Asymptotics of Long Standing Waves in One-Dimensional Pools with Shallow Banks: Theory and Experiment

We construct time-periodic asymptotic solutions for a one-dimensional system of nonlinear shallow water equations in a pool of variable depth D(x) with two shallow banks (which means that the function D(x) vanishes at the points defining the banks) or with one shallow bank and a vertical wall. Such solutions describe standing waves similar to well-known Faraday waves in pools with vertical walls. In particular, they approximately describe seiches in elongated bodies of water. 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 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.