The linear field induced by the sudden starting of a wave on an infinite plane surface is found exactly. An acoustic wavefront, on which the pressure remains constant and finite, moves outwards from the surface at the sound speed c. Behind this front the pressure field consists of two distinct components. The first is recognizable as the field due to steady motion over a wavy wall, while the second is an acoustic transient which propagates through the fluid to accelerate it into its steady asymptotic state. We find that whenever the surface phase speed is subsonic, there is an equipartition of the energy between the steady evanescent field attached to the surface and the outward-travelling sound. If the surface has sonic phase speed then the pressure on the surface grows like t½, and becomes unbounded. For subsonic waves the final momentum of the fluid parallel to the surface is shown to be equal to the total energy radiated divided by the surface phase speed. In the asymptotic state the ratio of momentum in the far field to that in the near field is ½m2/(1 − ½m2), where m is the ratio of the surface phase speed to the sound speed. The transient field on the surface can be identified as consisting of two travelling sound waves. One travels in the same direction as the surface wave, and the other in the opposite direction. They have amplitudes which are proportional to (1 − m)−l and (1 + m)−1 respectively, and decay in time as t−½. Their wavelength parallel to the surface is, of course, the same as that of the surface wave that induced them. We show that when the surface wave is very subsonic, the evanescent field is established very slowly, settling down only after the surface wave has travelled about m−1 wavelengths.

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.

The nonlinear development of the Batchelor–Nitsche instability (of a periodically stratified fluid) is considered, utilizing the disparity between vertical and horizontal scales of motion. The resulting evolution equation is used to show that the preferred pattern of convection takes the form of rolls, and that the motion evolves to larger and larger horizontal scales as time increases.

The results of Ursell (1964) are confirmed by proving that there are no additional contributions to Ursell's integral from singularities of the integrand at infinity. The method consists of proving that the asymptotic expansion of a force coefficient Δ(N) is uniformly valid in a finite sector of the complex N-plane. This in turn requires that the kernel of an integral equation remains small in this sector.

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.

An error in a recent paper on bubble and drop oscillations by Temkin (J. Fluid Mech. vol. 380 (1999), pp. 1–38) is pointed out and corrected. In this way, his results are shown essentially to reduce to earlier ones in the literature. A concise derivation of these earlier results is presented for the case of a gas bubble.

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.

We study vibrational instabilities of the thermocapillary return flow driven by a constant temperature gradient along the free surface of an infinite layer that vibrates in its normal direction with acceleration of amplitude $g_1 $ and frequency $\omega _1 $. The layer is unstable to hydrothermal waves in the absence of vibrations beyond a critical Marangoni number $M$. Modulated gravitational instabilities with $M\,{=}\,0$ are also possible beyond a critical Rayleigh number $R$ based on $g_1 $. We employ two-time-scale high-frequency asymptotics to derive the equations governing the mean field. The influence of vibrations on the hydrothermal waves is found to be characterized by a dimensionless parameter $G$ that is proportional to $R^2.$ The return flow at $G\,{=}\,0$ is also a mean field basic flow and we study its linear instability at different Prandtl numbers $P$. The hydrothermal waves are stabilized with increasing $G$ and reverse their direction of propagation at particular values of $G$ that decrease with increasing $P$. At finite frequencies, a time-periodic base state exists and we study its linear instability by calculating the Floquet exponents. The stability boundaries in the $(R,M)$-plane are found to be composed of two intersecting branches emanating from the points of pure thermocapillary or buoyant instabilities. Three-dimensional modes are always preferred and the region of stability, while anchored at the point of hydrothermal waves corresponding to $R\,{=}\,0$, is found to grow without bound along the $R$-axis with increasing frequencies. Results from the two approaches are shown to be in asymptotic agreement at large frequencies.

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 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.

We calculate the drag coefficient of a circular liquid domain in a flat fluid membrane surrounded by three-dimensional fluids on both sides. The coefficient of a rigid disk is well known, while that of a circular liquid domain is also well known when the membrane viscosity inside the domain equals the one outside the domain. As the ratio of the former viscosity to the latter increases to infinity, the drag coefficient of a liquid domain should approach that of the disk of the same size in the same ambient viscosities. This approach has not yet been shown explicitly, however. When the ratio is not unity, the continuity of the stress makes the velocity gradient discontinuous across the domain perimeter in the membrane. On the other hand, the velocity gradient is continuous in the ambient fluids, whose velocity field should agree with that of the membrane as the spatial point approaches the membrane. This means that we need to assume dipole singularity along the domain perimeter in solving the governing equations unless the ratio is unity. In the present study, we take this singularity into account and obtain the drag coefficient of a liquid domain as a power series with respect to a dimensionless parameter, which equals zero when the ratio is unity and approaches unity when the ratio tends to infinity. As the parameter increases to unity, the sum of the series is numerically shown to approach the drag coefficient of the disk.

The flow around a tethered cylinder subjected to an incoming flow transverse to its main axis and confined in a channel is investigated by means of global stability analysis of the full coupled body–fluid problem. When the cylinder is strongly confined (ratio of cylinder diameter to cell height, $D/H=0.66$) we retrieve the confinement-induced instability (CIV) discovered by Semin et al. (J. Fluid Mech., vol. 690, 2011, pp. 345–365), which sets in at a Reynolds number below the vortex-induced vibration threshold. For a moderately confined case ($D/H=0.3$), a new steady static instability is discovered, referred to as confinement-induced divergence (CID). This instability saturates into an asymmetric steady solution through a supercritical pitchfork bifurcation. In addition, the CIV and CID instabilities are studied via a reduced model obtained by considering a quasi-static response of the fluid, allowing for tracing back the physical mechanisms responsible for the instabilities.

This note provides numerical values for the long-wave limit of the virtual-mass coefficient relating to the heaving motion of a half-immersed circular cylinder on water of finite depth, found analytically by Ursell in the preceding paper; some preliminary analysis is needed, however.

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.