An analytic solution for combined wave diffraction and refraction around a vertical cylinder with idealized scour pit

An analytic solution of the mild slope wave equation for wave propagating over a vertical cylinder with a scour pit has been derived. The scour pit is idealized to be geometrically axisymmetrical, and the water depth inside the pit is assumed to decrease in proportion to an arbitrary power of the radial distance from the center of the cylinder. By separation of variables, solution of the mild slope wave equation is reduced to find the general solution of an ordinary differential equation, of which the coefficients are functions of the wavenumber or the wave celerity, and can eventually be transformed into explicit functions of the independent variable by means of Hunt's approximate formula for the dispersion relation of small amplitude waves. By a tactical mapping of the coefficients into polynomials, the ordinary differential equation is successfully solved in the form of Frobenius series. The solution obtained is a significant extension to the long wave solution previously presented by the same authors. The long wave assumption is now removed and the shape of the scour pit is also generalized to a large extent. Based on the solution obtained, the wave run up around the cylinder is carefully studied, and the effects of the incident wave conditions, the extent, the depth, and the shape of the scour pit on wave transformation are also discussed.

► Combined diffraction and refraction around a cylinder with scour pit are studied. ► Mild slope wave equation is analytically solved with the Frobenius method. ► The solution obtained reduces to existing ones under special conditions. ► Effects of scour pit on wave run-up against the cylinder are discussed.