In media with characteristics possessing finite discontinuities, the field problem can be treated by solving a single wave equation, the effect of discontinuities being contained in the coefficients in the equation. This provides an alternative to the conventional approach, in which field quantities are matched at the discontinuities. Two examples with discontinuities in the transverse and longitudinal directions are investigated: one is a periodic structure loaded with dielectric discs, each having a central hole, and the second is a cavity with a solid dielectric disc partially filling the central region. For both examples, a general solution is described and a 1st-mode approximation given in detail.