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We investigate Gravity Waves (GWs) in the lower atmosphere of Mars based on pressure timeseries acquired by the InSight lander. We compile a climatology showing that most GW activity detected at the InSight landing site takes place after the sunrise and sunset, they are almost absent during the aphelion season, and more prominent around the equinoxes, with variations during dust events and interannual variations. We find GWs with coherent phases in different sols, and a previously unnoticed coincidence of GW activity with those moments in which the diurnal cycle (of tidal origin) exhibits the fastest increases in absolute pressure. We explore the possibility that some of these GWs might actually be high-order harmonics of thermal tides transiently interfering constructively to produce relevant meteorological patterns, and discuss other interpretations based on wind patterns. The so-called Terminator Waves observed on Earth might also explain some of our observations.