Unihemispheric sleep is advantageous to mother dolphins and their calves as well. Dolphin calves are especially vulnerable to predators such as sharks, and also need to be near their mothers to nurse, so it would be dangerous for dolphin mothers and calves to fall into a full deep sleep as humans do.
Dolphins sleep by resting one half of their brain at a time. This is called unihemispheric sleep. The brain waves of captive dolphins that are sleeping show that one side of the dolphin's brain "awake" while the other in a deep sleep ("slow-wave sleep"). Also, during this time, one eye is open (the eye opposite the sleeping half of the brain) while the other is closed.
A 2005 study on captive bottlenose dolphin and orca mothers and calves showed that, at least at the surface, during the first month of the calf's life, both mom and calf appeared awake 24 hours a day. Even during this time, both eyes of the mom and calf were open, indicating that they weren't even sleeping 'dolphin-style'. Gradually, as the calf grew, sleep would increase in the pair. This study was questioned later, as it involved pairs that were only observed at the surface. A 2007 study, though, showed a "complete disappearance of rest at the surface" for a minimum of 2 months after the calf was born, although occasionally the mother or calf were observed with an eye closed.