It is shown that the Orr-Sommerfeld equation, which governs the stability of any mean shear flow in an unbounded domain which approaches a constant velocity in the far field, has a continuous spectrum. This result applies to both the temporal and the spatial stability problem. Formulae for the location of this continuum in the complex wave-speed plane are given. The temporal continuum eigenfunctions are calculated for two sample problems: the Blasius boundary layer and the two-dimensional laminar jet. The nature of the eigenfunctions, which are very different from the Tollmien-Schlichting waves, is discussed. Three mechanisms are proposed by which these continuum modes could cause transition in a shear flow while bypassing the usual linear Tollmien-Schlichting stage.

The expansion of an arbitrary two-dimensional solution of the linearized stream-function equation in terms of the discrete and continuum eigenfunctions of the Orr-Sommerfeld equation is discussed for flows in the half-space, y ε [0, ∞). A recent result of Salwen is used to derive a biorthogonality relation between the solution of the linearized equation for the stream function and the solutions of the adjoint problem.For the case of temporal stability, the orthogonality relation obtained is equivalent to that of Schensted for bounded flows. This relationship is used to carry out the formal solution of the initial-value problem for temporal stability. It is found that the vorticity of the disturbance at t = 0 is the proper initial condition for the temporal stability problem. Finally, it is shown that the set consisting of the discrete eigen-modes and continuum eigenfunctions is complete.For the spatial stability problem, it is shown that the continuous spectrum of the Orr-Sommerfeld equation contains four branches. The biorthogonality relation is used to derive the formal solution to the boundary-value problem of spatial stability. It is shown that the boundary-value problem of spatial stability requires the stream function and its first three partial derivatives with respect to x to be specified at x = 0 for all t. To be applicable to practical problems, this solution will require modification to eliminate disturbances originating at x = ∞ and travelling upstream to x = 0.For the special case of a constant base flow, the method is used to calculate the evolution in time of a particular initial disturbance.

Correction of an error in the matrix elements used by Salwen & Grosch (1972) has brought the results of the matrix-eigenvalue calculation of the linear stability of Hagen–Poiseuille flow into complete agreement with the numerical integration results of Lessen, Sadler & Liu (1968) for azimuthal index n = 1. The n = 0 results were unaffected by the error and the effect of the error for n > 1 is smaller than for n = 1. The new calculations confirm the conclusion that the flow is stable to infinitesimal disturbances.Further calculations have led to the discovery of a degeneracy at Reynolds number R = 61·452 ± 0·003 and wavenumber α = 0·9874 ± 0·0001, where the second and third eigenmodes have equal complex wave speeds. The variation of wave speed for these two modes has been studied in the vicinity of the degeneracy and shows similarities to the behaviour near the degeneracies found by Cotton and Salwen (see Cotton 1977) for rotating Hagen-Poiseuille flow. Finally, new results are given for n = 10 and 30; the n = 1 results are extended to R = 106; and new results are presented for the variation of the wave speed with αR at high Reynolds number. The high-R results confirm both Burridge & Drazin's (1969) slow-mode approximation and more recent fast-mode results of Burridge.

The linear stability of plane Poiseuille flow has been studied both for the steady flow and also for the case of a pressure gradient that is periodic in time. The disturbance streamfunction is expanded in a complete set of functions that satisfy the boundary conditions. The expansion is truncated after N terms, yielding a set of N linear first-order differential equations for the time dependence of the expansion coefficients.For the steady flow, calculations have been carried out for both symmetric and antisymmetric disturbances over a wide range of Reynolds numbers and disturbance wave-numbers. The neutral stability curve, curves of constant amplification and decay rate, and the eigenfunctions for a number of cases have been calculated. The eigenvalue spectrum has also been examined in some detail. The first N eigenvalues are obtained from the numerical calculations, and an asymptotic formula for the higher eigenvalues has been derived. For those values of the wave-number and Reynolds number for which calculations were carried out by L. H. Thomas, there is excellent agreement in both the eigenvalues and the eigenfunctions with the results of Thomas.For the time-dependent flow, it was found, for small amplitudes of oscillation, that the modulation tended to stabilize the flow. If the flow was not completely stabilized then the growth rate of the disturbance was decreased. For a particular wave-number and Reynolds number there is an optimum amplitude and frequency of oscillation for which the degree of stabilization is a maximum. For a fixed amplitude and frequency of oscillation the wave-number of the disturbance and the Reynolds number has been varied and a neutral stability curve has been calculated. The neutral stability curve for the modulated flow shows a higher critical Reynolds number and a narrower band of unstable wave-numbers than that of the steady flow. The physical mechanism responsible for this stabiIization appears to be an interference between the shear wave generated by the modulation and the disturbance.For large amplitudes, the modulation destabilizes the flow. Growth rates of the modulated flow as much as an order of magnitude greater than that of the steady unmodulated flow have been found.