Extreme waves in shallow and intermediate water depths

This paper concerns the description of large transient waves in shallow and intermediate water depths. It builds upon recent advances in the description of deep water waves, and provides the first quantitative comparisons with a new kinematic model which has hitherto only been validated in deep water conditions. A new series of experimental observations is presented in which the wave components within a random (or irregular) sea state are focused to produce a large transient wave group. Comparisons with both linear and second-order solutions suggest that non-linear wave-wave interactions produce a steeper wave envelope in which the central wave crest is higher and narrower, while the adjacent wave troughs are broader and less deep. Spectral analysis of the measured water surface elevation suggest that in addition to the development of significant long-wave energy, there is also a transfer of energy to the shorter, high frequency, harmonics. This energy redistribution has a significant effect upon the underlying kinematics. In particular, the near-bed velocities are shown to be in good agreement with the second-order solution which includes the long-wave (or frequency-difference terms) first identified by Longuet-Higgins and Stewart. Furthermore, the near-surface kinematics are shown to be highly non-linear. Indeed, a linear wave theory, based upon a correct representation of the freely propagating linear harmonics, is generally found to under-estimate the near-surface horizontal velocity. However, if the linear calculations are based upon a measured wave spectrum, the kinematic predictions suffer from high frequency contamination and require empirical correction. In contrast, the non-linear numerical model presented by Baldock and Swan provides a good description of the measured kinematics over the entire water depth.