Brown dwarfs are massive analogs of extrasolar giant planets and may host types of atmospheric circulation not seen in the solar system. We analyzed a long-term Spitzer Space Telescope infrared monitoring campaign of brown dwarfs to constrain cloud cover variations over a total of 192 rotations. The infrared brightness evolution is dominated by beat patterns caused by planetary-scale wave pairs and by a small number of bright spots. The beating waves have similar amplitudes but slightly different apparent periods because of differing velocities or directions. The power spectrum of intermediate-temperature brown dwarfs resembles that of Neptune, indicating the presence of zonal temperature and wind speed variations. Our findings explain three previously puzzling behaviors seen in brown dwarf brightness variations.