The propagation of electromagnetic waves in the plasma sheath is investigated theoretically and numerically using the variational method, which handles the inhomogeneity of the plasma sheath. We derive the variation of the action integral, which can be numerically represented by expanding the velocity and electromagnetic fields to the piecewise polynomial function space. We analyze the transmissivity of electromagnetic waves propagating through the plasma sheath with the barrier and the barrier-parabolic electron number density profile. We give the frequency-dependent transmissivity of electromagnetic wave propagation in different inhomogeneous plasma sheaths. The electromagnetic energy density derived from the Lagrangian density is also given, and the result shows that the collision between electrons and neutral particles can be used to enhance the transmission of electromagnetic waves in the plasma sheath.