Exponentially Reduced Reflection (ERR) for Water Waves: Elliptical Theory of Ocean Waves

In coastal areas with decreasing water depth, surface wave dynamics are strongly affected by seabed topography, particularly by the local inclination. Traditional linear wave theories (Airy/Laplace, Stokes) are limited to flat sea beds and disregard the mass conservation theorem and phase differences between incident and reflected waves, leading to conservative design estimates for maritime structures. This paper presents the extended linear wave theory - Exponentially Reduced Reflection (ERR) -, which analytically accounts for seabed inclination (0° ≤ α ≤ 90°) and corresponding phase shifts (Δ) in irregular wave reflection. It is based on a compilation of research results concerning the topics of wave resonance and anomalous dispersion in theory and nature, as well as the empirical and theoretical investigation of reflection on technically smooth inclined surfaces. The theory utilizes complex reflection coefficients and spectral analysis of energy densities to characterize partially standing waves formed by interference between incident and reflected components. Empirical studies using high-resolution wave gauge arrays and physical models on slopes confirm the ERR theory's capacity to capture variable phase jumps and energy distribution, revealing elliptical orbital particle paths whose major axes parallel the seabed. The ERR method enables accurate estimation of wave heights, orbital velocities, and accelerations along inclined and complex coastal profiles, facilitating more precise and less over-conservative design of offshore structures. Furthermore, analogies to electromagnetic reflection phenomena underscore the phase-dependent nature of coastal wave reflection. The paper offers practical vector-based graphical procedures for engineers to estimate local hydrodynamic loads and demonstrates the ERR theory's enhanced physical consistency, especially in shoaling and breaking zones. Applications of ERR support both structural design and prediction of extreme coastal water levels in the context of climate change-induced sea level rise.