The dispersion relation of Tollmien-Schlichting waves

It is well-known that shear flows in a strip or in the half plane are unstable for the Navier-Stokes equations if the viscosity $\nu$ is small enough, provided the horizontal wave number $\alpha$ lies in a small interval, between the so called lower and upper marginal stability curves. The corresponding instabilities are called Tollmien-Schlichting waves. In this letter, we give a simple presentation of the dispersion relation of these waves and study its mathematical properties.