A deep surface-piercing wedge-ended hull model was towed through still water. Measurements of the surface wave pattern confirmed earlier findings for ship models, that the measured bow-wave cusp line often lies well forward of the position predicted by thin-ship theory, and that this shift increases with bow water-line angle and with decreasing model speed. Two possible explanations are considered here in terms of changes of wave phase speed with wave convection and steepness. Calculations based on a transformation method due to Guilloton predict more realistic wave profiles than linear theory, but account for less than half the observed shift. Some tentative conclusions are drawn.

The singularity in the Green's function double integral is removed by an improved method, which simplifies the numerical integration. The new integrand decays within one oscillation.

Solutions are obtained for the baroclinic instability problem for situations in which the static stability and mean shear vary geminately with height. The simple solution given by Eady is shown to be a special limiting case of a class of exact solutions for flows whose basic states have a vanishing interior potential vorticity gradient. The generalized solutions show that the temperature amplitude distribution is particularly sensitive to vertical variations in static stability but that phases and other amplitudes are only slightly influenced by such variations. When the static stability and shear increase (decrease) with height an enhanced temperature maximum occurs at the upper (lower) surface in comparison with the standard Eady solution.

The generalized solutions also help to explain the character of annulus waves and predict a short-wave cut-off that is the same as that given by Eady's theory provided that it is based on the vertically averaged gravitational frequency.

By the ‘Poincaré problem’ is meant the determination of the free oscillations of a contained rotating fluid, its velocity being linearized around a state of solid rotation. Compressibility requires one to introduce a basic thermodynamic profile as well as a basic velocity distribution. Here the temperature gradient has been supposed proportional to the adiabatic gradient, by introduction of a proportionality constant α (α = 0 in the isothermal case; α = 1 in the adiabatic case). In this formulation the system is reducible to a single second-order ordinary differential equation and its boundary condition.

It is proved that if α = 1 the oscillation frequencies in the rotating system cannot equal plus or minus twice the rotation frequency. The negative case is pathological in the sense that there are solutions arbitrarily near the forbidden solution, and a solution curve of frequency as a function of rotation rate crosses the forbidden frequency.

The basic system is expanded in terms of a power series in γ − 1, where γ is the ratio of specific heats. The zeroth-order set of equations is solved in terms of confluent hypergeometric functions, and a solvability condition on the first-order set gives frequency shifts as functions of α. Several zeroth-order frequencies have been calculated, together with four first-order frequency shifts.

The effects of both viscous and pressure stress work are considered in laminar natural convection on flat surfaces. Whereas in previous investigations only viscous stress work effects were studied, it is established here that pressure work effects are generally rather more important both for gases and liquids. Variations in the properties of the fluid outside the natural convection boundary layer are shown to occur inevitably in this problem and are found to lead to effects comparable with those produced by pressure work. Property variations in the boundary layer itself are also found to be of importance. Such variations, both inside and outside the boundary layer, have been ignored in previous studies. Examples of flows are discussed in which the fluids are either perfect gases or fluids undergoing small changes in temperature and pressure. For both of these fluids, the effects of conditions of either constant temperature or constant entropy in the fluid outside the boundary layer are examined.

In this paper we consider the flow field induced by a periodic magnetic field in and about a conducting liquid drop immersed in an incompressible insulating fluid. It is assumed that at infinity the magnetic field varies with time t as cos ωt, where ω is a constant. This magnetic field is associated with a periodic electric field which produces a net electric stress, dependent on the spatial variables, normal to the drop surface. This stress sets up a flow field, in and about the liquid drop, that creates an appropriate viscous stress so that there is stress balance at the drop surface. The flow field is periodic with angular frequency 2ω. For small drop deformations the drop shape at any instant is a spheroid. It is shown that for large ω the amplitude of the velocity field for a conducting drop is approximately independent of ω and for a non-conducting drop it is proportional to ω. The larger velocity amplitude for a non-conducting drop is probably due to the fact that in this case there is no dissipation of electromagnetic energy. The electric stress over the drop surface increases with ω and it is suggested that the drop will burst at large ω unless the amplitude of the applied magnetic field is suitably decreased.

Internal Alfvén-gravity waves of small amplitude propagating in a Boussinesq, inviscid, adiabatic, finitely conducting fluid in the presence of a uniform transverse magnetic field in which the mean horizontal velocity U(z) depends on height z only are considered. We find that the governing wave equation is singular only at the Doppler-shifted frequency Ωd = 0 and not at the magnetic singularities Ωd = ± ΩA, where ΩA is the Alfvén frequency. Hence the effect of ohmic dissipation is to prevent the resulting wave equation from having magnetic singularities. Asymptotic solutions of the wave equation, which is a fourth-order differential equation, are obtained. They show the presence of the magnetic Stokes points Ωd = ± ΩA. The interpretation of upward and downward propagation of waves is also discussed.

To study the combined effect of electrical conductivity and the magnetic field on waves at the critical level, we have used the group-velocity approach and found that the waves are transmitted across the magnetic Stokes points but are completely absorbed at the hydrodynamic critical level Ωd = 0. The general expression for the momentum flux is mathematically complicated but will be simplified under the assumption \[ \frac{\partial^2h}{\partial x^2}+\frac{\partial^2h}{\partial y^2}\gg \frac{\partial^2h}{\partial z^2}, \] where h is the perturbation magnetic field. In this approximation we find that the momentum flux is not conserved and the waves are completely absorbed at Ωd = 0.

The general theory is applied to a particular problem of flow over a sinusoidal corrugation and asymptotic solutions are obtained by applying the Laplace transformation and using the method of steepest descent.

We consider axisymmetric steady centrifugally driven thermal convection in a compressible fluid in a rapidly rotating circular cylinder. The Boussinesq approximation is not used, because it is not valid for the case of practical interest. We clarify the importance of the effect of the flow-induced volume change of a fluid particle, and propose a widely applicable method of solution.

A fluid layer that lines the inner surface of a circular tube has motion induced by axial surface-tension gradients. The lubrication equations for the system are analysed and it is found that even for thin layers the motions differ markedly from those in planar layers. The planar case serves as a class of outer solutions. These approximate solutions are modified by a boundary-layer correction where the mean surface tension is important.

The stability of the oscillatory Stokes layers is examined using two quasi-static linear theories and an integration of the full time-dependent linearized disturbance equations. The full theory predicts absolute stability within the investigated range and perhaps for all the Reynolds numbers. A given wavenumber disturbance of a Stokes layer is found to be more stable than that of the motionless state (zero Reynolds number). The quasi-static theories predict strong inflexional instabilities. The failure of the quasi-static theories is discussed.

Edge waves travelling along a straight coastline are examined in the case when the water depth approaches a constant value at large distances from the coast. Only the fundamental mode is examined in the limit as the ratio of the water depth at infinity to the edge-wave wavelength tends to zero. Two comparison theorems are used to obtain upper and lower bounds for the dispersion relation. A long-wave approximation procedure is used to obtain the leading terms in the dispersion relation for a wide class of bottom topographies. The results obtained are compared with an exact result for the case when the bottom topography is a rectangular step.

The equations of conservation of momentum, energy and mass together with the equations of state are examined for free convection from a vertical paraboloid. A transformation due to Saville & Churchill is applied to the first- and second-order boundary-layer equations, which are then solved using series about the stagnation point, using asymptotic series far up the body and in between by a method due to Merk. The second-order outer inviscid flow is given in terms of infinite integrals as a solution of Laplace's equation in paraboloidal co-ordinates.

Eight second-order effects are distinguished, depending on longitudinal and transverse curvatures, the displacement flow, heat flux into the boundary layer and the variation of density, viscosity, thermometric conductivity and the coefficient of expansion with temperature. Expressions for the skin friction, heat-transfer coefficient and various flux thicknesses are obtained and a comparison of the second-order effects is made.

This paper examines the features of the flow field off the surface of an oscillating flat-plate airfoil immersed in a two-dimensional supersonic flow Although the exact linearized solution for a supersonic unsteady airfoil has been known for a long time, its expression in the form of an integral is not convenient for a physical interpretation. In the present paper, the quintessential features of the flow field are extracted from the exact solution by obtaining an asymptotic expansion in descending powers of a frequency parameter through the repeated use of the stationary-phase and steepest descent methods. It is found that the flow field consists of two dominant and competing signals: one is the acoustic ray or that component arising from Lighthill's ‘convecting slab’ and the other is the leading-edge disturbance propagating as a convecting wavelet. The flow field is found to be divided into several identifiable regions defined by the relative magnitude of the signals, and the asymptotic expansions appropriate for each flow region are derived along with their parametric restrictions. Such intimate knowledge of the flow field in unsteady, supersonic flow is of interest for interference aerodynamics and related acoustic problems.