Leaky waves in spatial stability analysis

Linear stability analysis of flat plate boundary layers implies, for wave like perturba- tions, to solve the so called Orr-Sommerfeld equations which solution can be expressed in terms of a continuous, and discrete spectrum. As the number of discrete modes change with the Reynolds number, and further seem to disappear behind the continuous spectrum at certain Reynolds num- bers, it is of interest to investigate if an all-discrete representation of the solution is possible. This can be done solving the response of the flat plate boundary forced instantaneously in space. Since the solution of the forced and homogeneous Laplace transformed problem both depend on the free stream boundary conditions, it is shown here that an opportune change of variables can remove the branch cut in the complex eigen value plane. As a result integration of the inversed Laplace trans- form along the new path corresponding to the continuous spectrum, which is now given by a straight line, equals the summation of residues of additional discrete eigen values appearing to the left of it. It is further shown that these additional modes are computed accounting for solution which grow in the wall normal direction. A similar problem is found in the theory of optical waveguides, such as optical fibers, where so called leaky waves are attenuated in the direction of the wave-guide, while it grows unbounded in a direction perpendicular to it. The theory is here applied to the case of two-dimensional flat plate boundary layers, of incompressible flows, subject to a pressure gradient.