Analytic study of the continuous spectrum of three-dimensional weak shock layers
Three-dimensional linear stability of a weak, compressible shock layer is analyzed using exact Reynolds-number-independent forms of the governing base flow and linear stability equations, obtained by rescaling the problem onto the shock's own natural viscous length and time scales. For any base flow with zero transverse velocity components, depending on the shock-normal coordinate alone, the resulting eigenvalue problem is proved covariant under rotation of the transverse wavenumber vector: the eigenvalue depends on the transverse wavenumbers only through their magnitude, and every three-dimensional eigenmode, throughout the continuous spectrum, decouples exactly into two independent constituents. A two-dimensional, in-plane acoustic--entropy branch carries the disturbance's entire dilatation, pressure and thermodynamic coupling, and a one-dimensional, out-of-plane purely solenoidal vortical branch is governed by a single scalar shear-diffusion equation. Three-dimensional shock-layer stability is thus resolved exactly into its compressible and vortical constituents, confirmed to machine precision by the Grosch--Salwen far-field companion spectra. Both decoupled operators carry an explicit damping term growing with the total transverse wavenumber, so a three-dimensional, oblique disturbance is never less stable than the two-dimensional disturbance sharing its in-plane wavenumber alone; the vortical branch, whose spectrum is controlled directly by this term, is shown to retreat from the imaginary axis in close proportion to the wavenumber squared. This finding is an exact, Reynolds-free analogue of Squire's classical theorem, with the stabilizing role of Reynolds number played here by the transverse wavenumber magnitude.