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

Abstract

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 method of separation of variables and the method of Frobenius. 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 twodimensional 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.

Analytical study on long wave refraction over a dredge excavation pit

“…In principle, the mild-slope wave equation is valid when | ∇ h|/kh ≪ 1. For a long wave problem, since kh is intrinsically small, the actual restriction on the absolute value of the bottom slope seems to be rather critical, but, in fact, the conventional mild-slope equation has been shown reasonably accurate under the long wave condition if the bottom slope is less than 1/3 (e.g., [3]). Namely, if the largest slope that occurs at r = r 1 is less than 1/3, or, if h 1 /r 1 b 1/(3α) is satisfied, the validity of the mild-slope assumption is guaranteed in the present study.…”

“…Suh et al [7] derived a solution for long waves propagating over a circular bowl pit. Recently, Jung and Suh [3] derived a new solution of the extended mild-slope equation for an axisymmetric pit, considering the effect of rapidly varying bathymetry.…”