In this Lecture we present some recent results of a mathematical nature, not well known, on the mechanics of water waves. Three separate subjects are considered. In the first, the transport of energy in uniform and modulated water waves of sinusoidal form has been considered and it has been shown that each of the three forms of energy constituting the waves (i.e. kinetic, gravity potential, and surface tension energy) travel at different speeds, and that the group velocity, cg, is the energy weighted average of these speeds, depth and time averaged in the case of kinetic energy. It has also been shown that the time averaged kinetic energy travels at every depth horizontally at a speed equal (deep water) or faster than the wave itself. In contrast, the propagation speed along the surface of the gravity potential energy is null, while the surface tension energy travels forward along the wave surface everywhere at twice the wave velocity. Finally, the propagation of both sinusoidal waves and wave group envelopes are shown to be made possible by the vertical transport of kinetic energy to, or in the case of capillaries, from the free surface, where it provides the balance in surface energy just necessary to allow the propagation of the wave. In the second, the mathematical criterion for the inception of the breaking process in a modulated sinusoidal wave in water of arbitrary uniform depth is precisely derived, and shown to be identical to the simple condition earlier (1994) determined from numerical experiments on the evolution of modulated waves (Yao, Wang, and Tulin): ”that upon passing through the peak of a modulation group, when the orbital velocity at the wave crest, u(crest), exceeds the wave group velocity, cg, then the wave crest and trough both rise, the front face steepens, the wave crest sharpens, and eventually a jet

forms at the crest, leading finally to splashing and a breakdown of the wave.” The present mathematical demonstration of this failure of progressive motion reveals that the modulated sinusoidal wave in water of any depth is incompatible with kinematical requirements while passing through the peak of a modulation, once, u(crest) ¿ cg. Experimental verification of this result is also shown, based on stereo video measurements of surface velocities in modulating breaking waves in a large wave tank. In the third, the necessary modification to the cubic NLS model of the wave evolution equations is shown which accounts properly for both the loss of energy and momentum in the organized wave motion due to breaking, and the importance of these terms is discussed.