The elephant's ears are large and thin, with lots of blood vessels. Flapping its ears is a way for the elephant to cool itself.
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In captivity it is likely a sign of deep frustration and boredom. They are intelligent creatures that need a very large habitat to be comfortable.
Elephants wave their trunks in the air and side-to-side to improve their sense of smell. They also use their trunks as "snorkels" while swimming.