An analytical solution to the extended mild-slope equation for long waves propagating over an axi-symmetric pit

In this study, an analytic solution to the extended mild-slope equation is derived for long waves propagating over an axi-symmetric pit, where the water depth decreases in proportion to a power of radial distance from the pit center. The solution is obtained using the separation of variables method and the Frobenius method. By comparing the extended and conventional mild-slope equations for waves propagating over conical pits with different bottom slopes, it is shown that for long waves the conventional mild-slope equation is reasonably accurate for bottom slopes less than 1:3 in horizontal two-dimensional domains. The effects of the pit shape on wave scattering are discussed based on the analytic solutions for different powers. Comparison is also made with an analytic solution for a cylindrical pit with a vertical sidewall. Finally, wave attenuation in the region over the pit is discussed.