Two of the simpler nonlinear wave systems on water of uniform depth are permanent waves and wave groups of permanent envelope. The interaction of each of these wave systems with swell of much smaller amplitude and greater wavelength, propagating in the same direction, is investigated analytically and numerically. A linear-stability analysis of the modulation of these systems by swell shows that they are unstable over short times. Calculations of their evolution over longer times confirms that the initial exponential growth of the modulations is not sustained, and that cyclic recurrence of the modulations occurs in some cases. The modulation of a wave train by swell is found to concentrate the energy of the wave train into single waves in turn, a process which may cause irreversible nonlinear changes such as wave breaking. In contrast, the only observable effect in the modulation of a wave group by swell is a small slow oscillation of the envelope of the group as it propagates. The conclusion is that wave trains on the ocean, generated for example by a wind system of long fetch and duration, disintegrate under the modulation of swell. Wave groups, however, either wind-generated or resulting from the breakdown of wave trains, propagate almost unchanged by the presence of swell.

A description is given of the principle and a preliminary embodiment of a flow-visualization technique in which the test body is pulled through water.

Initially, dyed and undyed parts of the water are separated by a plane horizontal interface; as the body emerges from below this interface, it carries dyed fluid upwards in its wake. A succession of photographs reveals the nature of the flow.

In principle, the flow is steady in a frame fixed relative to the body, and unsteady in one relative to the tank and the initial interface. Interpretation of the experimental results must therefore proceed by way of an analysis in which the steady-state equations are solved for velocity and turbulence quantities but an unsteady-state equation is solved for the dye concentration.

Preliminary results are reported for the flows behind a wedge and behind a bar. They reveal that these particular flows are unsteady even in the body-fixed coordinate frame, being strongly oscillatory.

The motion of an initially quiescent, incompressible, stratified and/or rotating uid of semi-infinite extent due to surface forcing is considered. The stratification parameter N and the Coriolis parameter f are constant but arbitrary and all possible combinations are considered, including N = 0 (rotating homogeneous fluid), f = 0 (non-rotating stratified fluid) and the special case N = f. The forcing is suction or pumping at an upper rigid surface and the response consists of geostrophic flows and inertial-internal waves. The response to impulsive point forcings (Green's functions) is contrasted with the response to finite-sized circularly symmetric impulsive forcings. Early-time and large-time behaviour are studied in detail. At early times transient internal waves change the vortices that are created by pumping/suction at the surface. The asymptotically remaining vortices are determined, a simple expression for what fraction of the initial energy is converted into internal waves is derived, as well as wave energy fluxes and the dependence of the flux direction on the value of N/f. The internal wave field is to leading order in time a distinct pulse, and rules for the arrival time of the pulse, its amplitude, its motion along a ray of constant frequency and decay with time, are given for the far field. A simple formula for the total wave energy distribution as a function of frequency is derived for when all waves have propagated away from the forcing.

The instability due to resonant interactions of finite-amplitude water waves is examined in the long-wave limit. In contrast to the well-known case of a small-amplitude limit in which the resonance is considered for a flat surface, here we consider a periodic approximate of the finite-amplitude solitary wave which is the long-wave limit of the periodic wave. The resonance conditions for the corresponding perturbations yield a new family of resonance curves that are totally different from those of the small-amplitude limit obtained by Phillips and Mclean. Under these resonance conditions, we conduct a systematic asymptotic analysis for small wavenumbers to obtain the growth rates of the perturbations explicitly and clarify whether each resonance curve is associated with instability. These results are verified numerically by showing that the instability bands for finite-amplitude periodic waves in shallow water are located along these unstable resonance curves.

The effect of a parallel magnetic field upon the stability of the plane interface between two conducting viscous fluids in uniform relative motion is considered. A parameter reduction, which has not previously been noted, is employed to facilitate the solution of the problem. Neutral stability curves for unrestricted ranges of the governing parameters are found, and the approximate solutions of other authors are examined in this light.

When a semi-infinite flat plate is immersed parallel to an unbounded, plane, steady, asymmetric, constant shear flow of an incompressible viscous fluid, an interaction occurs between the surface-generated vorticity and the external vorticity. A physical assumption was made in a previous paper (Mark 1962) concerning this problem that the pressure field in a thin layer adjacent to the top-side of the plate may be accurately approximated by the undisturbed constant-pressure field—that which exists before the insertion of the plate into the flow. This means that the vorticity interaction is assumed to have no effect on the undisturbed pressure field. That this is a valid first approximation far downstream along the plate where the interaction is intense is given rigorous support in this paper.

The flow below the plate is examined on a heuristic basis. It is found that there is a strong possibility for the flow to separate from the lower surface near the leading edge of the plate. However, far downstream the flow settles down to a Stokes-type flow near the plate.

Observations, numerical simulations and theoretical scaling arguments suggest that in mid-latitudes, away from the source regions and the equator, the meridional transport of abyssal water masses along a continental slope corresponds to planetary geostrophic flows that are gravity- or density-driven and topographically steered. We investigate these dynamics using a nonlinear reduced-gravity model that can describe grounded abyssal meridional flow over sloping topography that crosses the planetary vorticity gradient. Exact nonlinear steady and time-dependent solutions are obtained. The general steady theory is illustrated for a non-parallel equatorward flow that possesses a single along-slope grounding along the upslope flank of the current (complementing previous work). Four specific nonlinear time-dependent solutions are described. Two initial-value problems are solved exactly. The first initial configuration corresponds to an equatorward abyssal flow that has no cross-slope shear in the along-slope velocity and possesses a single grounding along the upslope flank of the current. The nonlinear time-dependent evolution of this initial current into a non-parallel shear flow is described. The second initial condition corresponds to an isolated radially symmetric grounded abyssal pool or dome. The nonlinear time-dependent evolution of this abyssal dome, which propagates equatorward with unsteady along- and cross-slope velocities while deforming into an elliptically shaped abyssal dome with $\unicode[STIX]{x1D6FD}$-induced diminishing height, is described. Finally, the nonlinear time-dependent boundary-value problem can be solved exactly in which the in-flow boundary condition on the poleward boundary of the mid-latitude domain corresponds to a time-dependent abyssal current with both an upslope and downslope grounding. Two specific time-dependent boundary conditions are examined. The first corresponds to a time-limited surge in the equatorward volume transport in the abyssal current along the poleward boundary. The second configuration corresponds to the nonlinear evolution of a finite-amplitude downslope plume or loop that forms in the abyssal current that is reminiscent of those seen in baroclinic instability simulations.

The paper presents a new analytical and experimental study of transonic flow around spheres. The results of the analytical study, which employs the method of orthogonal collocation for simultaneous solution of the momentum equations, the equation of continuity and the energy equation, are compared with hitherto unpublished measurements obtained on spheres of various sizes (1·02, 2·54 and 3·81 cm in diameter) in air, in dry steam and in wet steam with free-stream Mach numbers in the transonic range (0·58 < M∞ < 0·97). The relationship θ h = 91·78 + 8·59 M∞ between the attached-shock angle and the free-stream Mach number was obtained by fitting the theoretical pressure distributions to the experimental ones.

This paper presents analytical and numerical solutions for steady flow past long obstacles on a β-plane. In the oceanographically-relevant limit of small Rossby and Ekman numbers nonlinear advection remains important but viscosity appears only through the influence of Ekman pumping. A reduced boundary-layer-type equation is derived giving the long-obstacle limit of an equation described in Page & Johnson (1990). Analytical solutions are presented or described in various asymptotic limits of this equation and compared with previous results for this or related flows. A novel technique for the numerical solution of the boundary-layer equation, based on a downstream–upstream iteration procedure, is described. Some modifications of the asymptotic layer structure described in Page & Johnson (1991) and Johnson & Page (1993) for the weakly nonlinear low-friction regime are outlined for the case of a lenticular obstacle.

The flow field in the nose region of a blunt body in hypersonic flow is studied by considering the transport of vorticity and enthalpy. The entire region between body and shock is considered to be viscous, not necessarily thin in comparison with the nose radius of the body and to be of slowly varying density. The (given) post-shock vorticity need not be small and the density ratio ρ∞/ρs may either be small or near unity, the analysis being valid asymptotically at both limits.

It is found that the vorticity equation may be uncoupled from the total enthalpy equation if μ√ρ is constant. While the equations are not expected to be necessarily restricted to the immediate vicinity of the stagnation line, only there can the solution be written down explicitly; elsewhere, numerical integration is required.