with the requirement that

Here is called the horizontal wavenumber. In general, a disturbance excites components for each real value of . Now equations (12), (16) and (17) becomes

subject to boundary conditions

which can be written as using (31)

This gives three conditions at each of the end point for the sixth order equation (32) to determine the countable infinity of eigen values and associated eigen functions . Let . For a given values of and , a complete set of solutions satisfying the boundary conditions (BCs) is needed to represnt an arbitrary initial disturbance. These are called modes of the solution. For given , the flow is unstable if for any mode with any real value of and stable if for all modes. Hence the critical value of denoted by is such that for some whenever and for all whenever .