It is shown that weak harmonic waves can propagate upstream in a supersonic stream along a plane wall over distances that are very large in comparison with the boundary-layer thickness. These waves are characterized by small reduced frequencies, large wavelengths and small phase velocities. Results are presented for the flow along a single plane wall and also for the flow between two plane parallel plates (channel flow).

The progressive-wave solutions are found by asymptotic expansions for small disturbances, large Reynolds numbers of the basic flow, and small reduced frequencies of the unsteady disturbances. It turns out that, as in the corresponding steady theory, four flow regions have to be distinguished: a middle layer which embraces most of the boundary layer; an inner layer near the wall; the outer flow field; and a transition layer between the middle layer and the outer layer. A quasi-steady treatment of the middle, transitional and outer layers is appropriate. Unsteady effects originate in the inner layer.

The relative importance of viscosity and unsteady effects with regard to the waves is characterized by a dimensionless parameter N which is the product of certain powers of the reduced frequency and the Reynolds number. For N → ∞ and a single wall, Lighthill's steady theory of upstream influence in supersonic boundary layers is recovered as a limiting case of the present theory.