In this paper, it is shown how the author's general theory of slowly varying wave trains may be derived as the first term in a formal perturbation expansion. In its most effective form, the perturbation procedure is applied directly to the governing variational principle and an averaged variational principle is established directly. This novel use of a perturbation method may have value outside the class of wave problems considered here. Various useful manipulations of the average Lagrangian are shown to be similar to the transformations leading to Hamilton's equations in mechanics. The methods developed here for waves may also be used on the older problems of adiabatic invariants in mechanics, and they provide a different treatment; the typical problem of central orbits is included in the examples.

Turbulent mass transfer to a wall at high Schmidt numbers is controlled by the velocity field within the viscous sublayer. Measurements have been obtained of the root-mean-square fluctuating mass transfer coefficient and the frequency spectrum of the fluctuating mass transfer coefficient for a Schmidt number of about 2300. From an order-of-magnitude analysis it is concluded that flow fluctuations in the direction of mean flow have little effect on the mass transfer fluctuations. A comparison of the mass transfer spectrum with the spectrum of the component of the velocity gradient in the transverse direction sz reveals that the high-frequency portion of the sz spectrum is not effective in transferring mass. Approximate relations between the mass transfer spectrum and the sz spectrum are developed for high frequencies and for low frequencies.

A three-dimensional barotropic model of the wind-driven ocean circulation is examined and the flow near the east coast of the ocean is considered in detail. The model is linear and has constant coefficients of eddy viscosity. It is shown that a strong current may exist in the eastern boundary layer when upwelling or downwelling is present. No net northwards transport is produced as an equal deep counter current also occurs. A consequence of the downwelling is that the principal interior gyre is prevented from reaching the east coast and a secondary gyre is formed in this region.

A solid long slender body is considered placed in a fluid undergoing a given undisturbed flow. Under conditions in which fluid inertia is negligible, the force per unit length on the body is obtained as an asymptotic expansion in terms of the ratio of the cross-sectional radius to body length. Specific examples are given for the resistance to translation of long slender bodies for cases in which the body centre-line is curved as well as for those for which the centre-line is straight.

Measurements were made of some of the properties of grid-generated, turbulence for several concentrations of a drag-reducing polymer additive in water. Reference measurements of the grid pressure drops, streamwise intensities and one-dimensional spectra in pure water agreed with previous measurements obtained in Newtonian fluids. Corresponding results for the polymer solutions showed that the grid pressure drops were generally lowered, the turbulence intensity levels were increased and the one-dimensional energy spectra were unchanged compared to the results in water. A dimensionless rate of decay correlation was introduced which removed the dependence of the rate of decay on the initial conditions of grid-generated turbulence for Newtonian fluids: the polymer solution decay results did not follow this correlation, indicating that the decay process is different in these fluids. The one-dimensional energy spectra of the turbulence in the polymer solutions were of the same shape as those in Newtonian fluids, and were normalized with modified Kolmogoroff variables using an effective viscosity (Lumley 1964) and the viscous dissipation in order to provide a quantitative comparison to Newtonian spectra. All of the normalized spectra in the polymer solutions collapsed to a single curve which coincided with the normalized Newtonian spectral curve. It was also found that the rate of decay was less than the viscous dissipation in the polymer solutions, in accordance with recent theoretical predictions.

The meniscus profiles determined numerically by White & Tallmadge (1965) have been found to be in error. Accordingly, the reported values of meniscus height and curvature at the top of the meniscus are also in error. These errors were initially suspected by observing that top curvatures calculated from the reported values of height fell below the minimum possible (flat plate) value at large radii.

Electrochemical techniques are used to measure the circumferential component of the velocity gradient sz at the wall of a pipe through which a turbulent fluid is flowing. The ratio of the root-mean-squared value of sz to the time-averaged velocity gradient is found to be 0·09 or 0·1, depending on whether corrections are made for frequency response. The frequency spectrum is similar to that for the component of the wall velocity gradient in the direction of mean flow. The amplitude distribution function for sz is very roughly approximated by a Gaussian distribution.

Quarter-wavelength resonators for harbour entrances and similar applications are usually designed and built without regard to the end effect. This paper describes tests on rectangular resonators using water waves of unique frequency in a long narrow wave channel fitted with a wave generator at one end, a wave absorber at the other end and suitable wave filters. In these tests the wavelength was kept constant while the geometry of the rectangular resonant branch canal was systematically varied. Wave transmission across the resonator, as well as wave reflexion upstream of the resonator was measured. The results clearly indicate a considerable effect of main channel width on optimum resonator width and resonator length, invalidating the usual resonance theory (which predicts complete reflexion of the incident wave train when the length of the branch canal is one-quarter of a wavelength). For example, it is found that, under certain conditions, quarter-wavelength resonators may not have any effect on the incident wave. The work described is limited to the first resonant mode and to semi-infinite domains on each side of the resonator.

The breakdown of shallow water waves into forms exhibiting several secondary crests is analyzed by numerical computations based on approximate equations accounting for the effects of non-linearity and dispersion. From detailed results of two cases it is shown that when long waves are such that the parameter σ = ν*L*2/h*3 is of moderate magnitude, either due to initially steep waves generated at a wave-maker or due to forced amplification by decreasing depth, waves periodic in time do not remain simply periodic in space. Numerical results are compared with experiments for waves propagating past a slope and onto a shelf.

A uniformly valid asymptotic solution for large Reynolds number is constructed for plane steady laminar flow of a liquid into the channel between two semi-infinite parallel plates. The entry condition is taken as either that for a cascade of plates in a uniform oncoming stream, or uniform flow directly at the inlet. A paradox in the standard solution of Schlichting—that near the inlet the flow due to displacement would not be the accelerated uniform core on which his expansion is based—is resolved by showing that his series for small as well as large distance actually applies only to conditions far downstream, and matches with another expansion valid near the inlet. Good agreement is found with three independent numerical solutions of the full Navier-Stokes equations, except for a discrepancy in one solution for uniform entry that is traced to erroneous neglect of inlet vorticity.