Parabolic equation models are constrained by their fixed principal propagation direction, limiting wave fields to small angles. To overcome this limitation, this study proposes two modeling approaches based on a new dispersive nonlinear mild-slope equation model that enable wave propagation across a broad range of directions. The first approach integrates a minimax approximation for linear terms with nonlinear summation under a specialized ordering system, resulting in a higher-order parabolic model. The second approach extends the parabolic equation by incorporating alongshore wavenumber components through Fourier decomposition and modifies the inverse Fourier transform terms with additional forcing to account for interactions between lateral bottom variations and the wave field. We validate the proposed models through comparisons against laboratory experiments involving wave focusing by a topographical lens, an elliptic shoal, and a circular shoal. Overall, the proposed models enhance the prediction of wave propagation under a variety of conditions.

The model for ocean surface wave propagation can be formulated either in the form of deterministic models or stochastic models. The stochastic models appear to be particularly attractive in the global domain due to their computational efficiency. However, in the nearshore region, the phase becomes highly correlated, and the phase information therefore becomes critical. Therefore, a simplified consistent nonlinear mild-slope equation model has been developed in order to take advantage of the deterministic model for handling phase information, as well as the stochastic model for numerical simplicity. We demonstrate the advanced performance of the present model for random waves by comparing it with laboratory data and previous models.

A new nonlinear frequency-domain model based on the mild-slope equation is outlined. The model is an enhancement over previous work in that a closer correspondence between scaling of nonlinearity and horizontal variation of bathymetry is made relative to earlier models. This results in additional terms in the nonlinear summation terms of the model, as amplitude gradient terms are required in order to formulate a consistent model. From the resulting elliptic model, a parabolic approximation is developed in order to efficiently model the equations. Comparisons between the present model, previously-formulated models, and experimental data show that the present model does evidence improvement in performance over previous models.

A large-scale physical model was created in Oregon State University's Large Wave Flume to collect an extensive dataset measuring wave-induced horizontal and vertical forces on an idealized coastal structure. Water depth was held constant while wave conditions included regular, irregular, and transient (tsunami-like) waves with different significant wave heights and peak periods for each test. The elevation of the base of the test specimen with respect to the stillwater depth (air gap) was also varied from at-grade to 0.28 m above the stillwater level to better understand the effects of raising or lowering a nearshore structure on increasing or decreasing the horizontal and vertical wave forces. Results indicate that while both horizontal and vertical forces tend to increase with increasing significant wave height, the maximum and top 0.4% of forces increased disproportionally to other characteristic values such as the mean or top 10%. As expected, the horizontal force increased as the test specimen was more deeply submerged and decreased as the structure was elevated to larger air gaps above the stillwater level. However, this trend was not true for the vertical force, which was maximized when the elevation of the base of the structure was equal to the elevation of the stillwater depth. Small wave heights were characterized by low horizontal to vertical force ratios, highlighting the importance of considering vertical wave forces in addition to horizontal wave forces in the design of coastal structures. The findings and data presented here may be used by city planners, engineers, and numerical modelers, for future analyses, informed coastal design, and numerical benchmarking to work toward enabling more resilient nearcoast structures.