On the linear stability of a compressible inviscid parallel shear flow

Summary By applying an approach similar to that used in the Miles-Howard theory [1], [2] we derive simple constraints on the phase speedc_r^* of the neutral three-dimensional (3-D) monochromatic disturbances in an inviscid compressible parallel two-dimensional (2-D) shear flow. It is shown that for a boundary layer flow [a₀^*(y^*)]² —[U₀^*(y^*)—c_r^*]² must have a zero in [y₁^*,y₂^*) for the neutral 2-D modes whose phase speedc_r^* does not belong to the range ofU₀^*(y^*). For the unstable waves the argument of Chimonas [3] applies leading to the Howard semi-circle theorem. HereU₀^*(y^*) anda₀^*(y^*) are the dimensional base velocity and local sonic speed respectively. It is suggested that hypersonic flows possess vertically highly undulated unstable normal modes.