The evolution of large non-breaking waves in intermediate and shallow water. I. Numerical calculations of uni-directional seas

Abstract

In deep water it is well known that the evolution of the largest waves in realistic, broad-banded frequency spectra is governed by dispersive focusing. However, as the water depth reduces this process weakens and the relative significance of wave modulation is shown to be increasingly important. This leads to very different extreme wave groups, the properties of which are critically dependent upon the local nonlinearity. To explore these effects, and to provide a physical explanation for their occurrence, two complementary wave models are employed. The combined numerical results show that the nature of large uni-directional waves varies depending on the relative water depth. As the water depth reduces, both the bound and resonant interactions become more significant. However, the third-order resonant terms have the most profound influence. By modifying both the amplitude and phase of the underlying linear wave components, the largest waves arise as a local instability within a truncated quasi-regular wave train; the latter appearing because of an initial narrowing of the underlying frequency spectrum. Furthermore, the numerical calculations show that, with large changes in both the spectral shape and the phasing of the wave components, both the maximum crest elevations and wave heights are less than those predicted by linear theory.