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The propagation of small- and finite-amplitude surface acoustic waves (SAWs) in stressed thin-film systems is modeled. Results are presented for an initially monofrequency, plane wave traveling in the  direction of systems composed of either Ge (loading) or diamond (stiffening) epitaxial films under compressive stress on an unstressed  Si substrate. Cases are considered for both thinner and thicker films in terms of their ratios of dispersion to nonlinearity ratios. For finite-amplitude waves, comparison between unstressed and stressed films indicates that larger effects occur at longer propagation distances and for higher harmonics.