A simple wave is a flow in a region adjacent to a region of constant state. In the language of Riemann invariant, the simple wave can also be defined as the zone where one of the Riemann invariant is constant in the region of interest, and consequently, a simple wave zone is covered by arcs of characteristics that are straight lines.
Simple waves occur quite often in nature. There is a theorem (see Courant and Friedrichs) that states that a non-constant state of flow adjacent to a constant value is always a simple wave. All expansion fans including Prandtl–Meyer expansion fan are simple waves. Compressive waves until shock wave forms are also simple waves. Weak shocks (including sound waves) are also simple waves up to second-order approximation in the shock strength.
Simple waves are also defined by the behavior that all the characteristics under hodograph transformation collapses into a single curve. This means that the Jacobian involved in the hodographic transformation is zero.