0$. Flux lines of a divergenceless field cannot "begin" or "end"; they must loop if $\nabla\cdot B=0$ holds at all points. Otherwise, a nonzero divergence betokens charge density, as with the electric field lines which are described by $\nabla\cdot E=\rho/\epsilon_0$.

Why is the electric field diverging towards the walls?

The walls are assumed to be perfect conductors. That means that, if there were a tangential component of the electric field at the walls, charge would instantly shift, thus giving rise to a cancellation field. Equilibrium is reached when the electric field lines pierce the walls at right angles. You assume equilibrium because you must assume a good enough conductivity that the charge shifting is a great deal faster than the wave's frequency. The field lines end on the wall, in keeping with $\nabla\cdot E=\rho/\epsilon_0$: there is a time varying charge density in the walls.

You may even want to try graphing some of those field lines: for the magnetic field for example, the last two equations tell you that $\frac{H_y}{H_x} = \frac{m\,b}{n\,a}\, \tan\left(\frac{m\,\pi\,x}{a}\right)\,\cot\left(\frac{n\,\pi\,y}{b}\right)$. $\frac{H_y}{H_x}$ defines the direction of the tangent to the field line, so you have a differential equation:

$$\frac{\mathrm{d}\,y}{\mathrm{d}\,x}=\frac{m\,b}{n\,a}\, \tan\left(\frac{m\,\pi\,x}{a}\right)\,\cot\left(\frac{n\,\pi\,y}{b}\right)$$

which is readily integrable and which you can plot in something like Mathematica.