Nonlinear concentric water waves of moderate amplitude

TL;DR

This paper derives and compares reduced models for nonlinear concentric water waves, demonstrating that the extended cKdV model accurately describes moderate amplitude waves better than simpler models.

Contribution

It introduces the extended cKdV model for axisymmetric water waves and validates its improved accuracy over traditional models through numerical comparisons.

Findings

Extended cKdV model outperforms cKdV in accuracy.

Reduced models extend the validity range for moderate amplitude waves.

Numerical simulations confirm the extended model's effectiveness.

Abstract

We consider the outward-propagating nonlinear concentric water waves within the scope of the 2D Boussinesq system. The problem is axisymmetric, and we derive the slow radius versions of the cylindrical Korteweg - de Vries (cKdV) and extended cKdV (ecKdV) models. Numerical runs are initially performed using the full axisymmetric Boussinesq system. At some distance away from the origin, we use the numerical solution of the Boussinesq system as the "initial condition" for the derived cKdV and ecKdV models. We then compare the evolution of the waves as described by both reduced models and the direct numerical simulations of the axisymmetric Boussinesq system. The main conclusion of the paper is that the extended cKdV model provides a much more accurate description of the waves and extends the range of validity of the weakly-nonlinear modelling to the waves of moderate amplitude.