When a concentrated pressure travels with constant velocity over the free surface of water, it carries with it a familiar pattern of ship waves. Let viscosity and surface tension be neglected, let the free-surface condition be linearized, let the depth of water be assumed infinite, and let initial transient effects be ignored. Then, as is well known, the wave motion everywhere can be found by standard methods in the form of a double integral. The wave pattern at a great distance behind the disturbance can be found by an application of the ordinary method of stationary phase, which shows that the wave amplitude is considerable inside an angle bounded by the two horizontal rays θ = ± θc from the disturbance, where $\theta_c = \rm {sin}^{-1} {\frac{1}{3} \eDot 19{\frac{1}{2}\deg$. But the method fails in two regions, near the track θ = 0 of the pressure point, and near the critical lines θ = ±θc.

These two regions are treated in the present paper. It is shown that near θ = 0 the linearized surface elevation oscillates with indefinitely increasing amplitude and indefinitely decreasing wavelength. (This result holds only when the pressure is concentrated at a point and applied at the free surface.) Near the critical lines the surface elevation at a greater distance behind the pressure point can be expressed in terms of Airy functions, and this expression goes over into the known wave pattern inside the critical angle. It is shown that near the critical lines the crest length increases as the cube root of the distance, and that the separation between crests remains constant. Contour maps of the wave surface are given for three distances behind the moving pressure point.

The steady motion of an infinitely long solid cylinder parallel to its length in a conducting fluid in the presence of a uniform magnetic field is discussed. Due to Alfvén waves originating at the cylinder we find two opposite ‘wakes’ parallel to the applied magnetic field.

A formula which relates the total drag on the cylinder to the electric potential difference δΦ between the two undisturbed regions outside these two wakes is derived $D|\;|\delta \Phi| = 2\surd {\rho}v \sigma$ where ρν is the viscosity and σ is the conductivity of the fluid.

The reduction to a classical boundary-value problem is made for the case of an insulating cylinder.

Exact solutions are obtained for the case of a perfectly conducting or an insulating flat strip of semi-infinite width. These give a clear picture of the field, especially in the transition region near the edge of the strip.

The case of a strip of finite width is also discussed with special reference to the viscous and the magnetic drags, Df and Dm. We find that Df + ½Dm, on a perfectly conducting strip, is equal to the viscous drag on an insulating strip for which Dm is zero. Precise values of these drags are given.

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.

This paper is concerned with the spherically symmetric collapse of an empty cavity in water. The effects of viscosity and surface tension are neglected, but the compressibility of the water is allowed for and a suitable equation of state for the water assumed. The object of this is to clarify the effect of compressibility on the flow by considering it in isolation and thus to describe the formation of a pressure pulse by the collapse and its subsequent propagation.

The exact flow equations are integrated numerically and it is found that very high flow speeds develop in the neighbourhood of the collapse point. The radius of the cavity is found to be proportional to (-t)n for small t, where t = 0 is the instant of collapse, and n is some positive number less than unity.

The flow in the neighbourhood of the collapse point can be described by means of a similarity solution, and the value of n is determined by regularity properties of the similarity solution (the value of n depends on the equation of state assumed for the water). The similarity theory, which is valid only at very high pressures and velocities, can be continued beyond the instant of collapse to describe the formation of a shock wave after the collapse is completed, and the initial propagation of this shock.

The motion of bodies in a direction parallel to an applied magnetic field and through a perfectly conducting fluid is considered. It is shown that the perturbation in the state of the fluid cannot remain small except in the particular case when the velocity U of the body is much smaller than that of the Alfvén waves in the fluid. In this case, however, the perturbation is not confined to the neighbourhood of the body, and extends to infinity inside planes which touch the body and are parallel to the undisturbed magnetic field. In addition the body experiences a drag.

This paper describes a theory for a piston-operated compressor in which shock waves are used to heat the gas being compressed. Detailed calculations are given showing the use of such a compressor to heat air for use in a hypersonic wind tunnel. The effect of real gas properties on the compression process is included in this discussion.

The flow of a viscous incompressible electrically conducting fluid past a sphere is studied; the uniform ambient flow field is colinear with the ambient uniform magnetic field. The force exerted on the sphere is computed for various conductivities and Reynolds numbers; of particular interest is the distinction in behaviour between the flow with ambient particle speed greater than ambient Alfvén speed and that with particle speed less than Alfvén speed.

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 oscillating plate problem is investigated for the case of an incompressible, electrically conducting fluid in the presence of a magnetic field. The boundary conditions are examined in detail, and a solution is found with the aid of suitable approximations. The motion of the fluid is shown to consist mainly of magnetohydrodynamic waves, but there is also a viscous boundary layer in which the solution agrees with that given by other writers.

It is shown that the dispersion of a substance, with molecular diffusivity κ, in a stationary, homogeneous, turbulent velocity field can be formulated in terms of a ‘substance auto-correlation function’, this being a generalization of the well-known Lagrangian correlation between the velocity of a fluid particle at different times. It is found that the interaction between the molecular diffusion and the turbulent motion reduces the dispersion from the value it would have if the processes of molecular and turbulent diffusion were independent and additive. The conflict, between the results obtained in this paper and previous results which implied that the interaction increases the dispersion, is resolved. The ratio, of the contributions to the dispersion from the interaction term and the turbulent diffusion term, is obtained for comparatively large times by the use of intuitive arguments, and is found to be inversely proportional to the Prandtl number and a Reynolds number of the turbulence.

Gravity waves on the surface of an inviscid incompressible fluid of finite depth are considered. The waves are assumed to be periodic in time and in the horizontal direction. The surface profile, potential function, pressure and frequency of the motion are determined (to third order) as series in powers of the amplitude divided by the wavelength. It is found that the frequency increases with amplitude for depths less than a certain multiple of the wavelength and decreases with increasing amplitude for greater depths. Graphs of the surface profile and of the pressure as a function of depth are included.

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.

Exact one-dimensional solutions of the magnetohydrodynamic equations of an incompressible fluid are considered. It is shown that one class of plane wave solutions of the linearized equations is also a possible class of solutions of the general equations including the effect of displacement current. A similar result is also established for the solutions for a horizontally stratified fluid. For the particular case when the viscosity is equal to the magnetic diffusivity an exact solution is obtained for the magnetohydrodynamic Rayleigh problem for a semi-infinite plate. It is shown that this solution may be employed directly to give the solution for liquids of small, but not necessarily equal, viscosity and magnetic diffusivity.

The plane, steady, laminar vortex flow of a viscous, heat-conducting perfect gas is treated. Simple relations are obtained for the flow quantities in the irrotational vortex, for arbitrary Prandtl numbers. When the Prandtl number is ½, the irrotational vortex is also isentropic. When the temperature dependence of the viscosity coefficient is taken into account, the vortex flow is rotational. An exact solution for the rotational vortex is obtained which is suitable for numerical evaluation by successive approximations. Distributions of velocity, temperature, pressure, density, and stagnation temperature through the rotational vortex are given for a typical case.

By a consideration of the relationships holding along the characteristics in an unsteady motion involving plane, axially or spherically symmetrical flow of compressible inviscid fluid, it is shown that the existence of a region of compression anywhere in the flow must lead eventually to the breakdown of continuity. The paper generalizes and unites previous work on this topic, and discusses some recent numerical calculations in which the expected discontinuity was not found.

Starting with expressions for viscous stress, heat flux and diffusion flux, we formulate a continuum theory for steady flow of a binary mixture of chemically inert perfect gases through a normal shock of arbitrary strength. For shocks of vanishing strength, a solution by series expansion in Grad's (1952) shcok-strength parameter gives a result essentially the same as found previously by Dyakov (1954). For stronger shocks a straightforward numerical integration, quite analogous to that useful in the simpler pure-gas problem, is laid out.

The resulting problem has eight parameters: shock strength, ratio of specific heats, ration of bulk viscosity to shear viscosity, Prandtl number, Schmidt number, thermal diffusion factor, molecular mass ratio, and initial mixture concentration. A dozen examples were worked out on a simple desk calculator to exhibit the influence of some of these parameters. They involve the gas pairs argon40–argon36, argon–neon, argon–helium, and xenon–helium.

In discussing the results, special attention is paid to the degree of success with which the weak-shock theory may be extrapolated to arbitrary shock strength, and to the question of the accuracy of the Navier-Stokes approximation for a mixture of gases of very different molecular weights.

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.

An examination of the Newtonian inviscid theory of hypersonic flow past a sphere is made in the neighbourhood of the singular point which occurs at θ = 60°. A uniformly valid theory is developed in this neighbourhood which exhibits the limiting detached ‘free-layer’ behaviour postulated by Hayes and Lighthill in the limit as γ → 1. A complete solution is obtained which gives details of streamline and shock wave shape, pressure distribution, etc. for γ − 1 small.

Reasons are given why the empirical ‘modified’ Newtonian theory of Lees proves to be a good approximation to experimentally determined pressure distributions.

A novel check to the theory is provided by a simple power-law relation between the pressure on the sphere surface and the distance of the shock wave away from sphere at the same point.

In a previous paper (1953b) it was shown theoretically that just below the boundary layer at the surface of a free wave the mass-transport gradient should be exactly twice that given by Stokes's irrotational theory. The present paper describes careful experiments which confirm the higher value of the gradient.

The results have an implication for any oscillatory boundary layer at the free surface of a fluid; such a boundary layer must generate a second-order mean vorticity which diffuses inwards into the interior of the fluid.