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THE CUSP: FROM FINITE TIME SINGULARITIES TO BREAKING WAVES

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HAL Id: hal-01784609

https://hal.archives-ouvertes.fr/hal-01784609

Submitted on 3 May 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

THE CUSP: FROM FINITE TIME SINGULARITIES TO BREAKING WAVES

Patrice Le Gal

To cite this version:

Patrice Le Gal. THE CUSP: FROM FINITE TIME SINGULARITIES TO BREAKING WAVES.

Leonardo, Massachusetts Institute of Technology Press (MIT Press), 2018. �hal-01784609�

(2)

T HE CUSP: F ROM FINITE TIME SINGULARITIES TO BREAKING WAVES

Patrice Le Gal, Aix Marseille Université, CNRS, Centrale Marseille, Institut de Recherche sur les Phénomènes Hors Equilibre, 49 rue F. Joliot-Curie, 13384 Marseille, France Email: <legal@irphe.univ-mrs.fr>.

Submitted: <October 5

th

2017>

We all have in mind the sound of the ping-pong ball bounc- ing repeatedly faster and faster off the ground before it comes to a halt. This phenomenon of divergence in a finite duration of a physical quantity - here the frequency of rebounds - carries in physics the name of "singularity in finite time". Another exam- ple of a singularity in finite time comes from the wave break- ing phenomenon. In the vicinity of this subtle instant preceding breaking, the temporal derivative of the free surface of water diverges as the inverse of the square root of time. A geometric representation of this singular behavior is the cusp, the sim- plest catastrophe depending on two parameters as classified in his Catastrophes Theory by René Thom.

The drawing of the cusp shows that for certain time and space positions, the surface is a single valued function, whereas outside this domain, it takes three possible values.

Therefore, the cusp gives a good representation of the tilting of the wave before breaking. The deliminating curves between these two behavior are called caustics and merge at the singularity point. These lines share in fact the exact mathematical description of the pattern drawn by focusing light at the bottom of a cup after reflecting off the cup wall.

My wish during my SCIENTIFIC DELIRIUM MADNESS residency was to create an art piece with the intention to share the contemplation of this subtle and fragile instant where the singularity arises, where the curvature of forms changes sign, where waves tilt over. To extend this infinitely small duration, the singularity will be unfolded - as mathematicians say. The result of my creative process is an array of wooden rods whose arrangement makes straight rays to cooperate and interfere in an apparent continuous and smooth motion, giving rise to a regular but cusped surface that sifts and transforms our own perception of the world around it. Like a filter, the CUSP will let wind, sound, light and thoughts to go through its beams.

Fig. 1. The CUSP under the full moon in the Djerassi land.

(© Patrice Le Gal. Photo: Kristen Stipanov.)

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