The study of wave propagation by geometrical optics is applied to a consideration of propagation through a non-uniform, statistically homogenous medium. Following the trajectory of a phase point, two effects are examined; the retardation of the phase point by the variable local wave speed along its trajectory and the further retardation and dispersion resulting from its meandering. Mean and mean-squared displacements are obtained to describe the retardation of the wave front, and the dispersion of the phase point from the incident direction.

The theory has application as a correction to the use of geometrical optics wherever the latter can be employed. In particular, it is shown by an estimate of the magnitude of the pertinent parameters that an application may well be found in the study of the tsunami (a long ‘shallow’ ocean wave).

Henry & Roux have recently conducted extensive numerical studies on the interaction of Soret separation with convection in cylindrical geometry. Many of their solutions exhibit parallel flow away from endwalls. We show that their parallel flow results can be matched by closed-form solutions. We find that solutions are non-unique in some parameter regions. Disappearance of one branch of solutions correlates with a sudden transition of Henry & Roux's results from a separated to a well-mixed flow.

Consider the flow of an incompressible fluid between two infinite concentric circular cylinders. The outer cylinder is at rest whilst the angular velocity of the inner cylinder has a steady part and also a harmonically oscillating component. We examine the situation where, for a suitable choice of parameters, two types of vortex instability can occur simultaneously; first a short-wavelength mode which is essentially trapped in a thin ‘Stokes’ layer near the inner cylinder and, secondly, a long-wavelength mode which fills the whole region between the cylinders. We investigate the problem in which two short-wavelength vortices and one long-wavelength vortex coexist and are such that each pair interacts to drive the third. Additionally, the short-wavelength disturbances are nonlinear in their own right. Coupled amplitude equations for the three modes are derived and their solution discussed.

This form of interaction may also take place in a boundary layer. Such a situation is more complex than that under consideration here as it would be necessary to take into account the growth of the boundary layer. However, this simplified problem gives an insight into the behaviour of the more difficult situation.

Vorticity distributions in axisymmetric vortex rings produced by a piston–pipe apparatus are numerically studied over a range of Reynolds numbers, $Re$ , and stroke-to-diameter ratios, $L/D$ . It is found that a state of advective balance, such that $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r\approx F(\unicode[STIX]{x1D713},t)$ , is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $\unicode[STIX]{x1D713}$ is the Stokes streamfunction in the frame of the ring. Some, but not all, of the $Re$ dependence in the time evolution of $F(\unicode[STIX]{x1D713},t)$ can be captured by introducing a scaled time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D708}t$ , where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. When $\unicode[STIX]{x1D708}t/D^{2}\gtrsim 0.02$ , the shape of $F(\unicode[STIX]{x1D713})$ is dominated by the linear-in- $\unicode[STIX]{x1D713}$ component, the coefficient of the quadratic term being an order of magnitude smaller. An important feature is that, as the dividing streamline ( $\unicode[STIX]{x1D713}=0$ ) is approached, $F(\unicode[STIX]{x1D713})$ tends to a non-zero intercept which exhibits an extra $Re$ dependence. This and other features are explained by a simple toy model consisting of the one-dimensional cylindrical diffusion equation. The key ingredient in the model responsible for the extra $Re$ dependence is a Robin-type boundary condition, similar to Newton’s law of cooling, that accounts for the edge layer at the dividing streamline.

In this paper, results of laboratory experiments simulating buoyancy-driven coastal currents produced by estuarine discharges into the ocean, are discussed. The responses of the propagation speeds of the currents to increases and decreases of the volumetric discharge rate at the source are investigated. For increasing discharge rate, we find that the mean speed of the current head displays a sharp rise some time after the source discharge condition has changed. In contrast, a decrease of the current speed following a decreasing discharge rate proceeds gradually. The current speed after acceleration or deceleration is found to be equal to the speed that would be expected had the discharge been at the higher or lower rate from the start of the experiment. The relative speed at which the information of the changed discharge condition at the source approaches the advancing current head from upstream, for both increasing and decreasing discharge rates, is found to be approximately one to three times the mean speed of the current. Further, we find that this transmission speed is 0.82±0.20 times the propagation speed of a linear, long interfacial Kelvin wave.

Consider a wing-jet combination placed in a wind tunnel; the measured thrust on the wing due to a high speed jet emerging from it will require to be corrected to give the infinite stream value. This paper provides a theory of these wind tunnel corrections, and incidently establishes that the ideal thrust is almost independent of the jet exit angle.

The equation of motion is derived for the steady flow under gravity of a liquid in a uniform channel, including the effects of the friction of the bed and of the eddy viscosity for lateral mixing. The equation is solved to find the distribution of mean velocity across the channel when the cross-section is rectangular, triangular or trapezoidal. Numerical values are also given to indicate the extent to which eddy viscosity may affect the lateral distribution of mean velocity.

The related questions concerning the transmission of electromagnetic waves are considered:

1. The reflexion and transmission of plane waves at a perfectly conducting layer of gas in an otherwise non-conducting atmosphere, when there is a uniform external magnetic field perpendicular to the layer. Here the main result is that a layer of finite depth h is an almost perfect filter, being transparent to waves of frequency nπA0/h (A0 = Alfvén velocity, n an integer).

2. The existence of plane surface waves for such a finite layer. There is always one such wave and, for certain ranges of frequency, two. The first becomes ‘choked’ at the filter frequencies, its velocity first tending to zero and then jumping to a finite value. The second chokes at the frequencies nπA0 a0/h √(a20 + A20 (a0 = acoustic velocity).

It has been shown that it is possible to measure the turbulent diffusivity of a magnetic field by a method involving oscillatory sources. So far the method has only been tried in the special case of two-dimensional fields and flows. Here we extend the method to three dimensions and consider the case where the flow is thermally driven convection in a large-aspect-ratio domain. We demonstrate that if the diffusing field is horizontal the method is successful even if the underlying flow can sustain dynamo action. We show that the resulting turbulent diffusivity is comparable with, although not exactly the same as, that of a passive scalar. We were not able to measure unambiguously the diffusivity if the diffusing field is vertical, but argue that such a measurement is possible if enough resources are utilized on the problem.