An inviscid fluid is bounded above by a horizontal plane and below by finite-amplitude ridges. The fluid is rotating and stratified. A uniform transport is forced across the ridges at small dynamical Rossby number, although the boundary conditions are such that motion cannot remain geostrophic. The most significant parameter is found to be a thermal Rossby number based upon the vertical density difference and both vertical and horizontal length scales, but independent of the transport. Conditions determining whether or not effects of the bottom topography will penetrate vertically throughout the fluid are discussed. Some numbers characteristic of flow in the deep ocean are presented.

A steady slightly non-uniform flow with a free surface is subject to a concentrated surface pressure which gives rise to a pattern of surface waves. (For gravity waves on deep water this is the well-known Kelvin ship-wave pattern.) The motion is assumed inviscid, and the waves are assumed small. A theory is developed for the wave pattern, based on the following assumptions:

1. The stream velocity component normal to a wave crest is equal to the phase velocity based on the local wavelength;

2. the separation between consecutive crests is equal to the local wave-length. These assumptions are expressed in mathematical form, and the existence of a set of characteristic curves (associated with the group velocity) is deduced from them. These characteristics are not identical with the crests. Let the additional assumption be made that

3. the characteristics all pass through the point disturbance; the characteristics are then completely defined and may be constructed by a step-by-step process starting at the point disturbance. The same construction gives the direction of the wave crests at all points. The wave crests can then be deduced.

Assumptions of the same type as (1) and (2) have long been familiar in various applications of ray tracing. For uniform flows the present theory gives the same pattern as the method of stationary phase.

The kinematic approach to group velocity given in Lighthill & Whitham (1955) for one-dimensional waves is extended to cover the general three-dimensional case. The ideas have particular bearing on the theory developed by Ursell (1960) for treating steady wave patterns on non-uniform steady fluid flows.

Although this note was written in ignorance of the fact, all the main ideas presented here are implicit in §§66 and 67 of the book by Landau & Lifshitz (1959). However, these ideas do not seem to be well known to fluid dynamicists, and it was suggested to the author by the editor that a useful purpose would still be served by publishing this note as an expository article amplifying the paragraphs in Landau & Lifshitz. It also serves the original purpose of providing a supplement to Ursell's paper.

This paper considers the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small. Two types of problem arise: (i) to follow the growth of an unstable, infinitesimal disturbance (supercritical problem), possibly to a state of stable equilibrium; (ii) for values of the wave-number and Reynolds number for which no unstable infinitesimal disturbance exists, to follow the decay of a finite disturbance from a possible state of unstable equilibrium down to zero amplitude (subcritical problem). In case (ii) the existence of a state of unstable equilibrium implies the existence of unstable disturbances. Numerical calculations, which are not yet completed, are required to determine which of the two possible behaviours arises in plane Poiseuille flow, in a given range of wave-number and Reynolds number.

It is suggested that the method of this paper (and of the generalization described by Part 2 by J. Watson) is valid for a wide range of Reynolds numbers and wave-numbers inside and outside the curve of neutral stability.

In Part 1 by Stuart (1960), a study was made of the growth of an unstable infinitesimal disturbance, or the decay of a finite disturbance through a stable infinitesimal disturbance to zero, in plane Poiseuille flow, and that paper gave the most important terms in a solution of the equations of motion. The greater part of the present paper is concerned with a re-formulation of this problem which readily yields the complete solution. By the same method a solution for Couette flow is obtained. This solution is only a formal one for the present because the conditions imposed in deriving the solution may not be valid for Couette flow; this flow is believed to be stable to infinitesimal disturbances of the type considered.

The electrical conductivity of shock-ionized argon has been measured at temperatures between 6000 °K and 13,000 °K and electron number densities in the range 1016 to 1018 cm−3. The results are compared with theoretically computed values and show sufficient agreement to be plotted on a linear scale. The experimental technique is based on the measurement of voltages generated in a search coil by the flux from currents induced in a plasma moving rapidly through a magnetic field.

An experimental investigation of the explosions of 2 in. diameter glass spheres under high internal pressure has been made. The spheres were initially filled with air or helium at 400 and 326 p.s.i., respectively, and were exploded in air at atmospheric pressure. Experiments on the simulation of high-altitude explosions are also described. Schlieren and spark shadowgraph records of explosion phenomena, and pressure records of the reflexion of the spherical shock wave at various radii, are presented.

An account of some initial experiments on the implosion of 5 in. diameter glass spheres is given. The results were not very satisfactory because of the failure of the spheres to shatter in a desirable manner while under an external pressure of 65 p.s.i.

Numerical solutions to the air and helium sphere explosions are described and the experimental wave phenomena are shown to be in good quantitative agreement with the theoretical predictions, in that they exhibit all the main features that were predicted and are modified only by the physical limitations of the glass diaphragm. A formation process is associated with the spherical shock waves in practice, resulting in initial shock velocities which are lower than the theoretical values.

The expansion method introduced by Friedrichs (1948) for the systematic development of shallow-water theory for water waves of large wavelength was used by Keller (1948) to obtain the first approximation for the finite-amplitude solitary wave of Boussinesq (1872) and Rayleigh (1876), as well as for periodic waves of permanent type, corresponding to the cnoidal waves of Korteweg & de Vries (1895).

The present investigation extends Friedrich's method so as to include terms up to the fourth order from shallow-water theory for a flat horizontal bottom, and thereby obtains the complete second approximations to both cnoidal and solitary waves. These second approximations show that, unlike the first approximation, the vertical motions cannot be considered as negligible, and that the pressure variation is no longer hydrostatic.

An approximate solution is presented to the problem of the propagation of a temperature wave through a stratified medium which both diffuses and radiates heat. The solution is a combination of two waves whose relative amplitudes vary with the distance from the lower boundary. Apparent diffusivities computed from phase lag and attenuation coefficients can differ greatly from each other and can vary with height even if the actual diffusivity does not.

An example, using parameters simulating the earth's atmosphere, suggests that analysis of the propagation of the diurnal wave upward from the earth's surface is likely to be inadequate if radiative effects are not considered.

An exact solution is presented of the equations and boundary conditions governing the steady longitudinal motion of a semi-infinite non-conducting plate in an oblique magnetic field. The discussion covers the distortion of the boundary layer, the structure of the induced electric and magnetic fields, the current-density distribution, and the behaviour of the fields near the edge of the plate.

Most treatments of magnetohydrodynamics waves have confined physical interpretation to cases when the Alfvén velocity a1 is small compared with the sound velocity a0. Here we consider the ‘low-beta situation’, in which a1, the only possible waves are longitudinal ones, propagated unidirectionally along lines of magnetic force with velocity a0. These can be interpreted as sound waves, confined to effectively rigid magnetic tubes of force. Hall-current effects do not alter these conclusions (in contrast to the high-beta situation), and finite conductivity introduces only small dissipation.

An application is made to the flow pattern around a body moving through the F2 layer of the ionosphere, where, although neutral particles have a very large mean free path, charged particles interact electrostatically and, it is argued, may be regarded as forming a continuous fluid whose movement is independent of that of the neutral particles. A body moving at satellite speed or below would then excite the above-mentioned unidirectional sound waves, but no waves at much faster Alfvén velocity. These considerations suggest that its movement would be accompanied by a [xvee ]-shaped pattern of electron density (figure 2), which might be in part responsible for some anomalous radar echoes that have been reported.

Low Reynolds number flow of an incompressible fluid past an axially symmetric body in the presence of a uniform magnetic field is studied using a perturbation method. It is found that for small Hartmann number M an approximate drag formula is given by $ D^ \prime = D^\prime_0 \left(1 + \frac {D^\prime_0} {16\pi \rho vaU}M\right) + O(M^2),$ where D′0 is the Stokes drag for flow with no magnetic effect.