Internal-wave billiards in trapezoids and similar tables

TL;DR

This paper studies internal-wave billiard systems in trapezoid-like tables, proving the existence of three asymptotic regimes and analyzing parameter space, including the measure of these regimes, with implications for internal gravity wave dynamics.

Contribution

It introduces a new class of internal-wave billiard models with non-standard reflections and characterizes their asymptotic behaviors, including measure-theoretic properties of parameter regimes.

Findings

Three asymptotic regimes identified: attractor, beam, and dense trajectories.

Set of parameters with global attractor has positive measure, confirming Arnol'd tongues existence.

Sets for other regimes are non-empty but measure zero.

Abstract

We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this rule: reflections are standard Fresnel reflections but with the pretense that the boundary at any collision point is either horizontal or vertical (relative to a predetermined direction representing gravity). These systems are point particle approximations for the motion of internal gravity waves in closed containers, hence the name. For a class of tables similar to rectangular trapezoids, but with the slanted leg replaced by a general curve with downward concavity, we prove that the dynamics has only three asymptotic regimes: (1) there exist a global attractor and a global repellor, which are periodic and might coincide; (2) there exists a beam of periodic trajectories, whose boundary (if any) comprises an attractor and a repellor for all the other trajectories; (3) all trajectories are dense (that is, the system is minimal). Furthermore, in the prominent case where the table is an actual trapezoid, we study the sets in parameter space relative to the three regimes. We prove in particular that the set for (1) has positive measure (giving a rigorous proof of the existence of Arnol'd tongues for internal-wave billiards), whereas the sets for (2) and (3) are non-empty but have measure zero.