# gst: apropos of waves that escape trapping; on internal wave whispering gallery modes in channels and critical-slope wave attractors.

<< ️Internal waves are an important feature of stratified fluids, both in oceanic and lake basins and in other settings. Many works have been published on the generic feature of internal wave trapping onto planar wave attractors and super-attractors in 2D & 3D and the exceptional class of standing global internal wave modes. >>

<< ️However, most of these works did not deal with waves that escape trapping. By using continuous symmetries (AA) analytically prove the existence of internal wave Whispering Gallery Modes (WGMs), internal waves that propagate continuously without getting trapped by attractors. WGMs neutral stability with respect to different perturbations enable whispering gallery beams, a continuum of rays propagating together coherently. The systems' continuous symmetries also enable projection onto 2D planes that yield effective 2D billiards preserving the original dynamics. >>

<< ️By examining rays deviating from these WGMs in parabolic channels (They) discover a new type of wave attractor which is located along the channel instead of across it as in previous works. This new wave attractor leads to a re-understanding of WGMs as sitting at the border between the two basins of attraction. >>

<< ️Finally, both critical-slope wave attractors and whispering gallery beams are used to propose explanations for along-channel energy fluxes in submarine canyons and tidal energy intensification near critical slopes. >>

Nimrod Bratspiess, Eyal Heifetz, Leo R. M. Maas. On internal wave whispering gallery modes in channels and critical-slope wave attractors. arXiv: 2510.07218v1 [physics.flu-dyn]. Oct 8, 2025.

https://arxiv.org/abs/2510.07218

Also: waves, billiard, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, internal waves, internal wave whispering gallery modes (WGMs), wave attractors, basins of attraction, billiards.

# life: apropos of Trump-Musk dipole, an immediate MJQ apoptotic rift

<< In the last two days, the once powerful relationship between President Donald Trump and billionaire Elon Musk has unraveled as they took jabs at each other from their respective social media sites, X and Truth Social. Musk left his official role in the administration late last month. See how the digital confrontation devolved from complaints over the president’s tax bill to Musk alleging Trump was “in the Epstein files.” . >>

Rachel Lerman. A timeline of Musk and Trump’s devolving relationship. washingtonpost.com. Jun 6, 2025.

https://www.washingtonpost.com/politics/interactive/2025/trump-musk-x-truth-social-timeline/

FonT: an Ivy League Academic (e.g. a Harvard academic) could, as a funny hypothesis (and just to check if and how it works), to try to restore this form of sadic intermittent acyclic pseudo MJQ dipole; in this regard He/She/? could be immediately operational as a catalytic fractal epizeusis entity (obviously without ideological prejudices) to play Parrondo game to drive a quasi-stochastic tit-for-tat cooperative strategy. 

Soloperosservarecosasuccedeanzicheforse ...

Also: tit-for-tat, game, Parrondo, 

game, apoptosis, Mr. Donald, in https://www.inkgmr.net/kwrds.html

Also: MJQ, in FonT https://flashontrack.blogspot.com/search?q=mjq. in Notes (quasi-stochastic poetry) Attrattore cooperativo bipolare (nell' elastici spigoli), Mar 5, 2007. https://inkpi.blogspot.com/2007/03/2124-attrattore-cooperativo-bipolare.html

Also: Trump-Musk, in FonT https://flashontrack.blogspot.com/search?q=trump-musk

Also: sadico, in Notes (quasi-stochastic poetry) https://inkpi.blogspot.com/search?q=sadico

Keywords: life, Parrondo, Tit-for-Tat, MJQ, epizeusis, POTUS, Mr.Donald, Trump-Musk, sadistic, apoptosis, billiards, dance.

# gst: pensive billiards

AA << define a new class of plane billiards - the `pensive billiard' - in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting, while preserving the billiard rule of equality of the angles of incidence and reflection. This generalizes so called `puck billiards' (..), as well as a `vortex billiard', i.e. the motion of a point vortex dipole in 2D hydrodynamics on domains with boundary. (AA) prove the variational origin and invariance of a symplectic structure for pensive billiards, as well as study their properties including conditions for a twist map, the existence of periodic orbits, etc. (AA) also demonstrate the appearance of both the golden and silver ratios in the corresponding hydrodynamical vortex setting. Finally, (AA) introduce and describe basic properties of pensive outer billiards. >>

Theodore D. Drivas, Daniil Glukhovskiy, Boris Khesin. Pensive billiards, point vortices, and pucks. arXiv: 2408.03279v1 [math.DS]. Aug 6, 2024.

https://arxiv.org/abs/2408.03279

Also: billiard, in https://www.inkgmr.net/kwrds.html 

FonT: 'pensive billiard' evokes images in me that could inspire a series of quasi-stochastic short poems ( https://inkpi.blogspot.com ), but (for now) I will abstain.

Keywords: gst, billiards, pensive billiard, puck billiard, vortex billiard