For Couette flow, it is possible to make mathematical progress in the solution of the Orr–Sommerfeld equation.  In this section, a demonstration of this method is given for the case of free-surface flow, that is, when the upper lid of the channel is replaced by a free surface.  Note first of all that it is necessary to modify upper boundary conditions to take account of the free surface.  In non-dimensional form, these conditions now read
 at ,
 at .
The first free-surface condition is the statement of continuity of tangential stress, while the second condition relates the normal stress to the surface tension.  Here
are the Froude and Weber numbers respectively.
For Couette flow , the four linearly independent solutions to the non-dimensional Orr–Sommerfeld equation are,
where  is the Airy function of the first kind.  Substitution of the superposition solution  into the four boundary conditions gives four equations in the four unknown constants .  For the equations to have a non-trivial solution, the determinant condition
must be satisfied.  This is a single equation in the unknown c, which can be solved numerically or by asymptotic methods.  It can be shown that for a range of wavenumbers  and for sufficiently large Reynolds numbers, the growth rate  is positive.