O. M. Phillips

The equilibrium range in the spectrum of wind-generated waves

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

On the generation of waves by turbulent wind

\Phi (\omega) \sim \alpha g^2\omega^{-5}

On the dynamics of unsteady gravity waves of finite amplitude Part 1. The elementary interactions

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

On the penetration of a turbulent layer into stratified fluid

\rho _w u^3_*

On the incipient breaking of small scale waves

, where ρ w 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).

On turbulent entrainment at a stable density interface

A theory is initiated for the generation of waves upon a water surface, originally at rest, by a random distribution of normal pressure associated with the onset of a turbulent wind. Corrlations between air and water motions are neglected and the water is assumed to be inviscid, so that the motion of the water, starting from rest, is irrotational. It is found that waves develop most rapidly by means of a resonance mechanism which occurs when a component of the surface pressure distribution moves at the same speed as the free surface wave with the same wave-number. The development of the waves is conveniently considered in two stages, in which the time elapsed is respectively less or greater than the time of development of the pressure fluctuations. An expression is given for the wave spectrum in the initial stage of development (§ 3.2), and it is shown that the most prominent waves are ripples of wavelength λ cr = 1·7 cm, corresponding to the minimum phase velocity c = (4 gT /ρ) 1/4 and moving in directions cos -1 ( c / U c ) to that of the mean wind, where U c is the ‘convection velocity’ of the surface pressure fluctuations of length scale λ cr or approximately the mean wind speed at a height λ cr above the surface. Observations by Roll (1951) have shown the existence under appropriate conditions, of waves qualitatively similar to those predicted by the theory. Most of the growth of gravity waves occurs in the second, or principal stage of development, which continues until the waves grow so high that non-linear effects become important. An expression for the wave spectrum is derived (§ 4.1), from which follows the result

WAVE BREAKING IN THE PRESENCE OF WIND DRIFT AND SWELL

\overline {\xi ^2} \sim \frac {\overline{p^2}t} {2 \surd 2\rho ^2 U_c g},

The intnesity of Aeolian tones

where

The dispersion of short wavelets in the presence of a dominant long wave

\overline {\xi ^2}

Wave interactions - the evolution of an idea

is the mean square surface displacement,