A theory of the generation of dispersive waves by travelling forcing effects, that may be steady, oscillatory or transient in character, is given for a general homogeneous system. Small disturbances to the system are supposed stable, and governed by a linear partial differential equation with constant coefficients which admits solutions in the form of plane waves satisfying an, in general, anisotropic dispersion relation P(σ, k) = 0 between frequency σ and wave-number vector k.

If the forcing region, supposed of limited extent, travels with constant velocity U, then oscillatory forcing terms of frequency σ0 (which would be replaced by 0 in the limiting case of a steady forcing effect, while taking, for a typical transient one of duration T, values from 0 to about 10/T) produce waves of frequency σ0 + U. k (the Doppler effect). For any such waves, the wave-number k satisfies the equation P(σ0 + U.k,k) = 0, representing a surface in wave-number space here called S(σ0), and their position relative to that of the forcing region is determined by having been generated when that region was in an earlier position, and having subsequently progressed with the group velocity. This implies the rule, also derived analytically in §§2 and 3, that waves with a particular value of k on S(σ0) are found in a direction, stretching out from the forcing region, which is one of the directions normal to S(σ0) at k, namely the one pointing towards S(σ0 + δ). This rule is supplemented by results on wave amplitudes and shapes of crests.

The theory is applied (§§4 and 5) to Rossby waves excited in a beta-plane ocean by travelling patterns of wind stress. If a steady wind-stress pattern moves westward, semicircular waves of length 2π √(U/β) trail behind it, but signals are found also directly ahead, consisting of the disturbance integrated in the west-east direction and subjected to a ‘low-pass filter’ with respect to its north-south components of wave-number. An eastward-travelling pattern, by contrast, produces only a wake-like disturbance, calculated in detail in §4. The waves generated for intermediate directions of travel are identified, and the strong tendencies in all cases for westward intensification of transient currents are noted.

For example, a wind-stress pattern travelling 30° N. of E. leaves a trailing wedge of currents from W. to 30° S. of W. in the steady case. The influence on this conclusion of a finite duration T of such a pattern is investigated in §5 by Fourier analysis in time. The fate of Fourier components of frequency σ0 depends on the ratio L =σ0/√(Uβ). If this is less than 1 for all (T, up to about 10/T, then the disturbance retains its trailing character; on the other hand, any components with L > 1 have a much greater directional spread. Tidal terms make fairly small changes to the results, except that (in an ocean of depth H) they make the directional spread disappear for L greater than about √(βgH/4f2U).

Excitation of gravity waves in non-rotating fluid is briefly considered, includ-ing generation on deep water by a travelling oscillating disturbance (§6), and generation in a uniformly stratified fluid by a vertically moving obstacle (§7). The predicted wave shapes in the latter case, with cusps at a finite distance behind the obstacle, agree excellently (figures 7 and 8) with experiments by Mowbray (1966).

An exceptional case, in that part of S(σ0) is doubly covered, is generation by steady (σ0 = 0) motion of an obstacle along the axis of uniformly rotating homo-geneous fluid, the surface S(0) being a sphere and two coincident planes. Whereas waves corresponding to points on the sphere trail behind the obstacle, the appro-priate normals on the two planes point in opposite directions inside the sphere (§8), permitting the well-known formation of the ‘Taylor column’ ahead of the obstacle at low Rossby numbers.

Still more complicated, because fully three-dimensional, is the case when the obstacle moves at right angles to the axis of rotation (§9). At finite though small Rossby number it is impossible for the Taylor column formed near the body to extend to large distances from it, where on the contrary the disturbance is shown to take the form of slightly trailing cones, shown in cross-section in figure 12, containing waves whose crests have cusps on the boundaries of the cones. An estimate of the length of the Taylor column, as body dimension divided by Rossby number (for small enough kinematic viscosity), is made by considering the fit between the Taylor-column and wave-cone regions.

Using a hot-wire system developed for this purpose, measurements of the velocity were made in a circular liquid-mercury jet issuing into a low-speed secondary flow and subject to a uniform axial magnetic field. The Reynolds number of the jet was about 10,000 while the magnetic interaction parameter varied from zero to slightly over one. The jet was strongly turbulent under all conditions investigated. The radial distribution of the mean and fluctuating component of the meridional velocity was measured at four axial stations located between 2 and 34 diameters from the nozzle exit. The results indicate that the rate of spreading of the jet is decreased, that the shape of the velocity profile changes and that the turbulent intensity decreases with increasing magnetic-field strength. The high-frequency components of the fluctuations in the second flow seem to be damped more strongly than fluctuations at low frequencies, while the reverse is observed within the core of the jet.

A 17 in. diameter shock tube was coupled to a 1 in. tube by a 10° half-angle conical convergence. Timing measurements showed that the shock waves emerged from the convergence at 2–3 times their entrance speeds, and then decelerated downstream. After the application of a viscous correction, the shock speeds at the convergence exit agreed to within 5% with equilibrium real-gas calculations of the model of Chester (1954), Chisnell (1957) and Whitham (1958). The downstream deceleration, due to viscosity and to higher-order interactions not included in the theory, was also briefly studied.

The theory of Fay & Riddell (1958) is used to calculate stagnation temperatures from stagnation-point heat-transfer rates measured in the working section of a hypersonic gun tunnel at a Mach number of 9·8. Measurements using both thin-film gauges and calorimeters are described. The temperatures measured using this technique are found to be lower than predicted by Lemcke (1962) from measurements of shock strengths and final pressures in the gun barrel. This discrepancy is attributed to heat losses in the barrel during the initial shock compression cycle. A simple theory is developed to take into account these losses. There is good agreement between this theory and the experimental results.

Theoretical solutions based on the expansion scheme for large x and large M∞, as proposed by Freeman (1962), are obtained for the asymptotic inviscid flow over plane bodies of the shape y/d = (x/d)m in the range $\frac{2}{3}/\gamma < m < \frac{2}{3}$ where blast wave theory applies as a first approximation. In particular, the second-order terms, which are necessary to satisfy the body boundary conditions for the normal velocity are computed. The magnitude of the second-order terms is found to increase from zero at $m=\frac{2}{3}/\gamma $ to infinity at $m = \frac{2}{3}.$

As a comparison with theory, experiments at M∞ = 8·2 were made with two plane power-law bodies in the range $\frac{2}{3}/\gamma < m < \frac{2}{3} $ and on a plane parabola with a tangent wedge nose. These consisted of the determination of shock-wave shapes, surface pressure distributions and detailed investigations of the distribution of pitot and static pressure across the shock layer.

The experimental results are in good agreement with the theory in the case m =1/2, where the second-order effects are small. At m = 5/8 the region of validity of the theory is limited to much larger distances from the nose of the body and larger Mach numbers. Accordingly, the prediction for the deviation from firstorder theory, although being correct in sign, is too small. Shock-wave shapes on bodies of the same power but of different size are correlated by the similarity theory when scaled with respect to the dimension d.

The experimental results obtained with the wedge-parabola are in very good agreement with a characteristics solution by C. H. Lewis (1965, unpublished).

Internal gravity waves of small amplitude propagate in a Boussinesq inviscid, adiabatic liquid in which the mean horizontal velocity U(z) depends on height z only. If the Richardson number R is everywhere larger than 1/4, the waves are attenuated by a factor $\exp\{-2\pi(R - \frac{1}{4})^{\frac{1}{2}}\}$ as they pass through a critical level at which U is equal to the horizontal phase speed, and momentum is transferred to the mean flow there. This effect is considered in relation to lee waves in the airflow over a mountain, and in relation to transient localized disturbances. It is significant in considering the propagation of gravity waves from the troposphere to the ionosphere, and possibly in transferring horizontal momentum into the deep ocean without substantial mixing.

The axisymmetric flow of fluid, confined within a rapidly rotating annulus and caused by a horizontally applied temperature gradient, is discussed under the assumptions of small thermal Rossby number and small Taylor number. It is initially assumed that the transfer of heat is purely conductive, and the flow is calculated for when the upper fluid surface is either rigid or free. The analysis of the problem in which the upper surface is rigid is extended so as to allow con-vective as well as conductive transfer of heat.

A power-law relation is derived between the Stanton number and the Reynolds number, expressing the law of heat transfer for a wall adjacent to a region of turbulent separated flow. The derivation is based on Prandtl's (1945) proposal for the laws of dissipation, diffusion and generation of turbulent kinetic energy. The constants appearing in these laws are determined by reference to experimental data for the hydrodynamic properties of the constant-stress and the linear-stress layers.

The agreement between the resulting predictions and the experimental data of other workers is sufficiently good to suggest that the actual mechanism of heat transfer from separated flows has much in common with that which is postulated. Closer agreement can be expected only after the present one-dimensional analysis has been superseded by a two-dimensional one.

Measurements of the low-speed flow up a step of height equal to 1·75 times the initial boundary-layer thickness show that the flow satisfies Stratford's (1959) condition for rapid separation, the extra stress gradients being confined to the first one-eighth of the boundary-layer thickness. The increase in turbulence intensity up to separation is small, and attributable to low-frequency fluctuations in separation position. Townsend's (1962) criterion predicts the separation point fairly accurately. A simple expression is found for the additional pressure rise that can be withstood by a boundary layer already fairly near separation, which gives tolerable results at any point in the flow up a step.

A wind tunnel in which an arbitrary negative pressure gradient could be developed has been built for boundary-layer studies. The effects of selected pressure gradients on boundary layers grown on one of the walls of the tunnel were studied. It was possible to obtain equilibrium boundary layers of the type first suggested by Clauser: that is, layers whose non-dimensional velocity-defect distribution is invariant along the direction of flow. The velocity-defect distributions for two such boundary layers were established, corresponding to values of Clauser's dimensionless pressure-gradient parameter β of −0·35 and −0·53. These velocity profiles are compared with the profiles predicted theoretically by Mellor & Gibson (1966). The agreement between the two is very good.

The velocity field generated in a fluid of viscosity, v, by impulsively starting at time t = 0, a sphere of radius a spinning with angular velocity Ω about a diameter is described using a new expansion variable 2 √vt/r. It is first shown how the standard time-dependent boundary-layer equations can be modified to give series solutions satisfying all the boundary conditions. Next, that these new solutions are relevant when the Reynolds number R = a2Ω/v goes to infinity in such a way that $R^{\frac{1}{3}} \Omega t$ is large. Lastly, solutions are given, applicable at small times for non-zero Reynolds numbers. These last expansions show that the velocity components decay algebraically rather than exponentially at large distances.

The principal aim of this paper is to show that the variation of viscosity in a fluid can cause instability. Plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosities between two horizontal plates is considered, and it is found that both plane Poiseuille flow and plane Couette flow can be unstable, however small the Reynolds number is. The unstable modes are in the neighbourhood of a hidden neutral mode for the case of a single fluid, which is entirely ignored in the usual theory of hydrodynamic stability, and are brought out by the viscosity stratification.

A mechanism is proposed for the manner in which the turbulent components support Reynolds stress in turbulent shear flow. This involves a generalization of Miles's mechanism in which each of the turbulent components interacts with the mean flow to produce an increment of Reynolds stress at the ‘matched layer’ of that particular component. The summation over all the turbulent components leads to an expression for the gradient of the Reynolds stress τ(z) in the turbulence \[ \frac{d\tau}{dz} = {\cal A}\Theta\overline{w^2}\frac{d^2U}{dz^2}, \]where ${\cal A}$ is a number, Θ the convected integral time scale of the w-velocity fluctuations and U(z) the mean velocity profile. This is consistent with a number of experimental results, and measurements on the mixing layer of a jet indicate that A = 0·24 in this case. In other flows, it would be expected to be of the same order, though its precise value may vary somewhat from one to another.

The wakes formed behind sharp-edged, bluff, elliptical bodies have elliptical cross-sections, but the major axis of the wake is aligned with the minor axis of the body. This effect was observed in both mean velocity and turbulent intensity data in wakes throughout the range of the experiment, from several body diameters to distances of 250 diameters downstream of the body. The turbulent energy in the wake flow near the body displayed a periodicity which was correlated using a Strouhal number. Over the Reynolds-number range from 8 × 103 to 7 × 104, the Strouhal number depended only on the body eccentricity.

A method for obtaining analytic solutions to the problem of blunt bodies in the supersonic stream of an ideal gas is presented. The solutions are written in terms of power series whose coefficients are elementary functions. These solutions are approximate, but the approximation is rational, i.e. any higher approximation can, in principle, be obtained. Some of these higher approximations have been calculated. Examples are presented for various free-stream conditions and prescribed body shapes. These are compared with results from standard numerical procedure and with available experimental measurements.

The motion induced by sources and sinks distributed along the vertical side wall of a cylinder filled with fluid and rotating about a vertical axis is considered. Vertical motion, and hence vertical transport, is confined to a vertical boundary layer of thickness E1/3, where E is the Ekman number. The horizontal transport occurs through the interior of the fluid. The Ekman layers do not play any active role in the transport process. The results of a very simple experiment confirm these conclusions.

An analytical model of the interaction of a fibre tangle with an airflow is proposed. This model replaces the discrete fibres by a continuum medium with a non-linear stress-strain law. The governing equations have been examined for one-dimensional unsteady flow configurations and have been found to possess five characteristic directions.

A numerical-solution procedure, based upon the method of characteristics, has been outlined and applied to the flow within a dilation chamber. A fibre sample is located at the centre of the chamber, which is alternately pressurized and depressurized.

This paper extends the work of Wilson (1964) to include the effect of compressibility in the recurrence of dual solutions in the flow in the boundary layer on a semi-infinite, thermally insulated, flat plate placed at zero incidence to a uniform stream of electrically-conducting gas with an aligned magnetic field at large distances from the plate. Numerical integration of the boundary-layer equations has been performed for several values of the ratio, β, of the square of the Alfvén speed to the fluid speed in the undisturbed fluid, the conductivity parameter ε = 0·1 and ∞ and the square of the Mach number M2 = 0, 1/2, 1, 2, 2·5, 4 and 5. The effect of compressibility is to increase the value of β for which a solution can exist such that the skin friction at the plate is greater than zero. Dual solutions are seen to occur for non-zero Mach number and all values of ε but no attempt here has been made to explain this phenomenon. An analytic argument indicates that no solutions of the equations exist if the skin friction at the plate is greater than zero and if the vorticity and current decay exponentially, the condition for which is \[ M^2 < 1,\quad \beta > 1/(1-M^2). \] Nothing specific has been proved if this condition is not satisfied.

The boundary conditions at the surface of a small bubble rising in a liquid are examined theoretically, and it is shown by order-of-magnitude arguments, which are confirmed by detailed calculation in a special case, that although surfacetension gradients must always exist around the bubble, they are too small to affect the motion appreciably unless surface-active substances are present. This is because gradients of surface tension imply gradients of temperature in a pure liquid, and these turn out to be always small near the bubble if they tend to zero at large distances from it. The same is true for drops of one fluid in another.