This paper studies the second-order currents and changes in mean surface level which are caused by gravity waves of non-uniform amplitude. The effects are interpreted in terms of the radiation stresses in the waves.

The first example is of wave groups propagated in water of uniform mean depth. The problem is solved first by a perturbation analysis. In two special cases the second-order currents are found to be proportional simply to the square of the local wave amplitude: (a) when the lengths of the groups are large compared to the mean depth, and (b) when the groups are all of equal length. Then the surface is found to be depressed under a high group of waves and the mass-transport is relatively negative there. In case (a) the result can be simply related to the radiation stresses, which tend to expel fluid from beneath the higher waves.

The second example considered is the propagation of waves of steady amplitude in water of gradually varying depth. Assuming no loss of energy by friction or reflexion, the wave amplitude must vary horizontally, to maintain the flux of energy constant; it is shown that this produces a negative tilt in the mean surface level as the depth diminishes. However, if the wave height is limited by breaking, the tilt is positive. The results are in agreement with some observations by Fairchild.

Lastly, the propagation of groups of waves from deep to shallow water is studied. As the mean depth decreases, so the response of the fluid to the radiation stresses tends to increase. The long waves thus generated may be reflected as free waves, and so account for the 'surf beats’ observed by Munk and Tucker.

Generalle speaking, the changes in mean sea level produced by ocean waves are comparable with those due to horizontal wind stress. It may be necessary to allow for the wave stresses in calculating wind stress coefficients.

The solution is obtained to the problem of the steady one-dimensional flow of an incompressible, viscous, electrically conducting fluid through a circular pipe in the presence of an applied (transverse) uniform magnetic field. A no-slip condition on the velocity is assumed at the non-conducting wall. The solution is exact and thus valid for all values of the Hartmann number. Excellent agreement exists between the present theoretical results and the experimental values obtained by Hartmann & Lazarus (1937) in the low to medium Hartmann number range. The high Hartmann number case is treated by Shercliff (1962) in the following paper.

The paper presents an improved, second approximation for the laminar motion of a conducting liquid at high Hartmann number in non-conducting pipes of arbitrary cross-section under uniform transverse magnetic fields. A satisfactory comparison with the author's previously experimental pressure gradient/flow results is made for the case of a circular pipe.

It is the purpose of this paper to solve a boundary-value problem posed by induction electromagnetic pumps and generators. Solutions are obtained by an expansion technique and a momentum method for the laminar, incompressible flow problem. For large values of the interaction parameter (μ2σH20λ/ρμe viscous effects are shown to be restricted to periodic boundary layers. In regions of high-field strength a local Hartmann solution is valid. Where the applied field is weak an inertial boundary layer is present which thickens in the upstream direction.

A logical explanation of this phenomenon is given. The condition that a boundary-layer type flow exist is obtained and is shown to be in general satisfied. The results show that inviscid theory may be used to calculate the overall performance of electromagnetic pumps and generators while the boundary-layer theory developed here may be used to obtain the wall shear stress.

The flow of air in the lowest few metres of the atmosphere is examined in the case when a neutrally stratified logarithmic profile encounters a sudden change in surface roughness. Examination of the available evidence suggests that the ratio of new to old friction velocity is given by the ratio of the roughness lengths raised to a power about equal to 0.09. Some field observations and wind-tunnel measurements over a change in roughness indicate that vertical displacement of the streamlines at all heights within the range considered commences very near the surface transition. Calculations are made of the fetch necessary for the new wind profile to be established up to a given height and indicate that its ratio to the height is not constant but varies with the height and amount of change in roughness. Over a fairly wide range this ratio is about 100-150.

The vertical motion (heave) of a freely floating sphere is studied under the action of incident sine waves. Forced heave is defined as the vertical motion of sphere in still water, while free heave is defined as the vertical motion of sphere in the sine-wave sea. The total velocity potential describing the motion is decomposed into three terms: incident wave potential, diffracted wave potential (potential as if the sphere were fixed in sine-wave sea), and forced heave potential (potential as if the sphere were in forced vertical motion in still water). The forced heave and diffraction problems are solved separately. The linearized equation of motion is then used to effect the synthesis of the free motion via two unknowns: amplitude of vertical displacement and phase difference.

Both the radiation and diffraction problems are solved by expansions of non-orthogonal functions (wave-free potentials). These functions are trivial solutions of the Sommerfeld radiation condition in the sense that they attenuate faster than O(r−½) and it is necessary to add multipole terms which have the proper behaviour at infinity. Infinite systems of linear equations are obtained for the unknown expansion coefficients and the unknown source strengths of the multipole. The added mass, damping coefficient and wave-making coefficient in forced heave are studied as well as the force on the fixed sphere. In addition the added mass and damping coefficient in free heave are obtained.

The horizontal motion (surge) of the freely floating sphere can be obtained by methods similar to those employed for the vertical motion. The author is presently working on this problem. Surge and heave are not independent motions but can be treated separately because of the symmetry of the sphere.

An investigation is made into the validity of the Oseen equations, for incom-pressible, viscous flow past a body, as an approximation to the Navier-Stokes equations. It is shown that, when the body is such that a reversal of the uniform flow at infinity merely reverses any component of the force on the body without changing its absolute magnitude, that component can be determined correctly to the first order in the Reynolds number, though the detailed velocity field is not correct to this order. Moreover, this force can be deduced simply from a knowledge of the force on the body according to Stokes's approximation.

The analysis is also generalized to include the magneto-hydrodynamic effects when the fluid is conducting and the flow takes place in the presence of a magnetic field.

This paper contained several extensions of the theory of hydromagnetic waves in a partially ionized gas. The gas is confined in a cylindrical tube through which passes an axial magnetic field. The tube wall is assumed to be either a perfect conductor or a material of small or zero conductivity.

The effects of the viscosity and compressibility of the ionized and neutral gases are included in the theory, as also are the contributions of finite conductivity and the ion-cyclotron term. The non-isotropic character of the viscosity and conductivity coefficients of the ionized gas is taken into account. New boundary conditions are derived for tubes with insulating walls, and it is shown that the only pure modes that can be propagated along such tubes are just two special cases of those waves having no azimuthal dependence.

The principal result of the paper is a new dispersion relation which allows for all the dissipative effects just mentioned and which is valid for a range of frequencies which extends well beyond the ion cyclotron frequency, but falls short of the frequency at which electron inertia and displacement currents became effective.

In a previous paper (Woods 1962) a theory is presented which leads to a general dispersion relation for hydromagnetic waves in a dissipative plasma contained in a cylindrical tube. In the present paper experimental observations are compared with the predictions of this theory.

An experiment is described in which torsional hydromagnetic waves are excited in a gas discharge. Measurements of the wave velocity and damping are compared with solutions of the dispersion equation which are computed from measured values of the plasma parameters. The results are consistent with the theory, and good numerical agreement is obtained by assuming a loss of particles to the tube walls. There is evidence of a cut-off in wave propagation in the region of ion cyclotron resonance.

Measurements were made of the mean velocity profiles in a jet immersed in a stream whose velocity was in the same direction as that of the jet. They showed a mean velocity on the jet axis falling inversely as the distance downstream of the jet origin, a behaviour demonstrably inconsistent with any theory which assumes that the turbulence at each section of the jet is in local equilibrium. A mixing length, LM, and a length, Ld, characterizing the rate of dissipation are defined and it is shown that LM increases with distance downstream more rapidly than the jet width, and Ld less rapidly.

The stability of small two-dimensional travelling-wave disturbances in an incompressible laminar boundary layer over a flexible surface is considered. By first determining the wall admittance required to maintain a wave of given wave-number and phase speed, a characteristic equation is deduced which, in the limit of zero wall flexibility, reduces to that occurring in the ordinary stability theory of Tollmien and Schlichting. The equation obtained represents a slight and probably insignificant improvement upon that given recently by Benjamin (1960). Graphical methods are developed to determine the curve of neutral stability, as well as to identify the various modes of instability classified by Benjamin as ‘Class A’, ‘Class B’, and ‘Kelvin-Helmholtz’ instability, respectively. Also, a method is devised whereby the optimum combination of surface effective mass, wave speed, and damping required to stabilize any given unstable Tollmien-Schlichting wave can be determined by a simple geometrical construction in the complex wall-admittance plane.

What is believed to be a complete physical explanation for the influence of an infinite flexible wall on boundary-layer stability is presented. In particular, the effect of damping in the wall is discussed at some length. The seemingly paradoxical result that damping destabilizes class A waves (i.e. waves of the Tollmien-Schlichting type) is explained by considering the related problem of flutter of an infinite panel in incompressible potential flow, for which damping has the same qualitative effect. It is shown that the class A waves are associated with a decrease of the total kinetic and elastic energy of the fluid and the wall, so that any dissipation of energy in the wall will only make the wave amplitude increase to compensate for the lowered energy level. The Kelvin-Helmholtz type of instability will occur when the effective stiffness of the panel is too low to withstand, for all values of the phase speed, the pressure forces induced on the wavy wall.

The numerical examples presented show that the increase in the critical Reynolds number that can be achieved with a wall of moderate flexibility is modest, and that some other explanation for the experimentally observed effects of a flexible wall on the friction drag must be considered.