Gravity waves occur on the surface of a liquid such as water, and the manner in which they propagate depends upon its depth. Although this dependence is described in principle by the equations of the ‘exact linear theory’ of surface waves, these equations have not been solved except in some special cases. Therefore, oceanographers have been unable to use the theory to describe surface wave propagation in water whose depth varies in a general way. Instead they have employed a simplified geometrical optics theory for this purpose (see, for example, Sverdrup & Munk (1944)). It has been used very successfully, and consequently various attempts, only partially successful, have been made to deduce it from the exact linear theory. It is the purpose of this article to present a derivation which appears to be satisfactory and which also yields corrections to the geometrical optics theory.

A mechanism is proposed by which cellular convective motion of the type observed by H. Bénard, which hitherto has been attributed to the action of buoyancy forces, can also be induced by surface tension forces. Thus when a thin layer of fluid is heated from below, the temperature gradient is such that small variations in the surface temperature lead to surface tractions which cause the fluid to flow and thereby tend to maintain the original temperature variations. A small disturbance analysis, analogous to that carried out by Rayleigh and others for unstable density gradients, leads to a dimensionless number B which expresses the ratio of surface tension to viscous forces, and which must attain a certain minimum critical value for instability to occur. The results obtained are then applied to the original cells described by Bénard, and to the case of drying paint films. It is concluded that surface tension forces are responsible for cellular motion in many such cases where the criteria given in terms of buoyancy forces would not allow of instability.

The initial value problem presented by mixing and chemical reaction in the wake of a flat plate is solved using the boundarylayer approximation. When a cool combustible mixture and its hot combustion products are separated by a finite, perfectly insulating flat plate, the velocity, temperature, and combustible concentration are determined in the vicinity of the trailing edge.

The mixing problem without chemical reaction is solved in terms of a ‘universal solution’ for a given initial temperature ratio and Prandtl number from which the solution for arbitrary temperature ratios can be obtained.

The mixing problem with chemical reaction is solved in terms of a ‘universal solution’ for the first two terms of an assumed series solution for the temperature. In this case the ‘universality’ is with respect to a parameter B characterizing the chemical and hydrodynamic initial conditions.

The axial distance from the trailing edge to the first local temperature maximum is given in terms of the initial conditions and is shown to be greatly shortened by the presence of the viscous wake as compared with non-viscous mixing.

A hydromagnetic shock tube has been used to obtain an approximate measurement of the time to reach equilibrium ionization behind shock waves of Mach number 11 to 17, moving into air at a pressure of about 1 mm of mercury. This ionization time decreases with increasing Mach number. Experimental results are presented as a graph of the ionization time vs Mach number. The principal source of error in the measurements is the attenuation of the shock and the results indicate a lower limit for the ionization time in air.

A shock wave is generated by a uniform compressive piston motion and passes into a channel of slowly varying cross-section. A relation in closed form is obtained between shock strength and the area of the channel and is used to discuss converging cylindrical and spherical shocks.

A numerical solution has been obtained for the starting flow of a viscous fluid past a circular cylinder at Reynolds numbers 40 and 100. The method used is the step-by-step forward integration in time of Helmholtz's vorticity equation. The advantage of working with the vorticity is that calculations can be confined to the region of non-zero vorticity near the cylinder.

The general features of the flow, including the formation of the eddies attached to the rear of the cylinder, have been determined, and the drag has been calculated. At R = 40 the drag on the cylinder decreases with time to a value very near that for the steady flow.

A theoretical treatment is presented for the slow flow of a viscous fluid through a cylindrical container within which a small spherical particle is confined. The sphere is situated in an arbitrary position within the cylinder and moves at constant velocity parallel to the walls. Approximate expressions are derived which give the frictional drag, rotational couple, and permanent pressure drop caused by the presence of this obstacle in the original Poiseuillian field of flow. The primary parameters involved are the ratio of sphere to cylinder radius and fractional distance of the particle from the longitudinal axis of the cylinder. With appropriate modifications, the results are also applicable to a sphere settling in a quiescent fluid. This yields the necessary boundary corrections to Stokes law arising in connection with devices such as the falling ball viscometer when the sphere is eccentrically located.

The effects of compressibility on a radial laminar wall jet are investigated. On the assumption that the coefficient of viscosity is proportional to the temperature, it is shown that a similarity solution for the velocity distribution exists, which is expressible directly in terms of the corresponding solution for an incompressible wall jet. For arbitrary Prandtl number the energy equation is studied in detail and solutions are obtained for a variety of temperature conditions.

A study is made of the propagation of sound in both a constant gradient shear flow and a turbulent shear flow above a flat surface. Curves are presented showing how, in the case of downstream propagation, the flow gradient tends to channel the sound energy into a narrow layer next to the wall. These results are used in estimating the effect of a flow on the attenuation of sound in a duct with absorbing side walls.

In this paper the propagation of long waves (tides) into a canal is studied. The canal is assumed to be rotating at constant angular velocity and the depth of the fluid is uniform.

The rate of rotation can have a considerable effect on the amplitude of the unattenuated modes in the channel. This is due partly to the modification of the known solution when the rotation is zero, and partly to the fact that even if there is only a single semi-infinite barrier unattenuated waves of a special type (Kelvin waves) may be propagated in the rotating system into what is normally called the ‘shadow’ region behind the barrier (Crease 1956).

The purpose of this theoretical investigation is to seek a partial explanation of the behaviour of tides and storm surges in the North Sea. For this reason two models of this region are discussed. In Example I the model is of two parallel semi-infinite barriers in the path of a plane progressive wave, and in Example II there are two parallel barriers, one semi-infinite and the other infinite.

The ratio of the observed height of the semi-diurnal M2 tide in the North Sea to the height of the incident tide from the Atlantic lies between the results predicted by the two models and is in closer agreement with the model of Example I.

This paper considers the stabilizing effect of density stratification on the horizontal shear layer between two parallel streams of uniform velocities. A simple continuous velocity distribution, $U = V\; \rm {tanh}(y|d)$, is used to represent the laminar shear layer. The density of the fluid is assumed to vary as exp (-βy), y being the vertical coordinate, with a small total change in density across the shear layer. The fluid is unbounded, and is assumed to be inviscid, incompressible and under the action of gravity.

By the methods of hydrodynamic stability theory, it is shown that a disturbance of small amplitude and wave-number α is neutrally stable if the Richardson number, defined as $J = g\beta d^2|V^2$, has the value $\alpha ^2 d^2 (1 - \alpha^2 d^2)$, and the form of the neutral disturbance is obtained. It follows that the critical Richardson number is $\frac{1}{4}$, so that the flow is stable if $J\;\; \textgreater \;\; \frac{1}{4}$. The relation between these results and Goldstein's derivation of the same critical Richardson number for a flow with discontinuous velocity and density gradients is discussed.

A large number of small spheres of the same size were injected successively into a horizontal pipe conveying water at constant mean velocity, and their times of transit were measured. The mean velocity of the spheres that were either somewhat heavier or lighter than water was less than that of those of neutral density; for those having a terminal velocity in water within ± 1% of the mean velocity of the water in the pipe, the discrepancy was only about 0.1%. The dispersion of the times of transit of the spheres was almost independent of their density. A theory is developed to show how the mean velocity of the spheres depends upon their relative density and size.

The theory of an ‘ideal dissociating’ gas developed by Lighthill (1957) for conditions of thermodynamic equilibrium is extended to non-equilibrium conditions by postulating a simple rate equation for the dissociation process (including the effects of recombination). This equation contains the ‘equilibrium’ parameter of the Lighthill theory plus a further ‘non-equilibrium’ parameter which determines the time scale of the dissociation phenomena.

The behaviour of this gas is investigated in flow through a strong normal shock wave and past a bluff body. The assumption is made that the gas receives complete excitation of its rotational and vibrational degrees of freedom in an infinitesimally thin region according to the familiar Rankine-Hugoniot shock wave relations before dissociation begins. The variation of the relevant thermodynamic variables downstream of this region is then computed in a few particular cases. The method used in the latter case is an extension of the ‘Newtonian’ theory of hypersonic inviscid flow. In particular, the case of a sphere is treated in some detail. The variation of the shock shape and the ‘stand-off’ distance with the coefficient Λ, which is the ratio of the sphere diameter to the length scale of the dissociation process, is exhibited for conditions extending from completely undissociated flow to dissociated flow in thermal equilibrium. Results would indicate that significant and observable changes from the undissociated values occur, although values for the non-equilibrium parameter are not, at present, available.

A simple mathematical model is proposed to describe the steady melting of a body of ice which presents a plane surface transverse to a stream of hot air; the temperature of the air is such that vaporization does not occur.

The analysis takes into account the convection of heat away from the surface by the water released in melting and the results show that the rate of transfer of heat to the body and thus the rate of melting, is reduced by as much as 46% by this convection.

Simple approximate expressions are obtained for the rate of melting, the thickness of the water layer, and the thickness of the thermal boundary layer in the ice, in terms of a basic parameter S which can be calculated in terms of known quantities. These results are compared with those obtained by a separate Pohlhausen calculation and are found to be in good agreement.

It is also shown that there exists a thermal boundary layer, in the body, of thickness much greater than that of the boundary layer in the air, in which the temperature changes rapidly from its value at the melting surface to its value in the far interior.

A formally exact solution for the thermal boundary layer on a non-isothermal surface subjected to non-uniform free stream velocity is presented in the form of a series. It is demonstrated that the solution can be recast in terms of universal functions, which are independent of the wall temperature data of particular problems, and which depend only on a single parameter characterizing the variation of the free stream velocity.

In this paper transonic small-disturbance theory is applied to a simplified model of the flow near the front of a ducted body. The body is assumed to consist simply of two parallel flat plates which extend from the inlet station to infinity downstream. The velocity far upstream is sonic, and the velocity far downstream in the duct, which is assumed to be known, is slightly subsonic. Air is therefore ‘spilled’ around the intake edges. An analytic solution is found for the resulting flow field up to the ‘limiting Mach wave’, and asymptotic solutions are found for the supersonic flow and for the shock wave far from, and near, the intake edges. The pressure distribution along the outside walls is then known at both ends, and its computation is completed by an empirical procedure. Distributions of pressure along the centre-line and along the inside and outside walls are shown. These results may be used to compute the drag of sharp-edged intakes with a very small frontal area.

A direct current electric arc has been developed as a heating device for argon gas. Negligible amounts of electrode material are consumed during an operating time of several minutes. Under normal operating conditions 50% to 90% of the input electric power is transferred directly to the gas. The remaining power is absorbed by the water-cooled electrodes. Measurements were made to determine the total gas enthalpy and the proportion of the enthalpy in directed kinetic energy, random particle motion, and ionization energy. From these measurements it is postulated that the gas is initially in a non-equilibrium state on leaving the arc, but approaches equilibrium relatively quickly when confined to a constant diameter jet outside the arc. The gas temperature range in these experiments varies from 5000°K to 15000°K.

Consideration of the structure of wind-generated waves when the duration and fetch of the wind are large suggests that the smaller-scale components of the wave field may be in a condition of statistical equilibrium determined by the requirements for attachment of the crests of the waves. A dimensional analysis, based upon the idea of an equilibrium range in the wave spectrum, shows that for large values of the frequency ω, the spectrum Φ(ω) is of the form $\Phi (\omega) \sim \alpha g^2\omega^{-5}$ where α is an absolute constant. The instantaneous spatial spectrum Ψ (k) is proportional to k−4 for large wave numbers k, which is consistent with the observed occurrence of sharp crests in a well-developed sea, and the loss of energy from the wave system to turbulence and heat is proportional to $\rho _w u^3_*$, where ρw is the water density and u the friction velocity of the wind at the surface. This prediction of the form of Φ(ω) for large ω with α = 7·4×10−3, agrees well with measurements made by Burling (1955).

The results of calculations of the compression of air by repeated shock waves are compared with the perfect-gas values given by Evans & Evans (1956). The comparison emphasizes the increasing divergence of real from perfect-gas results as shock strengths are raised. The equations relating conditions across a shock wave are obtained in a convenient form for solution using a Mollier diagram.

In a recent paper Carrier & Greenspan (1958) showed that, within the framework of the non-linear shallow-water theory, there exist waves which do not break as they climb a sloping beach. The formation of a shock or bore is dependent on a variety of factors (wave shape, particle velocity, etc.) and, as yet, no general criteria for breaking have been found. In this paper, we consider waves which propagate shoreward into quiescent water; it is shown that any compressive wave (a wave of positive amplitude) which has a non-zero slope at the wave-front eventually breaks before reaching the coastline. In fact, an explicit relation is obtained between the initial conditions and the position where breaking occurs.