  The flow of a liquid film over an inclined plane is frequently found in
nature and industry, and, under some conditions, instabilities in the free
surface may appear. These instabilities are initially two-dimensional surface
waves, known as Kapitza waves. Surface waves are important to many industrial
applications. For example, liquid films with surface waves are employed to
remove heat from solid surfaces. The initial phase of the instability is
governed by the Orr-Sommerfeld equation and the appropriate boundary
conditions; therefore, the fast and accurate solution of this equation is
useful for industry. This paper presents a spectral method to solve the
Orr-Sommerfeld equation with free surface boundary conditions. Our numerical
approach is based on a Galerkin method with Chebyshev polynomials of the first
kind, making it possible to express the Orr-Sommerfeld equation and their
boundary conditions as a generalized eigenvalue problem. The main advantages of
the present spectral method when compared to others such as, for instance,
spectral collocation, are its stability and its readiness in including the
boundary conditions in the discretized equations. We compare our numerical
results with analytical solutions based on Perturbation Methods, which are
valid only for long wave instabilities, and show that the results agree in the
region of validity of the long-wave hypothesis. Far from this region, our
results are still valid. In addition, we compare our results with published
experimental results and the agreement is very good. The method is stable,
fast, and capable to solve initial instabilities in free surface flows.
