The operational calculus mild-slope equation

This study presents the development and analysis of a new mild-slope wave equation derived using an operational-calculus framework. The derivation systematically incorporates both linear and nonlinear bottom variations through the introduction of a detuning-from-resonance operator, which enables a structured expansion of the governing equations. Although the formulation ultimately leads to a system of coupled second-order equations, retaining terms up to second order in this operator initially yields coupled fourth-order partial differential equations with bottom derivatives up to the fourth order. These equations are then reduced to a second-order form without compromising accuracy. This reduction retains the accuracy of the original model but subtly alters its mathematical structure, replacing the original elliptic formulation with a coupled second-order system. The proposed model is validated through numerical simulations of three benchmark cases: normally incident wave propagation over a sloping planar bathymetry, Roseau's non-planar bathymetry, and the quasi-three-dimensional case of oblique incidence, all of which have analytical solutions. The results demonstrate that the operational calculus-based equation improves accuracy relative to the complementary mild-slope equation, particularly for steeper bathymetric variations, making it a valuable tool for modeling wave behavior over varying topographies.