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/d chi D(chi)/d chi 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.