Trapped electromagnetic modes in a waveguide with a small discontinuity

We demonstrate that a small discontinuity (such as an enlargement or a hole) on a smooth waveguide can result in the appearance of trapped modes localized in the vicinity of the discontinuity. The frequencies of these modes lie slightly below the cutoff frequencies of the corresponding propagating modes in the waveguide. We find the distribution of the electromagnetic field in the modes and calculate their damping rate due to a finite conductivity of the walls. The contribution of the trapped modes to the longitudinal impedance is calculated.