Two new standing solitary waves in shallow water

TL;DR

This paper presents the first closed-form solutions for two new types of standing solitary waves caused by parametric resonance in vibrating shallow water, enhancing understanding of Faraday's wave phenomena.

Contribution

It introduces novel analytic solutions for two previously unreported standing solitary waves in shallow water, based on symmetry and linearized Boussinesq equations.

Findings

Derived closed-form wave elevations for the new waves

Explained experimental phenomena with the new solutions

Deepened understanding of Faraday's standing waves

Abstract

In this paper, the closed-form analytic solutions of two new Faraday's standing solitary waves due to the parametric resonance of liquid in a vessel vibrating vertically with a constant frequency are given for the first time. Using a model based on the symmetry of wave elevation and the linearized Boussinesq equation, we gain the closed-form wave elevations of the two kinds of non-monotonically decaying standing solitary waves with smooth crest and the even or odd symmetry. All of them have never been reported, to the best of our knowledge. Besides, they can well explain some experimental phenomena. All of these are helpful to deepen and enrich our understandings about standing solitary waves and Faraday's wave.