Analytic study on long wave transformation over a seamount with a pit

In this paper, an analytic solution is derived for linear long waves scattering over a submarine seamount landform with a pit. The seamount is axisymmetric with a pit on the top. The water depth is defined by a trinomial function in the radial direction. The governing linear shallow water equation for long waves is expressed in the polar coordination, which is solved through separation of variables. As the topography is axisymmetric, solutions can be written as Fourier-cosine series. Waves over the seamount are expressed using Frobenius series expansion, while the water surface elevation in the outer region is expressed as Fourier-Bessel series, and the final solution is obtained by matching them at the conjunction. The solution can be degenerated into the previous analytic solutions for waves propagation over an axisymmetric pit or a submerged hump by adjusting the topography parameters.