WATER-WAVES IN A DEEP SQUARE BASIN

The form and evolution of three-dimensional standing waves in deep wat er are calculated analytically from Zakharov's equation and computatio nally from the full nonlinear boundary value problem. The water is con tained in a basin with a square cross-section, when three-dimensional properties are significant because the natural frequencies of waves in the two directions perpendicular to pairs of sides are the same. It i s found that non-periodic standing waves commonly follow forms of cycl ic recurrence over long times. The two-dimensional Stokes type of peri odic standing waves (dominated by the fundamental harmonic) are shown to be unstable to three-dimensional disturbances, but over long times the waves return cyclically close to their initial state. In contrast, the three-dimensional Stokes type of periodic standing waves are foun d to be stable to small disturbances. New two-dimensional periodic sta nding waves with amplitude maxima at other than the fundamental harmon ic have been investigated recently (Bryant & Stiassnie 1994). The equi valent three-dimensional standing waves are described here. The new tw o-dimensional periodic standing waves, like the two-dimensional Stokes standing waves, are found to be unstable to three-dimensional disturb ances, and to exhibit cyclic recurrence over long times. Only some of the new three-dimensional periodic standing waves are found to be stab le to small disturbances.