Generation of the Tollmien–Schlichting wave in a supersonic boundary layer by two sinusoidal acoustic waves

The problem is solved using parabolized equations of stability for three-dimensional perturbations of a compressible boundary layer on a flat plate. Nonlinearity is taken into account by quadratic terms that are most significant in estimates of the viscous critical layer of the stability theory. These terms are determined by the total field of two acoustic perturbations, and the equations become linear and inhomogeneous. The calculations are performed for one acoustic wave being stationary in the reference system fitted to the plate for Mach numbers $\mathrm{M}=2$ and $5$. Solutions are presented, which are identified very accurately with Tollmien–Schlichting waves at a rather large distance from the plate edge.