text: A Mach wave propagates across the flow at the Mach angle μ, which is the angle formed between the Mach wave wavefront and a vector that points opposite to the vector of motion. It is given by
  where M is the Mach number.
  Mach waves can be used in schlieren or shadowgraph observations to determine the local Mach number of the flow. Early observations by Ernst Mach used grooves in the wall of a duct to produce Mach waves in a duct, which were then photographed by the schlieren method, to obtain data about the flow in nozzles and ducts. Mach angles may also occasionally be visualized out of their condensation in air, for example vapor cones around aircraft during transonic flight.