While the high-pitched whine from jet engines during ground maneuvers is mainly due to the movement of their mechanical parts (large masses rotating at very high speeds), the deep roar you hear during takeoff—when the engines are unleashing most of their power—is simply... air!That’s right: massive volumes of air being pushed at very high speeds behind the engines create the same effect as an incredibly powerful wind, resulting in that typical roar. It is precisely this air being hurled backward that pushes the plane forward (which is why they are called "jet" or "reaction" engines). This forward force is known as thrust.
Today, many design features are used to try and reduce the noise produced by engines. One of the most recent is the use of "chevrons"—a zig-zag shape on the back of the engine casing. By smoothing the mixing of the exhaust air with the outside air, they manage to significantly decrease the noise levels.
DontWorryFlyHappy!