Influence of Bed Variations on Linear Wave Propagation beyond the Mild Slope Condition

As water waves travel from deep to shallow waters, they experience increased nonlinearity and decreased dispersion due to the reduced water depth. While the impact of bed slope on wave propagation celerity is documented, it is often overlooked in commonly used depth-integrated wave models. This study uses the WKB approximation and solves higher-order slope-related terms to analyze the influence of varying depth, including the gradient and laplacian of water depth. One result is an extended longwave for linear wave reflection and transmission on a ramp to deeper waters. The main outcome and focus of this work, however, is a new, simple analytical expression for linear dispersion that includes bed variations. The results are applied to two cases: wind-generated water wave propagation in the nearshore, emphasizing corrections to bathymetry inversion methods, and tsunami propagation over the continental slope, highlighting the limitations of neglecting slope on dispersion and the significant role of the Laplacian of water depth.