Wave scattering by an infinite trench in the presence of bottom-mounted inverted <b>Π</b>-shaped or floating <b>Π</b>-shaped structure

This article examines the scattering of surface gravity waves by two types of structures: a bottom-mounted inverted Π-shaped structure or a floating Π-shaped structure, in the presence of an infinite trench. The physical phenomenon is formulated as a boundary value problem governed by the modified Helmholtz equation, which is transformed into a system of Fredholm-type integral equations through the eigenfunction expansion technique. To enhance numerical accuracy and convergence, basis functions (such as Chebyshev and ultraspherical Gegenbauer polynomials) multiplied by appropriate weights are incorporated into a multi-term Galerkin approximation. Thus, the model effectively captures the singular behavior of the horizontal fluid velocity near the sharp lower edges of the Π-shaped structure and the corners of the trench. Validation of the model is achieved by comparing its results with existing solutions from the literature on water wave problems in specific limiting scenarios. Numerical results for reflection and transmission coefficients, mean drift forces, and free surface elevation are presented graphically. These results provide a comprehensive understanding of the influence of various non-dimensional wave and structural parameters on these hydrodynamic characteristics. Notably, the floating Π-shaped structure achieves a more significant reduction in free surface elevation in the transmitted region compared to the bottom-mounted inverted Π-shaped structure.