Boundary element formulation of the Mild-Slope Equation for harmonic water waves propagating over unidirectional variable bathymetries

TL;DR

This work develops a boundary-element formulation for the Mild-Slope Equation in domains with unidirectionally varying bathymetry by leveraging Belibassakis' Green's function and a frequency-domain fundamental solution. The core approach combines a Fourier transform in the lateral direction with a 1D transformed problem in the along-bathymetry direction, solved via FEM, and enforces open-domain radiation conditions through a boundary-integral framework. The paper demonstrates accurate reproduction of refraction, diffraction, reflection and shoaling through validation against constant-depth theory and a suite of numerical experiments, including channel shoaling, cylinder scattering, elliptic shoal coupling, and harbor resonance, while highlighting the potential for FEM-BEM coupling to handle more complex bathymetries and absorbing boundaries. The results indicate high accuracy and practical applicability for coastal-engineering problems involving variable bathymetry, enabling robust open-boundary MSE simulations and integration with other numerical schemes for multi-domain problems.

Abstract

This paper presents a boundary element formulation for the solution of the Mild-Slope equation in wave propagation problems with variable water depth in one direction. Based on the Green's function approximation proposed by Belibassakis \cite{Belibassakis2000}, a complete fundamental-solution kernel is developed and combined with a boundary element scheme for the solution of water wave propagation problems in closed and open domains where the bathymetry changes arbitrarily and smoothly in a preferential direction. The ability of the proposed formulation to accurately represent wave phenomena like refraction, reflection, diffraction and shoaling, is demonstrated with the solution of some example problems, in which arbitrary geometries and variable seabed profiles with slopes up to 1:3 are considered. The obtained results are also compared with theoretical solutions, showing an excellent agreement that demonstrates its potential.

Figures (18)

• Figure 1: Wave number variation in the x-direction for a fixed wave frequency due to a monotonically decreasing water depth profile $h(x)$. Wave number is higher where water depth is lower as dictated by the dispersion relation
• Figure 2: Antisymmetric integration path $C$ in the complex plane avoiding the $\pm\hat{k}_1$ and $\pm\hat{k}_3$ roots lying on the real axis
• Figure 3: Approximation and decomposition into linear paths of the integration path in the complex $\xi$-plane used to compute the inverse Fourier transform of the function $\Psi$
• Figure 4: Comparison of analytical and numerical fundamental solution $\psi$ for constant water depth. Cross section of the 2D fundamental solution along the lines $y=0$ (top) and $x=0$ (bottom)
• Figure 5: Comparison of analytical and numerical fundamental solution derivatives $\psi_{,x}$ and $\psi_{,y}$ for constant water depth. Cross section of the $x$-derivative along the line $y=0$ (top) and the $y$-derivative along $x=0$ (bottom)