We only address linear media in introductory physics classes, and really only focus on infinite or semi-infinite sine-wave-like disturbances. In the real world waves have a finite duration; this type of wave is often called a wave packet. If the wave is large enough, it can be affected by the nonlinear properties of the medium. The most dramatic example of this is the soliton, a disturbance that propagates with no loss in speed, size or shape over large distances.
Though first observed more than 150 years ago, solitons weren’t understood until the last few decades and are still actively investigated in mathematics, physics and engineering. This site gives an overview of current research as well as a historical account of the first description of a soliton on a canal in England.