The mild-slope equation is an effective approximation for treating the combined effects of refraction and diffraction of infinitesimal water waves, for it reduces the spatial dimension of the linear boundary value problem from three to two. We extend this approximation to nonlinear waves up to the second order in wave steepness, in order to simplify the inherently three-dimensional task. Assuming that the geometrical complexity is restricted to a finite, though large, horizontal domain, the hybrid-element method designed earlier for linearized problems is modified for the two-dimensional elliptic boundary-value problems at the second order. This thesis consists of two parts. In Part I, the incident waves are monochromatic. Application is first made to the special case of a a semi-circular peninsula (or a vertical cylinder on a cliff). Effects of the angle of incidence are examined for the free surface height along the cylinder. Numerical results for three examples involving radially varying depth are discussed. In Part II the second-order mild-slope approximation will be further extended for random waves with a broad frequency spectrum. A stochastic approach of Sclavounos is generalized for the prediction of spectral response in harbors. Focuss is on the low-frequency harbor resonance, so the third-order solution is unnecessary. Numerical examples are given for a simple square harbor of constant depth. Effects of harbor entrance are examined. Possible extensions and other applications are discussed.
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