A unified treatment of wave motion in a stratified fluid, with or without density discontinuities, is achieved by reducing the governing differential system to a Sturm-Liouville system. With the aid of Sturm's comparison theorem, it is found (without detailed calculations) that, for any stratification, the phase velocity increases as the wave-number decreases and that, for the same wave-number, the phase velocity increases as the density gradient is increased everywhere and decreases as the density is increased everywhere by a constant amount. Sturm's oscillation theorem provides upper and lower bounds for the phase velocity for a given stratification, a given wave-number, and a given number of zeros of the eigenfunction (or a given number of stationary surfaces in the fluid). The inequalities giving these bounds are used to explain the well-known tendency for surfaces of density discontinuities to behave as rigid boundaries when the stratification in each layer is slight. The rigid-boundary behaviour of interfaces in such cases enables one to obtain the approximate eigenvalue spectrum by superimposing the spectra of the individual layers (with the interfaces treated as rigid) on the spectrum of the interfacial (or free surface) waves, obtained by ignoring the slight continuous stratification in each layer. It is pointed out that the Ritz method can be used for calculating the eigenvalues even when the density is discontinuous, and examples are given to show the accuracy of the Ritz method. The nature of the spectrum when the depth is infinite is also clarified.

In the course of the development of the theory, the effects of compressibility and of three-dimensionality are determined and given explicitly, the rate of growth of unstable stratifications is related to the phase velocity of waves in stable ones, and equipartition of energy is proved. Motion due to a wave-maker is discussed in order to bring out the connexion between the type of the governing partial differential equation and the nature (local or not local) of the disturbances. The effect of surface tension and the stability of a stratified fluid under vertical oscillation are also discussed.

The utility of piston theory as a means of solving insentropic two-dimensional aerodynamic problems has been aptly demonstrated by many writers in recent years. The present treatment removes the restriction of isentropic flows, extending the applicability of piston theory to flows with strong shocks. Sample calculations for a thin biconvex airfoil are carried out in which the local flow is assumed to be isentropic and non-isentropic. Comparison of the result is made with that of the shock expansion theory of Cole, Gazley & Williams (1956).

This paper concerns an investigation into the behaviour of a layer of dense salt solution on the floor of a sloping rectangular pipe in which there is a turbulent flow. The various phenomena which are observed are described qualitatively and by the presentation of typical concentration profiles.

Numerical values are obtained experimentally for the rates of spread of the edge of the layer in the case where the salt is moving entirely in the direction of the main stream. The rate of spread is found to depend mainly on the slope α and on the pipe Richardson number, defined by Rip = DΔd cos α/V2, where D is the depth of the pipe, Δd = g(ρd−ρa)/ρa, ρd is the density of the fully mixed discharge and ρa is the density of the ambient flow. In the range of Riρ from 0 to 0·005 the rate of spread of the layer decreases by a factor of about 3 at small slopes.

Some discussion is given of the factors determining the initial rate of spread just after the layer leaves the slit. Finally, it is shown how the depth measurements can be related to the determination of the concentration at the floor.

This paper continues an investigation into the mixing of a dense layer of salt solution in a turbulent pipe flow in order to obtain a more detailed understanding of the underlying physical processes. The effect of the density difference on the velocity profile in a sloping pipe is calculated using a simplified model, and the results compared with direct measurements obtained by timing streaks of dye at various levels in the pipe. These velocity profiles are also used in conjunction with density profiles to estimate the dependence of the transfer coefficients for salt and momentum KS and KM, on stability.

It is found that KS is much more greatly affected by the density gradient than KM, and that the ratio KS/KM may be represented, to the accuracy of the experiments, as a function of the local Richardson Number Ri. The results agree with what is known of KS/KM in neutral and very stable conditions, and they confirm an earlier prediction by Ellison that the critical flux Richardson number, at which KS becomes zero, is much less than unity.

Finally, a crude semi-empirical method is outlined which indicates how the new measurements of the transfer coefficients may be related to the overall properties of the flow discussed in the first part of the paper.

Dungey (1956) has shown that the number densities of electrons and positive ions, under the action of turbulence and the magnetic field in the ionosphere, are closely equal—in other words the only fluctuations in electron density are those controlled electrostatically by the ions—and he has estimated the magnitude of the fluctuations. The present paper uses Dungey's model and results, reduces his equations to a single equation for electron density, in each of several cases, and then investigates the possible spectra of fluctuations. It is concluded that, in the circumstances that commonly arise, the spectrum on this model should be nearly isotropic, though exceptionally there could be a strong elongation of irregularities at right angles to the magnetic field. Thus some other mechanism is required to account for the elongation that is observed, parallel to the field.

Below 110 km, where the magnetic effect on the ions’ motion is small, and at wave-numbers in the inertial subrange of the turbulence, dimensional argument shows that the spectrum function (integrated over all directions) is proportional partly to κ−1 and partly to $\kappa^{-\frac{5}{3}$. Above 120 km the magnetic effect is large; a more detailed study shows that when turbulence is present, which probably is not often, the spectrum function in the inertial subrange is proportional to $\kappa {\frac{2}{3}$, with considerable anisotropy.

Short gravity waves, when superposed on much longer waves of the same type, have a tendency to become both shorter and steeper at the crests of the longer waves, and correspondingly longer and lower in the troughs. In the present paper, by taking into account the non-linear interactions between the two wave trains, the changes in wavelength and amplitude of the shorter wave train are rigorously calculated. The results differ in some essentials from previous estimates by Unna. The variation in energy of the short waves is shown to correspond to work done by the longer waves against the radiation stress of the short waves, which has previously been overlooked. The concept of the radiation stress is likely to be valuable in other problems.

An investigation has been made of the flow over axisymmetric spiked bodies at a Mach number of 6·8. For some ranges of the ratio of spike length to body diameter the flow was found to be unsteady. The effect of the shape of the body nose on this unsteadiness was investigated and an explanation of the mechanism of the oscillation is given.

The stability problem for a thin film of liquid having a linear mean-velocity profile and bounded by a fixed wall and free surface is solved asymptotically for large values of the Reynolds number R. The analysis is similar to that for plane Couette flow, but instability occurs for sufficiently large values of R in accordance with Heisenberg's criterion that neutral disturbances having finite wave numbers and phase velocities for R = ∞ are necessarily unstable as R → ∞. It is found that a sufficient condition for stability is W < 3, where W is the Weber number based on the mean speed at the free surface and the depth of the film. The minimum critical Reynolds number, also based on free surface speed and film depth, is found to be R = 203. This last figure is in order-of-magnitude agreement with observation, but there remains considerable uncertainty as to whether the observed instability corresponds to that considered here. Neutral stability curves are presented in an R vs α (= wave-number) plane with W as the family parameter. Brief consideration also is given to the time-rate-of-growth of unstable disturbances and to the lighter fluid that, in actual configurations, is responsible for the shear in the film. An appendix gives extended and more accurate results for the function $\cal F$(z), introduced and calculated previously by Tietjens (1925) and Lin (1955).

The turbulent natural convection above fires in a dry calm atmosphere with a constant lapse rate has been the subject of several recent investigation (see references). The present paper presents solution curves from which the natural convection may be computed over a fire of arbitrary size in an atmosphere with arbitrary lapse-rate variation. The independent parameters of fire size, energy release rate (buoyancy), momentum release rate and atmospheric lapse rate are given over the expected range of values. The arbitrary variation of lapse rate is thus calculable as piecewise constant.

This paper is concerned with the propagation of a simple wave along a duct of initially constant and then slowly varying cross-section. The equations of motion are linearized in the deviations from the simple wave solution. The solution for the propagation of a centred simple rarefaction wave is obtained in closed form and compared with the results of step-by-step calculations. The results are also found to be good when the area change is not small. Some remarks on boundary-layer influence are included.