Most methods of calculating steep gravity waves (of less than the maximum height) encounter difficulties when the radius of curvature R at the crest becomes small compared with the wavelength L, or some other typical length scale. This paper describes a new method of calculation valid when R/L is small.

For deep-water waves, a parameter ε is defined as equal to q/2½c0, where q is the particle speed at the wave crest, in a frame of reference moving with the phase speed c. Hence ε is of order (R/L)½. Three zones are distinguished: (1) an inner zone of linear dimensions ε2L near the crest, where the flow is described by the inner solution found previously by Longuet-Higgins & Fox (1977); (2) an outer zone of dimensions O(L) where the flow is given by a perturbed form of Michell's solution for the highest wave; and (3) a matching zone of width O(L). The matching procedure involves complex powers of ε.

The resulting expression for the square of the phase velocity is found to be \[ c^2 = (g/k)\{1.1931-1.18\epsilon^3\cos(2.143\ln \epsilon + 2.22)\} \] (see figures 5a, b), which is in remarkable agreement with independent calculations based on high-order series. In particular, the existence of turning-points in the phase velocity as a function of wave height is confirmed.

Similar expressions, valid to order ε3, are found for the wave height, the potential and kinetic energies and the momentum flux or impulse of the wave.

The velocity field is extended analytically across the free surface, revealing the existence of branch-points of order ½, as predicted by Grant (1973).

It is well known that the vortices shed from a circular cylinder lock on in frequency to the vibrations when the cylinder is forced to vibrate or is naturally excited to sufficient amplitudes by flow-induced forces. This paper presents a model for a universal wake Strouhal number, valid in the subcritical range of Reynolds numbers, for both forced and vortex-excited oscillations in the locking-on regime. The Strouhal numbers thus obtained are constant at St* = 0·178 over the range of wake Reynolds numbers Re* = 700-5 × 104. This value is in good agreement with the results obtained by Roshko (1954a) and Bearman (1967) for stationary circular cylinders and other bluff bodies in the same range of Reynolds numbers. A correspondence between the amplification of the cylinder base pressure, drag and vortex circulation is demonstrated over a wide range of frequencies and for vibration amplitudes up to a full cylinder diameter (peak to peak). The fraction ε of the shed vorticity in the individual vortices is found to be dependent upon the base-pressure parameter K = (1 − Cpb)½. Consequently, ε is also a function of the amplitude and frequency of the vibrations in the locking-on regime.

A model of Wilson Cloud formation following a low-altitude nuclear detonation is developed. It is shown that, for detonation yields between 10−3 kt and 100 kt, simple scaling laws characterize the evolution and physical properties of the Wilson Cloud.