A theory is proposed to describe the generation of sound by turbulence at high Mach numbers. The problem is formulated most conveniently in terms of the fluctuating pressure, and a convected wave equation (2.8) is derived to describe the generation and propagation of the pressure fluctuations.

The supersonic turbulent shear zone is examined in detail. It is found that, at supersonic speeds, sound is radiated as eddy Mach waves, and as the Mach number increase, this mechanism of generation becomes increasingly dominant. Attention is concentrated on the properties of the pressure fluctuations just outside the shear zone where the interactions among the weak shock waves have had little effect. An asymptotic solution for large M is derived by a Green's function technique, and it is found that radiation with given frequency n and weve-number K can be associated with a coresponding critical layer within the shear zone.

It is found that $(\overline {p - p_0})^2 $ increases approximately as $\rm {M}^{\frac {3} {2}}$ for M [Gt ] 1 contrasting with the M8 variation found by Lightill for M [Lt ] 1. The acoustic efficiency thus varies as $\rm {M}^-{\frac {3} {2}}$ for M [Gt ] 1, and as M5 for M [Lt ] 1, indicating a maximum acoustic efficiency for Mach numbers near one. The directional distribution of the radiation is discussed and the direction of maximum intensity is shown to move towards the perpendicular to the shear zone as M increases. The predictions of the theory are supported qualitatively by the few available experimental observations.

The frequency dependence of the velocity and attenuation of sound waves in a gas which undergoes vibrational relaxation have been investigated theoretically. At low audible frequencies the attenuations due to viscosity, thermal conduction and relaxationin the gas, add linearly, while the velocity is the relaxation velocity diminished by the Helmholtz-Kirchhoff factor. The relations have been confirmed experimentally, and the free gas velocities of sound at zero frequency, one atmosphere pressure and 30 °C, found for carbon dioxide, air and oxygen, are 270·57 ± 0·04 m sec−1, 349·18 ± 0·02 m sec−1 and 331·33 ± 0·03 m sec−1, respectively. The corresponding specific heats are Cp/R = 4·537 ± 0·008 for carbon dioxide and Cp/R = 3·547 ± 0·003 for oxygen.

Experimental data are presented for the growth of vapour bubbles in various superheated liquids, such as carbon tetrachloride, benzene, ethyl alcohol, and methyl alcohol. These data are compared with the theoretical results obtained by Plesset & Zwick (1953) who derived these results by taking into account the heat diffusion across the bubble boundary. The agreement in all cases between experiment and theory is found to be good.

The growth of vapour bubbles in slightly superheated water is also presented in the form of experimental data for bubbles just beginning to grow from a point of equilibrium which is presumed to be dynamically unstable. The radii corresponding to the points of equilbrium are of the same order of magnitude as those predicted by theoretical considerations.

The structure of the steady magnetohydrodynamic switch-on shock wave is investigated for several orderings of the four diffusivities involved in the problem. The various orderings are approximated to by allowing one or more of the appropriate diffusivities to approach zero, and approximate solutions that are uniformly valid to order unity are sought. In general, singular perturbation problems are encountered, the number occurring (from zero to a maximum of three) depending upon the ordering of the diffusivities and the magnitude of the downstream velocity normal to the shock relative to certain critical velocities downstream of the shock. Where necessary, the approximate solutions are rendered uniformly valid to first order by the insertion of boundary layers, for which the approximate equations are determined to first order. For most of the cases considered, the limiting forms of the integral curves are determined and they are sketched in appropriate three-dimensional phase spaces.

A transformation has been found which reduces steady non-homentropic flows of a compressible fluid to homentropic flows, provided diffusive and gravity effect are negligible. With this transformation, the equation governing steady two-dimensional flows of a compressible fluid with variable entropy in a gravitational field is derived, which is then applied to the study of atmospheric waves in the lee of mountains. The corresponding equation governing swirling axisymmetric flows is also given.

A singularity is encountered in the flow field about two-dimensional and axisymmetric bodies characterized by a sharp corner, where the fluid velocity becomes sonic. Investigation shows that the problem in question belongs, as do many other discontinuity problems, to the family of asymptotic or ‘boundary-layer’ phenomena of mathematical physics. The solution of a first approximation to the flow equations is given by a series in powers of a variable measuring the distance from the corner, with coefficients depending on an appropriate similarity variable. The leading coefficient of the series is independent of three-dimensional and rotationality effects, in complete analogy to the well-known solution of the corner problem in supersonic flow. Detailed results are presented for the leading singularity and for the first two corrections due to rotationality and axial symmetry of the flow.

The formula for transfer of $\overline {\theta^2}$-stuff across the θ-spectrum, which was obtained by Batchelor, Howells & Townsend (1959) for wave-numbers at which molecular conduction is important, is extended here to smaller wave-numbers by means of a simplified general picture of the mechanism involved. The interaction of velocity and temperature fields is represented by a combination of eddy conductivity due to the smaller eddies, and a straining action due to the larger eddies, and this leads to an approximate equation for the θ-spectrum, for a fluid of arbitrary Prandtl number, over at least the equilibrium range of wave-numbers.

A similarity solution is obtained up to the first order in an effective Rayleigh number for the behaviour of very weak thermal vortex rings produced by the rapid release of heat at one point of a large region of fluid.

The laminar pattern of flow is similar to that in ordinary vortex rings, but the temperature decreases outwards from the centre in all directions with some asymmetry about the horizontal plane through the centre, and there is no accumulation of heat into the vortex ring. The vortex propagates slowly in relation to its rate of growth, and the process is dominated by viscous and thermal diffusion.

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 nπA0a0/h √(a20 + A20 (a0 = acoustic velocity).

An experimental study of the incompressible flow near the side edge of a finite flat plate at zero incidence is reported for the Reynolds number range 104 to 106, but comparison with data already published shows that the conclusions are quantitatively valid for Reynolds numbers up to 109. The laminar velocity field is everywhere convex and does not contain any secondary flow other than that of the normal diffusive growth of the layer, but has a logarithmic singularity at the edge where the stress is controlled by the local radius of curvature. The excess skin friction due to the edge is considerably greater than that given by calculations based on the Rayleigh approximation but agrees with a recent Pohlhausen calculation by Varley (1958). Near the edge the high stress and ease of momentum diffusion makes the flow very unstable and turbulent spots occur well upstream of the normal transition zone in the middle of the plate. The spots originate from a nearly point-like region at the edge of the plate and grow linearly in time at the same rate as ordinary turbulent spots to sweep out a narrow tongue of turbulent fluid near the edge until they merge with the normal transition zone. Within this tongue a weak secondary flow of Prandtl's second kind, driven by the anisotropic Reynolds stresses, begins to develop. In fully turbulent flow when the secondary flow is largely localized to within a few boundary-layer thicknesses of the edge the secondary flow velocities are everywhere less than 0.04 of the free-stream velocity. Nevertheless, the secondary flows from each of the side edges interact, regardless of the width of the plate, to increase the total drag coefficient by an amount 0·0004 which is independent both of the Reynolds number and the width of the plate except when it is very narrow. This simple result allows apparent discrepancies between various formulations of the drag coefficient of a finite plate to be reduced to less than ± 1% S.D. Of these formulations Schoenherr's (1932) empirical relation agrees best with the present data.