In a previous paper, Phillips (1960) showed that two or three trains of gravity waves may interact so as to produce a fourth (tertiary) wave whose wave-number is different from any of three primary wave-numbers k1, k2, k3, and whose amplitude grows in time. Such resonant interactions may produce an appreciable modification of the spectrum of ocean waves within a few hours. In this paper, by a slightly different method, the interaction is calculated in detail for the simplest possible case: when two of the three primary wave-numbers are equal (k3 = k1).

It is found that, when k1 and k2 are parallel or antiparallel, the interaction vanishes unless k1 = k2. Generally, if θ denotes the angle between k1 and k2, the rate of growth of the tertiary wave with time is a maximum when θ [eDot ] 17°; the rate of growth with horizontal distance is a maximum when θ [eDot ] 24°. The calculations show that it should be possible to detect the tertiary wave in the laboratory.

It is shown that, when two trains of waves in deep water interact, the phase velocity of each is modified by the presence of the other. The change in phase velocity is of second order and is distinct from the increase predicted by Stokes for a single wave train. When the wave trains are moving in the same direction, the increase in velocity Δc2 of the wave with amplitude a2, wave-number k2 and frequency α2 resulting from the interaction with the wave (a1, k1, σ1) is given by Δc2 = a21k1σ1, provided k1 < k2. If k1 > k2, then Δc2 is given by the same expression multiplied by k2/k1. If the directions of propagation are opposed, the phase velocities are decreased by the same amount. These expressions are extended to give the increase (or decrease) in velocity due to a continuous spectrum of waves all travelling in the same (or opposite) direction.

The convective diffusion of matter from a stationary object to a moving fluid stream is distinct from pure heat transfer because of the appearance of a finite interfacial velocity at the solid surface. This velocity is related to the rate of mass transfer by a dimensionless group B in such a way that for −1 < B < 0 the transfer is from the bulk to the surface while for 0 < B < ∞ the transfer is from the surface to the main stream. In this paper, asymptotic solutions to the two-dimensional laminar boundary-layer equations are developed for the case B [Gt ] 1, and for rather general systems. It is shown that in most instances the asymptotic expressions for the rate of mass transfer become accurate when B > 3 and that the transition region between the pure heat-transfer analogy (B ∼ 0) and the B [Gt ] 1 asymptote may be described by a simple graphical interpolation. These results may easily be extended to three-dimensional surfaces of revolution by the usual co-ordinate transformations of boundary-layer theory.

The shape of a bubble of one liquid inside a denser body of liquid which rotate rigidly together is determined, the effect of gravity being neglected. When the anular velocity of the liquids is zero the bubble assumes a spherical form, and with increasing angular velocity the bubble flattens at the equator and the length increases. It is found that the length of the bubble is asymptotically proportional to the four-thirds power of the angular velocity. If the speed of the rotation is held constant and the volume is increased, then the bubble elongates, the radius approaches a limiting value, and the bubble length increases almost linearly with the volume. This result suggests a method whereby the interfacial surface tension can be measured.

In the second part of the paper the stability of a long bubble subjected to small amplitude axisymmetric disturbances sinusoidal in the axial direction is investigated. The relation between the wave-number and angular velocity for neutral stability is elliptic. When account is taken of the decrease in the radius of the undisturbed bubble with increase in the angular velocity, it is found that the bubble is stable to all wave-lengths provided the radius attains at least 63% of the limiting value. A criterion is then found for the minimum length of the bubble consistent with stability.

The flow past circular cylinders of finite length, supported at one end and lying with their axes perpendicular to a uniform stream, has been investigated in a supersonic stream at Mach number 1.96 and also in a low-speed stream. In both stream it was found that the flow past the cylinders could be divided into three regions: (a) a central region, (b) that near the free end of the cylinder, and (c) that near the supported end. The locations of the second and third regions were found to be almost independent of the cylinder length-to-diameter ratio, provided that this exceeded 4, while the flow within and the extent of the first region were dependent on this ratio. Form-drag coefficients determined in the central region in the supersonic flow were in close agreement with the values determined at the same Mach number by other workers. In the low-speed flow the local form-drag coefficients were dependent on length-to-diameter ratio and were always less than that of an infinite-length cylinder at the same Reynolds number.

When using a hot wire for velocity measurements close to a solid boundary, errors may be introduced if the effect of the boundary on the rate of heat loss from the wire is ignored. An experimental determination of the effect is described, in which a hot wire was mounted at various distances from a metal surface forming one wall of a two-dimensional channel. The rate of heat loss was determined electrically, and the air velocity at the wire found from the known laminar velocity profile. The application of the results to turbulent flows is discussed briefly.

This paper considers the stability of a barotropic current on a beta earth. The motion is assumed to be horizontal, non-divergent and barotropic. The current is taken to be of the form U(y) = A sech2by+B. The perturbations are required to approach zero as y approaches ± ∞. We introduce the non-dimensional wave-number l and a parameter χ, which is a measure of the rotation effect. χ is inversely proportional to β.

There are only two kinds of perturbations: symmetric disturbances (those with maximum amplitude at y = 0) and antisymmetric disturbances (those with zero amplitude at y = 0). We find the neutral curve in the (χ, l2)-plane for both types of disturbances. The rates of amplification in the immediate vicinity of the neutral curves are also found. It is seen that the beta effect, which is due to the earth's rotation, tends to stabilize the current. For the symmetric disturbances we find a band of unstable wavelengths when χ > 1/2; and for large χ the estimated curve of the maximum value of the imaginary part of the phase velocity is asymptotic to the lower branch of the neutral curve. The antisymmetric disturbances are more stable than the symmetric disturbances.

The paper deals with the initial motion of a two-dimensional bubble starting from rest in the form of a cylinder with its axis horizontal. The theory is based on the assumptions of irrotational motion in the liquid round the bubble, constant pressure within the bubble, and small displacements from the cylindrical form. This theory predicts that the bubble should rise with the acceleration of gravity, over a distance of at least the initial bubble radius, and that a tongue of liquid should be projected up from the base of the bubble into its interior. These predictions are confirmed by experiments which also show how the vorticity necessary for steady motion in the spherical-cap form is generated by the detachment of two small bubbles from the back of the main bubble.

Coexistent processes of heat, mass and momentum transfer operative within a combustible turbulent boundary layer have been experimentally investigated. The boundary layer was established on a porous cylinder mounted in a low-speed wind tunnel with its long axis in the flow direction. Methane was transpired into the boundary layer and ignited. Results indicate that the dimensionless transfer numbers corresponding to the three transfer processes can be correlated by the formula 0.038Re−0.2 to within ± 30% of measured values so that a rough numerical analogy exists among all three processes. The effect of mass injection on the skin friction coefficient is reasonably well accounted for by available theory. No effect of mass injection was found on the values of heat and mass transfer parameters. Finally, there was a lack of evidence indicating any sort of reaction-generated turbulence or that the experimentally demonstrated disturbance of the viscous layer by mass injection substantially affected the transport phenomena.

It is shown that the draining of liquid from a vertical tank produces a small but positive damping of free-surface oscillations. This conclusion is consistent with the experimental observations of Brooks.

This paper describes the results of examining the influence of radiative heat transfer on turbulent natural convection above fires in an atmosphere of constant potential temperature, under both the ‘opaque’ and ‘transparent’ approximations. It turns out that on the basis of the over-all approximations introduced in this investigation, the former case reduces to that of no radiative transfer. For the latter case, the initial fire size, the energy release rate (initial temperature difference) and the absorption coefficient have been regarded as independent parameters. The solution curves presented cover a range over which these parameters are expected to vary in practice.

A study is made of the propagation of acoustic waves in a semi-infinite expanse of radiating gas on one side of an infinite, plane, radiating wall. A solution is found, in particular, for the case of sinusoidal oscillations in both position and temperature of the wall. The solution is based on a single linear integro-differential equation that plays the same role here as does the classical wave equation in equilibrium acoustic theory. The solution is applicable throughout the range from a completely transparent to a completely opaque gas and from very low to very high temperatures. The solution appears, in general, as the sum of two types of travelling waves: (1) an essentially classical sound-wave, but with a slightly altered speed and a small amount of damping and (2) a radiation-induced wave whose speed and damping may be either large or small, depending on the temperature and absorptivity of the gas. Since the waves are coupled, both types will usually be present together, even in the special cases of pure motion or pure temperature variation of the wall.