When studying the propagating modes in SIW, one should go to the classical rectangular waveguide theory. Every mode propagating in a waveguide has some surface currents associated to it. A SIW can be seen as a special rectangular waveguide filled with dielectric substrate and with some slots along its side walls.
In the figure, we can observe that the vertical slots do not interrupt the surface currents of the TE10 mode, so it can propagate inside the structure. The other TEn0 modes have similar surface current distribution, so they can also propagate without radiating significantly.
Therefore, in general, just the modes of the SIW corresponding to the TEm0 modes of the equivalent ideal rectangular waveguide are considered. This concept of the equivalent waveguide will be explained below.
Assuming that there is no variation of the field in the direction normal to the substrate, the circuit can be modeled as a bidimensional problem. So, if the circuit is placed in the horizontal plane, there are just vertical electric fields and horizontal magnetic fields in the problem. This kind of problem would be an inductive problem. The inductive problems are those whose structure geometry is invariant in height. The term inductive [Mar86] is applied to this kind of problems because in the circuital model of this kind of obstacles or discontinuities, an inductance appears; whereas, when the discontinuity or obstacle is invariant in width, a capacitance appears in the equivalent circuit, and so we talk about capacitive problems.