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Observations of oscillations of temperature and wind in planetary atmospheres provide a means of generalizing models for atmospheric dynamics in a diverse set of planets in the Solar System and elsewhere. An equatorial oscillation similar to one in the Earth's atmosphere has been discovered in Jupiter. Here we report the existence of similar oscillations in Saturn's atmosphere, from an analysis of over two decades of spatially resolved observations of its 7.8-microm methane and 12.2-microm ethane stratospheric emissions, where we compare zonal-mean stratospheric brightness temperatures at planetographic latitudes of 3.6 degrees and 15.5 degrees in both the northern and the southern hemispheres. These results support the interpretation of vertical and meridional variability of temperatures in Saturn's stratosphere as a manifestation of a wave phenomenon similar to that on the Earth and in Jupiter. The period of this oscillation is 14.8 +/- 1.2 terrestrial years, roughly half of Saturn's year, suggesting the influence of seasonal forcing, as is the case with the Earth's semi-annual oscillation.