An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance.

The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field.

The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).

Conservable quantities measuring ‘wave activity’ are discussed. The equation for the most fundamental such quantity, wave-action, is derived in a simple but very general form which does not depend on the approximations of slow amplitude modulation, linearization, or conservative motion. The derivation is elementary, in the sense that a variational formulation of the equations of fluid motion is not used. The result depends, however, on a description of the disturbance in terms of particle displacements rather than velocities. A corollary is an elementary but general derivation of the approximate form of the wave-action equation found by Bretherton & Garrett (1968) for slowlyvarying, linear waves.

The sense in which the general wave-action equation follows from the classical ‘energy-momentum-tensor’ formalism is discussed, bringing in the concepts of pseudomomentum and pseudoenergy, which in turn are related to special cases such as Blokhintsev's conservation law in acoustics. Wave-action, pseudomomentum and pseudoenergy are the appropriate conservable measures of wave activity when ‘waves’ are defined respectively as departures from ensemble-, space- and time-averaged flows.

The relationship between the wave drag on a moving boundary and the fluxes of momentum and pseudomomentum is discussed.

Quantitative velocity measurements of unsteady convection in a Bénard cell suddenly cooled from above to a supercritical Rayleigh number are reported. The results are obtained by a novel application of laser speckle photography adapted for the study of fluid dynamics. This technique allows two-dimensional full-field velocity information to be obtained solely from a selected plane within the volume of moving liquid at any given instant in time. The convective rolls are found to be regular and oriented parallel to the short side of the cell. The number of rolls within the cell is approximately twice as great as the number reported for steady conditions. The maximum vertical velocity distribution is observed to be a sinusoidal function of the horizontal distance with a detectable third-harmonic component.

The temperature dependence of the density of both pure and saline water, even to very high salinity and pressure levels, decreases at decreasing temperature toward an extremum. The nature of this variation precludes approximating the buoyancy-force density difference linearly with a temperature difference. This peculiar density variation of water has very significant effects, even at environmental temperature levels. A new equation has appeared which relates density to temperature, salinity and pressure with very high accuracy. Its form is especially suited to the analysis of convective motions. We consider here vertical boundary-layer flows. Analysis of flows arising from thermal buoyancy and from combined buoyancy effects shows the simplicity of the formulation. Relatively few new parameters arise. Extensive calculations for thermally buoyant flows show the large magnitude of the effects of the complicated density variation on transport. Buoyancy-force reversals and convective inversions are predicted. The latter are in close agreement with past experiments. A new Grashof number arises which is an accurate indication of the actual local flow vigour. The effects of specific temperature conditions are given in detail. The appreciable effect of the Prandtl number is calculated. Transport parameters are given for salinities and pressures up to 40 p.p.t. and 1000 bars, respectively.

Measurements have been made of the time-mean velocity, all three fluctuating velocity components and the turbulence shear stress in the plane mixing layer produced in an open wind tunnel between two nominally parallel air streams with fairly low turbulence levels. The hot-wire measuring system is described, the evaluation equations derived and the principal results presented.

The transmission of sound through a turbulent shear layer is reviewed both with respect to the mathematical theory of the distortion of signals during propagation and also as the basis of the analysis in part 2 of the spectral broadening evinced by experimental results relevant to the study of aircraft noise. The reflexion and transmission coefficients, which involve both amplitude and phase changes, are obtained for scattering by an irregular and unsteady interface convected between two media; diffraction of high frequency sound by small-scale turbulence in a shear layer is accounted for by means of a phase shift and conservation laws for energy, wavenumber and amplitude. These results may be used to construct the sound field transmitted through a turbulent shear layer from a source in the interior of a jet; multiple internal reflexions are accounted for in the case of transmission through two parallel shear layers, i.e. a jet of finite width shielding a noise source. A statistical description is given of the process of attenuation of the transmitted wave as energy is diffracted by the turbulence, and of the partial compensation by interference between correlated components of the refracted wave; the reference case is a monochromatic point source, which, when placed behind a system of shear layers, has its energy redistributed directionally and spread over a range of frequencies. The expressions obtained for the energy flux include as particular cases the results of Howe for the plane (1975) and impedance (1976) layers, the effects of turbulence being shown to be consistent with the experiments of Schmidt & Tilmann (1970) and Ho & Kovasznay (1976a, b).

It has been observed experimentally by Candel, Julienne & Julliand (1975) that a monochromatic test tone generated by a source inside a jet is received outside as a broad frequency band of definite shape. This phenomenon of spectral broadening occurs during transmission through the shear layer, which generally has a randomly irregular and unsteady shape, contains in addition distributed turbulence, and separates the jet and the ambient medium. We show in the first place that, in the audible range of frequencies, neither the approximation which treats the shear layer as a scattering interface with a convected undulating shape nor the opposite, high frequency limit obtained by means of asymptotic estimation of integrals derived for the diffraction of rays in turbulence is sufficient to provide a satisfactory theory of the observations. The refraction integrals obtained in part 1 have to be evaluated exactly in order to account for the phenomenon of spectral broadening, the methods used possibly being of interest in other branches of wave theory. The formation of the transmitted spectrum from an incident tone can be illustrated by representing a simple shear layer as an array of elements each re-radiating energy received from the source with its own characteristic attenuation and frequency shift. A computer program is used to obtain spectra under conditions corresponding to the experiments of Candel, Guédel & Julienne (1975) and gives encouraging agreement with their measurements, which were made with high frequency sources immersed in low speed jets. The theory can also be applied to the prediction of spectra received at various angles to the axis of high subsonic jets, but depends on extrapolation when supersonic exhausts are considered. We conclude with an example of the possible relevance of spectral broadening as a means of reducing the noise disturbance from current jet-powered aircraft, such as Concorde.

The propagation of internal Alfvén-inertio-acoustic gravity waves in a perfectly electrically conducting, stratified, inviscid, non-isothermal, rotating atmosphere permeated by a non-uniform magnetic field is investigated. These waves exhibit singular properties at the critical levels at which the magnetic field and the sound velocity are such that \[ (\omega^2 - S^2)\{(c^2+V^2)\omega^2-c^2S^2\}-(c^2+V^2)\overline{R}^2=0, \] where ω is the frequency of the waves, $S = kV_x + lV_y,\overline{R} = 2\Omega_z\omega$, Vx and Vy are the x and y components of the Alfvén velocity, k and l are the corresponding wavenumbers and c is the sonic velocity. These levels act like valves which permit waves to penetrate them from one side only and absorb them when they propagate from the other side. In contrast to the incompressible results of Acheson (1972), we show that the valve effect in compressible flow no longer requires the presence of non-zero components of rotation in the plane normal to the direction in which the medium varies. We find that the compressibility increases the probability of a valve effect and so increases the capacity of a hydromagnetic wave to propagate across a field line, rather than being absorbed at some critical level.