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.

Shock-like features of phase-change flows in porous media are explained, based on the generalized Darcy model. The flow field consists of two-phase zones of parabolic/hyperbolic type as well as adjacent or imbedded single-phase zones of either parabolic (superheated, compressible vapour) or elliptic (subcooled, incompressible liquid) type. Within the two-phase zones or at the two-phase/single-phase interfaces, there may be steep gradients in saturation and temperature approaching shock-like behaviour when the dissipative effects of capillarity and heat-conduction are negligible. Illustrative of these shocked, multizone flow-structures are the transient condensing flows in porous media, for which a self-similar, shock-preserving (Rankine–Hugoniot) analysis is presented.

Using a simple three-layer model of the ocean, we study a generation mechanism for the lowest internal-wave mode by nonlinear coupling to modulations of the surface-wave spectrum. We first examine the case of a narrow-band surface-wave spectrum, applying a method developed by Alber (1978) to derive a transport equation for the spectral density. Alber demonstrated that, when the spectral width (in the main wave direction) exceeds some critical value, the spectrum is stable against modulational perturbation (i.e. the Benjamin–Feir-type instability is suppressed). We show, however, that, for a stratified ocean, a modulational instability may persist because of a coupling between a ‘modulational mode’ of the surface-wave spectrum and an internal wave. The growth rate is calculated for a simple model of the angular distribution of the spectrum. It turns out that an important parameter is 〈(∇ξ)2〉/Δθ, the ratio between the averaged square of the wave steepness, and the angular width of the spectrum.

For appreciable growth one must have roughly \[ 2\times 10^{-3}\lesssim kd \langle (\nabla\zeta)^2\rangle / \Delta\theta, \] where k is a characteristic wavenumber for the surface-wave spectrum, and d is the depth of the thermocline (50-100 m). This condition is probably too limiting for the above-mentioned modulational instability to be of any practical interest in the oceans.

We also consider the broad-band case of modulational interaction, and show the connection with incoherent three-wave interactions.

The parabolic approximation which has recently been found to be useful in other physical contexts, is extended to head-sea diffraction of short waves by a slender raft on deep water. In particular, it is a much more direct way of getting the inner approximation of the outer solution in a scheme of matched asymptotics than the original method of Faltinsen (1971). The present results are compared with a more involved integral equation method and are found to be remarkably accurate even when the raft length is comparable to the wavelength. Finally, the asymptotic method is modified for a compliant raft which absorbs wave power by suitably controlled impedance. Optimum efficiency and other performance characteristics are predicted.