Analytical Solutions of Modified Shallow Water Equations for Atmospheric-Forced Seiches in Non-Uniform Basins

This study examines water wave oscillations in semi-enclosed basins using a modified shallow water equation (SWE) that includes atmospheric pressure forcing. We simplify the governing equations to a non-homogeneous Sturm–Liouville problem, addressing spatial and temporal variations in pressure. We present new analytical solutions for basins with non-uniform bathymetry, specifically rectangular, semi-parabolic, and parabolic profiles cases that have received limited attention. Our analysis of the eigenfunctions and eigenfrequencies highlights the modal structures and resonance periods. Our findings show that semi-parabolic bathymetry leads to an asymmetric oscillatory response, with greater amplitudes in deeper regions and rapid decay in shallower areas. This curvature acts as a spatial filter, redistributing modal energy. Notably, the semi-parabolic profile achieves the highest amplification under atmospheric pressure, suggesting stronger coupling between external forcing and depth-dependent modal structures compared to rectangular or moderately parabolic basins. These results enhance our understanding of pressure-driven oscillations in coastal and harbor environments, underscoring the significance of bathymetric non-uniformity in shaping resonance characteristics.