Long-Wave Scattering by a Vertical Cylinder with Idealized Scour Pit

The combined diffraction and refraction of long waves around a vertical cylinder with scour pit is studied analytically on the basis of a newly derived analytic solution of the mild-slope wave equation. The geometry of the scour pit around the cylinder is idealized to be axisymmetrical, and a simple power function is employed to represent its radial profile. The problem is solved in the polar coordinate system by separation of variables, and the resulting ordinary differential equation in the radial direction over the scour pit in this case can be transformed into the Bessel equation of real order. On the basis of the analytic solution obtained, the effect of the scour pit on wave scattering is investigated by comparing the cases with and without the presence of the scour pit. The scour pit contributes to reduce the wave run-up against the cylinder. In addition, the effects of the scour pit on wave run-up against the cylinder are more significant when the dimensions of the scour pit are larger or the incident wave is relatively shorter.