Modal fields inside a rectangular waveguide can be readily obtained by solving the standard lossless wave equations for **E** and **H**(∇^{2}**E**+k^{2}**E**=0, similarly for **H**), coupled with Faraday's law(∇X**E**=-jω**H**). We first note that rectangular waveguides can support either transverse-electric(**TE**) or transverse-magnetic(**TM**)modes(or hybrid modes) but not TEM modes. Here we consider a specific mode, namely TE_{10}.

As explained above, we solve the wave equation along with Faraday's law, and apply the boundary conditions - electric field must be zero at each face of the waveguide(since we assume perfect conductors). Coupling this with the **separation of variables** method, we can find all field compenents.

Here we focus on finding the components of the magnetic field (**H**), as they are responsible for introducing currents on the surfaces of the waveguide. For TE_{mn} mode, we have, in time domain:

**Note: **Here we are using the convention that wave propagation is in x-direction, and equations are written accordingly.

where a,b are the dimensions of the waveguide,H_{0} is the wave amplitude(assume unity) and h^{2}=(mπ/a)^{2}+(nπ/b)^{2}

We find the surface currents at each face using the relation **J _{s}**=

We plot these surface currents for **m=1** and **n=0**, on the y=0 and z=1 faces. The y=1 and z=0 faces' surface currents are simply a mirror image of their counterparts.

We observe that for the y=0 face, since n=0, the surface currents are always vertical, and vary in magnitude sinusoidally along the direction of propagation(x-direction).

For the z=1 face, we observe 2-D standing wave patterns in x and y that vary with time, as expected