Asymptotics of Long Standing Waves in One-Dimensional Basins with Shallow Coasts: Theory and Experiment

We construct time-periodic asymptotic solutions of the one-dimensional system of nonlinear shallow water equations in a basin of variable depth \(D\left( x \right)\) with two shallow coasts (which means that the function \(D\left( x \right)\) vanishes at the points defining the coast) or with one shallow coast and a vertical wall. Such solutions describe standing waves similar to the well-known Faraday waves in basins with vertical walls. In particular, they approximately describe seiches in elongated basins. The construction of such solutions consists of two stages. First, time-harmonic exact and asymptotic solutions of the linearized system generated by the eigenfunctions of the operator \(d{\text{/}}dxD(x)d{\text{/}}dx\) are determined, and then, using a recently developed approach based on the simplification and modification of the Carrier–Greenspan transformation, solutions of nonlinear equations are reconstructed in parametric form. The resulting asymptotic solutions are compared with experimental results based on the parametric resonance excitation of waves in a bench experiment.