Exact Solution to the Modified Mild-Slope Equation for Wave Scattering by a Cylinder with an Idealized Scour Pit

Abstract

In this paper, wave scattering by a vertical cylinder with a scour pit governed by the modified mild-slope equation (MMSE) is studied analytically. The scour pit around the cylinder is assumed to be axi-symmetric and idealized with its radial profile being a power function. This assumption permits transformation of the two-dimensional MMSE into an ordinary differential equation (ODE) in the radial direction through the technique of variable separation. By employing a newly derived explicit form of the resultant ODE of the MMSE in the scour pit region, an exact solution to the MMSE is constructed in terms of a Fourier-cosine series and Taylor series. To validate this new analytic solution to the MMSE, a comparison among the present solution, analytic solution to the long wave equation, and analytic solution to the Helmholtz equation is made and a good agreement is obtained. It is found that the present MMSE model is valid for a maximum bottom slope of approximately 0.927. Based on the present solution to the MMSE, the effect of dimensions of the scour pit, including both depth and width, on wave run-up around the cylinder is investigated. Finally, the influence of the wavelength of incident waves from shallow to deep water on wave run-up around the cylinder is also investigated.