Leaky waves in boundary layer flow

Linear stability analysis of boundary layer flow is traditionally performed by solving the Orr-Sommerfeld equation (OSE), either in a temporal or a spatial framework. The mode structure of the OSE is in both cases composed of a finite number of discrete modes which decay at infinity in the wall- normal direction $y$, and a continuous spectrum of propagating modes behaving as $\exp(\pm \mathrm{i} k y)$ when $y\rightarrow\infty$, with real $k$. A peculiarity of this structure is that the number of discrete modes changes with the Reynolds number, $Re$. They indeed seem to disappear behind the continuous spectrum at certain $Re$. This phenomenon is here investigated by studying the response of the Blasius boundary layer forced instantaneously in space and time. Since the solution of the forced and homogeneous Laplace-transformed problem both depend on the free-stream boundary conditions, it is shown here that a suitable change of variables can remove the branch cut in the Laplace plane. As a result, integration of the inverse Laplace transform along the two sides of the branch cut, which gives rise to the continuous spectrum, can be replaced by a sum of residues corresponding to an additional set of discrete eigenvalues. These new modes grow at infinity in the $y$ direction, and are analogous to the {\em leaky waves} found in the theory of optical waveguides, i.e. optical fibers, which are attenuated in the direction of the waveguide but grow unbounded in the direction perpendicular to it.