Rumbling is a deep, vibrating infra sound, very low, often impossible to hear for the human ear, which elephants produce in the near of their throat.
It is often used as a wide distance communication, since it carries kilometers, and it is less cut by bushes and similar. When an elephant rumbles, the forehead is vibrating, and when riding on a captive elephant, or touching the elephant skin, one feels a strong vibration through the whole body. It is often used between elephants when they are frightened, after the trumpet signal has been given. This sound has been known by the asian mahouts and others, for a long time, but was (re)discovered by western academics in the late 1970´s.
It is said that elephants talk to one another, mumbling with their mouths the speech of men.
But to not all is the speech of the beasts audible, but only the men who tame them hear it."
Oppian, Cynegetica, II, 529-555.