Boundary-value problem for two-dimensional fluctuations in boundary layers

The streamwise evolution of disturbances in a boundary layer is described as an asymptotic solution of the forced Orr-Sommerfeld equation. The velocity fluctuations and their derivations are specified along the y-axis. With these boundary conditions, the effects are included of vortical and irrotational free stream disturbances, fluctuations originating from leading edges, and discrete eigenmodes. A Fourier transform in time and a Laplace transform in the streamwise direction are used. Complementary and particular integrals are found and the inverse transforms are taken. Five families of 2-D fluctuations can exist in a parallel-flow, incompressible boundary layer. Three families have exponentially growing fluctuations, one of which is the Tollmien stability wave. Another is an exponentially-growth standing wave that oscillates in time and does not travel. This fluctuation appears as a mathematical pole in transform space, like the stability waves, but does not vanish far away from the boundary layer. A third growing fluctuation appears in Laplace space as a branch line. This continuous spectrum diffuses and travels upstream. The last two of these three growing fluctuations are excluded in our quarter-plane problem that extends forever downstream. Besides the Tollmien wave that can grow or decay in the streamwise direction, the other discrete modes appearing as mathematical poles are damped. Two other decaying fluctuations appear.