Numerical calculation of nonlinear axisymmetric standing waves in a circular basin

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Abstract
The solution of fully nonlinear first‐mode axisymmetric standing waves in a circular basin is obtained numerically by direct collocation of truncated Fourier and Dini series in time and radial coordinates, respectively. The method produces accurate results which offer substantial improvements, especially in shallow depth, over existing solutions based on perturbation expansions. Numerical results for the frequency, partition of energy, surface profiles, and particle trajectories are presented for both deep and shallow water for a range of steepnesses. It is found that salient features such as the nonmonotonic dependence of frequency on steepness at certain depths, characteristic of two‐dimensional standing waves, are also observed for axisymmetric standing waves.

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