Source code of analytical solution

Over the past three decades, analytical solutions for tide-induced groundwater hydraulics in coastal leaky aquifers have been predominantly limited to one- or two-layer systems, typically failing to address the complex heterogeneity patterns characterized by vertical layering and horizontal zonal variations in hydraulic properties. This study presents a generalized analytical model that accommodatesan arbitrary number ofleaky aquifer layers and horizontally segmented zones. To enhance its generality, the model integrates mixed boundary conditions at the lateral boundaries, zone-matching periodic-head boundary conditions at the top boundary, and tidal loading effects driven by top-boundary periodic head. The solution derivation begins with the formulation of a system of ordinary differential equations thatgovern either steady-state or periodic component of the tidal aquifer head within each horizontal zone.Matrix eigenvalue analysis approach is employed to derive the general solutions, and recursive relationships for involved constant coefficients are established to ensure head and flux continuity across zone interfaces.To circumvent matrix singularity arising from inversion computations, an innovative local coordinate system methodology is introduced. The accuracy of the analytical solutionis verified against numerical simulations. Theoretical analysis revealsthat the tidal hydraulic response is sensitive to both vertical layering and horizontal zonal heterogeneity. Most notably,whenaquitardwindows act as preferential conduits, concentrated cross-aquifer tidal wave propagation could cause an anomalous amplitude increase in the adjacent aquifer exhibiting a lower tidal headamplitude. Thepresented analytical framework demonstrates generalityand versatility in the investigation of groundwater-surface water interactions,offering valuable insights for groundwater management, protection, and engineering control in riparian regions.