When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.

This paper deals with the stability of a two-dimensional laminar jet against the infinitesimal antisymmetric disturbance. The curve of the neutral stability in the (α, R)-plane (α, the wave-number; R, Reynolds number) is calculated using two different methods for the different parts of the curve; the solution is developed in powers of (αR)−1 for obtaining the upper branch of the curve and in powers of αR for the lower branch.

The asymptotic behaviour of these branches is that for branch I,$\alpha \rightarrow 2, \;\; c \rightarrow \frac{2}{3}$ for $R \rightarrow \infty$; and for branch II, $R \sim 1\cdot12\alpha^{-1|2},\; c \sim 1\cdot 20 \alpha^2$ for α → 0. Some discussion is given on the validity of the basic assumption of the stability theory in relation to the numerical result obtained here.

The equations of motion for a chemically reacting gas in the absence of viscosity and heat conduction are set up. It is shown that the characteristic speed defined by this set of equations is the high-frequency limit of the phase velocity of sound waves as long as the reaction rate is finite. At infinite reaction rate (chemical equilibrium) the characteristics suddenly change to the lowfrequency sound speed. The nature of this transition is discussed in connection with a recent paper of Resler (1957).

A new water tunnel, incorporating a slotted wall working section, was found to suffer from severe vibration. A theoretical explanation is given for this, together with experimental evidence gleaned from this water tunnel and a small wind tunnel. It is shown that the oscillations are hydrodynamic in origin and are associated with the slotted wall design. Consideration is given to methods of elimination or reduction of the oscillations.

In this paper the propagation of long waves (tides) into a canal is studied. The canal is assumed to be rotating at constant angular velocity and the depth of the fluid is uniform.

The rate of rotation can have a considerable effect on the amplitude of the unattenuated modes in the channel. This is due partly to the modification of the known solution when the rotation is zero, and partly to the fact that even if there is only a single semi-infinite barrier unattenuated waves of a special type (Kelvin waves) may be propagated in the rotating system into what is normally called the ‘shadow’ region behind the barrier (Crease 1956).

The purpose of this theoretical investigation is to seek a partial explanation of the behaviour of tides and storm surges in the North Sea. For this reason two models of this region are discussed. In Example I the model is of two parallel semi-infinite barriers in the path of a plane progressive wave, and in Example II there are two parallel barriers, one semi-infinite and the other infinite.

The ratio of the observed height of the semi-diurnal M2 tide in the North Sea to the height of the incident tide from the Atlantic lies between the results predicted by the two models and is in closer agreement with the model of Example I.

A formally exact solution for the thermal boundary layer on a non-isothermal surface subjected to non-uniform free stream velocity is presented in the form of a series. It is demonstrated that the solution can be recast in terms of universal functions, which are independent of the wall temperature data of particular problems, and which depend only on a single parameter characterizing the variation of the free stream velocity.

In a recent paper Carrier & Greenspan (1958) showed that, within the framework of the non-linear shallow-water theory, there exist waves which do not break as they climb a sloping beach. The formation of a shock or bore is dependent on a variety of factors (wave shape, particle velocity, etc.) and, as yet, no general criteria for breaking have been found. In this paper, we consider waves which propagate shoreward into quiescent water; it is shown that any compressive wave (a wave of positive amplitude) which has a non-zero slope at the wave-front eventually breaks before reaching the coastline. In fact, an explicit relation is obtained between the initial conditions and the position where breaking occurs.