Analytical benchmark for linear wave scattering by a submerged circular shoal in the water from shallow to deep

An analytical study on linear wave propagation over a submerged circular shoal is conducted, where the water depth is quasi-idealized with the water depth within the shoal region being a positive constant plus a power function of the radial distance. By using two variable transforms, a new analytical solution in the form of Frobenius series to the modified mild-slope equation (MMSE) is constructed and the convergence condition of the series is analyzed and clarified. The present solution extends the validity scope of the existing analytical solutions for wave propagation over a circular shoal from the long-wave range to the whole wave range. Comparison among the present solution, two sets of experimental solutions and MMSE based numerical solutions is conducted and nice agreements is obtained. Based on the present model, the influence of shoal size such as the radius, height and concavity on wave amplification is investigated. It is shown that the maximal wave amplification increases with the increase of the shoal size. The study of the influence of the incident wavelength on wave amplification shows that the maximal wave amplification often occurs when the wavelength is about five times of the global water depth.