Second-order near trapping of water waves by a square array of vertical columns in bi-chromatic seas

Near trapping of water waves by an array of closely spaced columns is a hydrodynamic phenomenon with practical applications. A numerical investigation is conducted for the second-order interaction of bi-chromatic waves with a square array of vertical columns. The cross-sectional shape of the columns can be either circular or square with rounded corners. Numerical results indicate that a four-column cluster can amplify free-surface elevation through a near-trapping mechanism. Three lower-order near-trapped modes, namely p=0, 2 and 3 modes, are primarily concerned, among which the amplification corresponding to the p=2 mode is most pronounced. In addition, the four-column cluster can create both the linear and second-order near trappings. The latter could occur when the frequency sum of a pair of incident waves is consistent with the near-trapping frequency. The linear and second-order near-trapped modes own common characteristics. The second-order near-trapping response is predominated by the 'potential wave' component, while the 'quadratic wave' component plays a secondary role. Meanwhile, the effect of frequency difference on the mode shape in the second order is not evident. It is also noted that a decrease in the corner ratio can enhance the hydrodynamic interference effect among the columns and, in turn, evidently promote the near-trapping response for the p=2 mode. An approximate approach is developed for the second-order near-trapping response, in which the 'quadratic wave' component is evaluated exactly. In contrast, the complicated 'potential wave' component corresponding to a pair of incident waves with distinct frequencies is approximated by that in mono-chromatic seas with the mean frequency. This approach is based on the hypothesis of weakly scattering of linear wave fields. Numerical results indicate that such approximation offers a reasonable alternative for the near-trapped modes in the low-frequency region.