A series solution based on the mild-slope equation is produced in this study of wave scattering produced by a circular cylindrical island mounted on an axi-symmetrical shoal. The solution is presumed to be a Fourier cosine expansion with variable coefficients in the radial direction on account of the symmetric scattering field, which translates the original 2-D boundary-value problem to a 1-D one in which an ordinary differential equation is in effect treated. Approximations to the coefficients of the governing equation with the Taylor expansions enable the use of the Frobenius method, and consequently the solution is obtained in a combined Fourier and power series. For verification, the present method is mainly compared with Zhu and Zhang's [1] analytical solution of the linearised shallow water equation for a conical shoal, and with a different analytical solution of the mild-slope equation developed by Liu et al. [2] for a paraboloidal shoal. Fine agreements are achieved. The present method is then used to investigate the variation pattern of the wave run-up when the shoal profile varies from conical to paraboloidal, and some interesting phenomena are observed.

This paper considers a finite dam with independently and identically distributed (i.i.d.) inputs occurring in a Poisson process; the special cases where the inputs are (i) deterministic and (ii) negative exponentially distributed are considered in detail. The instantaneous release trate is proportional to the content, i.e., there is an exponential fall in conten except when inputs occur. This model may arise in several other situations such as a geiger counter or integrated shot noise. The distribution of the number of inputs, and of the time, to first overflowing is obtained in terms of generating functions; in Case (i) the solution is obtained through recurrence relations involving iterated integrals which can be evaluated numerically, and in Case (ii) using a series solution of a second order differential equation. Numerical results, in particular for the first two moments, are obtained for various values of the parameters of the model, and compared with a large number of simulations. Some remarks are also made about the infinite dam.