Interaction of ocean waves of nearly equal frequencies and the effect on pressure

Abstract

We study the superposition of a train of freely traveling waves in a form that includes the possibility for each wave of complex amplitude An to have a ‘sister’ wave of complex amplitude Bn with equal frequency and opposite direction. For an ideal, incompressible and homogeneous fluid, we consider three-dimensional flows that are irrotational and spaceperiodic. Through a weakly nonlinear analysis we obtain full second-order expressions for the free-surface elevation, the velocity potential and the dynamic pressure. Then we generalize and unify all related expressions in the literature, without any assumption on the water depth. When the frequencies of the surface waves of nearly opposite directions are nearly equal, a second-order pressure can be felt all the way to the sea bottom. Hence, in particular, we apply a theoretical analysis on the dynamic pressure obtained, and we quantify the degree of nearness in amplitude, frequency and incidence angle that must be reached to observe the phenomenon. Such phenomena of the second-order pressure, independent of the depth, have been supposed to be at the origin of so-called secondary microseisms. A comparison with real data for pressure induced by waves in the ocean is also presented.