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

This paper is concerned with the non-linear interactions between pairs of intersecting gravity wave trains of arbitrary wavelength and direction on the surface of water whose depth is large compared with any of the wavelengths involved. An equation is set up to describe the time history of the Fourier components of the surface displacement in which are retained terms whose magnitude is of order (slope) 2 relative to the linear (first-order) terms. The second-order terms give rise to Fourier components with wave-numbers and frequencies formed by the sums and differences of those of the primary components, and the amplitudes of these secondary components is always bounded in time and small in magnitude. The phase velocity of the secondary components is always different from the phase velocity of a free infinitesimal wave of the same wave-number. However, the third-order terms can give rise to tertiary components whose phase velocity is equal to the phase velocity of a free infinitesimal wave of the same wave-number, and when this condition is satisfied the amplitude of the tertiary component grows linearly with time in a resonant manner, and there is a continuing flux of potential energy from one wave-number to another. The time scale of the growth of the tertiary component is of order of the (−2)-power of the geometric mean of the primary wave slopes times the period of the tertiary wave. The Stokes permanent wave appears as a special case, in which the tertiary wave-number is the same as that of the primary, but its phase is advanced by ½π. The energy transfer to the tertiary component in this case is usually interpreted as an increase in the phase velocity of the wave. The dynamical interactions in water of finite depth are considered briefly, and it is shown that the amplitude of the secondary components becomes large (though bounded in time) as the water depth becomes smaller than the wave-length of the longest primary wave.