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Surface wave characteristics and radiation efficiency are calculated for a patch antenna with a given current distribution and located on top of an inhomogeneous, isotropic substrate. The substrate is infinite in the lateral directions and inhomogeneous with respect to the height coordinate. The surface waves in the substrate are analysed using the propagator technique and residue calculus. The propagators and the residues are found by solving two sets of systems of ordinary differential equations (ODEs). Several dispersion curves of the surface waves for inhomogeneous substrates are calculated. The radiation efficiency is compared with the corresponding quantities of a printed antenna on a homogeneous substrate.