This paper refers to the work of Moeckel (1952) on the interaction of an oblique shock wave with a shear layer in steady supersonic flow and the work of Chester (1955) and Chisnell (1957) on the propagation of a shock wave down a non-uniform tube. It is shown that their basic results can be obtained by the application of the following simple rule. The relevant equations of motion are first written in characteristic form. Then the rule is to apply the differential relation which must be satisfied by the flow quantities along a characteristic to the flow quantities just behind the shock wave. Together with the shock relations this rule determines the motion of the shock wave. The accuracy of the results for a wide range of problems and for all shock strengths is truly surprising.

The results are exactly the same as were found by the authors cited above. The derivation given here is simpler to perfom (although the original methods were by no means involved) and of somewhat wider application, but the main reason for presenting this discussion is to try to throw further light on these remarkable results.

In discussing the underlying reasons for this rule, it is convenient to use the propagation in a non-uniform tube as a typical example, but applications to a number of problems are given later. A list of some of these appears at the beginning of the introductory section.

Assuming local thermodynamic equilibrium in the fluid, an expression is derived for the rate of destruction of the mean square of the temperature fluctuations by radiative transfer of heat. This takes a particularly simple form (a) if the fluid is effectively transparent over distances equal to the scale of the turbulent motion, when the effect appears as a decay time for temperature fluctuations from the mean, and (b) if the fluid is effectively opaque, when the effect is of an increased conductivity due to radiation. A theory of the interaction of the temperature and velocity fields developed in a previous paper shows that, if the radiative effects are relatively weak, a sudden collapse of the turbulent motion occurs while the flux Richardson number is still less than one. If the radiative effects are strong, the turbulent intensity approaches zero as the flux Richardson number approaches one. The effects of radiation are always to increase the critical value of the ordinary Richardson number. Criteria for fully turbulent motion of an unrestricted flow are given in terms of the gradients of mean velocity and mean temperature and of the rate of radiative cooling. The relevance of these calculations to motions of the atmosphere is briefly discussed.

The plane steady decelerated flow of a dust-gas mixture is analysed in an approximate manner. The problem, which has a five-parameter family of solutions, is reduced to a form such that the analysis can be completed by the integration of a first-order non-linear differential equation and a quadrature. A few integral curves are given and the characterizing features of the flow field are discussed.

Four alternative theoretical treatments of ‘displacement thickness’, and, generally, of the influence of boundary layers and wakes on the flow outside them, are set out, first for two-dimensional, and then for three-dimensional, laminar or turbulent, incompressible flow. They may be called the methods of ‘flow reduction’, ‘equivalent sources’, ‘velocity comparison’ and ‘mean vorticity’.

The principal expression obtained for the displacement thickness δ1 in three-dimensional flow may be written $\delta_1 = \delta_x - \frac{1}{Uh_y} \frac {\partial}{\partial y}\int^x_0 \delta_y dx,$ if, as orthogonal coordinates (x, y) specifying position on the surface, we choose x as the velocity potential of the external flow, and y as a coordinate, constant along the external-flow streamlines, such that hydy is the distance between (x, y) and (x, y + dy); and if also δx and δy are the streamwise and transverse ‘volume-flow thicknesses’ $\delta_x = \frac {1}{U}\int _0^\infty (U - u)\;dz,\;\;\;\;\;\;\delta_y = \frac {1}{U}\int ^\infty_0 v\; dz,$z is the distance from the surface, u and v are the x and y components of velocity, and u takes the value U just outside the boundary layer.

A study is made of the propagation of sound in both a constant gradient shear flow and a turbulent shear flow above a flat surface. Curves are presented showing how, in the case of downstream propagation, the flow gradient tends to channel the sound energy into a narrow layer next to the wall. These results are used in estimating the effect of a flow on the attenuation of sound in a duct with absorbing side walls.

The theory of an ‘ideal dissociating’ gas developed by Lighthill (1957) for conditions of thermodynamic equilibrium is extended to non-equilibrium conditions by postulating a simple rate equation for the dissociation process (including the effects of recombination). This equation contains the ‘equilibrium’ parameter of the Lighthill theory plus a further ‘non-equilibrium’ parameter which determines the time scale of the dissociation phenomena.

The behaviour of this gas is investigated in flow through a strong normal shock wave and past a bluff body. The assumption is made that the gas receives complete excitation of its rotational and vibrational degrees of freedom in an infinitesimally thin region according to the familiar Rankine-Hugoniot shock wave relations before dissociation begins. The variation of the relevant thermodynamic variables downstream of this region is then computed in a few particular cases. The method used in the latter case is an extension of the ‘Newtonian’ theory of hypersonic inviscid flow. In particular, the case of a sphere is treated in some detail. The variation of the shock shape and the ‘stand-off’ distance with the coefficient Λ, which is the ratio of the sphere diameter to the length scale of the dissociation process, is exhibited for conditions extending from completely undissociated flow to dissociated flow in thermal equilibrium. Results would indicate that significant and observable changes from the undissociated values occur, although values for the non-equilibrium parameter are not, at present, available.

Consideration of the structure of wind-generated waves when the duration and fetch of the wind are large suggests that the smaller-scale components of the wave field may be in a condition of statistical equilibrium determined by the requirements for attachment of the crests of the waves. A dimensional analysis, based upon the idea of an equilibrium range in the wave spectrum, shows that for large values of the frequency ω, the spectrum Φ(ω) is of the form $\Phi (\omega) \sim \alpha g^2\omega^{-5}$ where α is an absolute constant. The instantaneous spatial spectrum Ψ (k) is proportional to k−4 for large wave numbers k, which is consistent with the observed occurrence of sharp crests in a well-developed sea, and the loss of energy from the wave system to turbulence and heat is proportional to $\rho _w u^3_*$, where ρw is the water density and u the friction velocity of the wind at the surface. This prediction of the form of Φ(ω) for large ω with α = 7·4×10−3, agrees well with measurements made by Burling (1955).