Popsicle-Stick Cobra Wave

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Popsicle-Stick Cobra Wave

Jean-Philippe Boucher, Christophe Clanet, David Quéré, and Frédéric Chevy

Phys. Rev. Lett. 119, 084301 – Published 25 August 2017

See Focus story: Video—Cobra Wave Explained

Abstract

The cobra wave is a popular physical phenomenon arising from the explosion of a metastable grillage made of popsicle sticks. The sticks are expelled from the mesh by releasing the elastic energy stored during the weaving of the structure. Here we analyze both experimentally and theoretically the propagation of the wave front depending on the properties of the sticks and the pattern of the mesh. We show that its velocity and its shape are directly related to the recoil imparted to the structure by the expelled sticks. Finally, we show that the cobra wave can only exist for a narrow range of parameters constrained by gravity and rupture of the sticks.

Received 23 February 2017

DOI:https://doi.org/10.1103/PhysRevLett.119.084301

Physics Subject Headings (PhySH)

Classical mechanics Continuum mechanics Scaling laws of complex systems

Nonlinear DynamicsInterdisciplinary Physics

Focus

Video—Cobra Wave Explained

Published 25 August 2017

High-speed video of the dramatic “cobra wave” produced by a latticework of popsicle sticks helped researchers explain the shape and speed of the wave.

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Authors & Affiliations

Jean-Philippe Boucher¹, Christophe Clanet¹, David Quéré², and Frédéric Chevy³

¹LadHyX, UMR 7646 du CNRS, École Polytechnique, 91128 Palaiseau Cedex, France
²PMMH, UMR 7636 du CNRS, ESPCI, 75005 Paris, France
³Laboratoire Kastler Brossel, ENS-PSL Research University, CNRS, UPMC, Collège de France, 24, rue Lhomond, 75005 Paris, France

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Issue

Vol. 119, Iss. 8 — 25 August 2017

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Images

Figure 1
(a) Picture of a wooden stick with its characteristic parameters: length $L$ , width $w$ , thickness $e$ , mass $M$ , density $ρ$ and Young’s modulus $E$ . (b) Schematics of the lattice with definition of the angle $θ$ of the lattice and the spatial period $a$ of the pattern. The blue and red sticks are the sticks that end the lattice. The construction of the lattice starts with the blue stick, then the sticks are added one after the other according to the numbering for the first four sticks. (c) Schematic side view of a stick deformed in the mesh.
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Figure 2
Time-lapse photographs of the cobra wave obtained with sticks of type 1 (see the Supplemental Material [19]) for $θ = 45 °$ , from videos taken at 1000 fps with a Photron-Fastcam high-speed camera. The different colors represent the wave at different instants. (a) At the beginning ( $Δ t = 100 ms$ between two consecutive images); (b) during the stationary phase ( $Δ t = 70 ms$ ), with $v_{0} ≃ 2.2 m / s$ the velocity of the wave front; (c) at the end ( $Δ t = 70 ms$ ). Scale bars are 10 cm long.
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Figure 3
Speed of the wave front $v_{0}$ (solid dots) and speed of the expelled sticks $v$ (open dots) as a function of a characteristic speed $c (e / L)^{2}$ for $θ = 45 °$ and six different kinds of sticks (see the Supplemental Material [19]). The black line corresponds to the fit $v_{0} = 3.95 c (e / L)^{2}$ and the dashed line to the fit $v = 5.46 c (e / L)^{2}$ . (i) Speed of the cobra wave $v_{0}$ as a function of the angle of the lattice $θ$ for sticks of type 1.
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Figure 4
Experimental cobra profile for sticks of type 2. The steady profile (red dashed line) is described theoretically by a parametric curve $r (s^{'})$ , where $s^{'}$ is the curvilinear abscissa. $α$ is the angle between the mesh and the horizontal axis, and $ψ$ the angle between the velocity of the expelled sticks and the tangent vector in $s^{'} = s_{\max}$ . The scale bar is 10 cm long.
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Figure 5
Dimensionless height $h_{ψ}$ as a function of the dimensionless number $Λ = g \sqrt{\bar{K} / (μ v_{0}^{3} v^{3})}$ which compares gravity and elasticity, for the six different kinds of sticks (see the Supplemental Material [19]). The solid line corresponds to the prediction of Eqs. (4)–(9) for $ψ = 60 °$ . The shaded band corresponds to the observed 10° variations of the ejection angle. The red dashed line represents the large $Λ$ expansion $h_{ψ} ≃ 2 \sin^{4} ψ / 3 Λ^{3}$ for $ψ = 60 °$ .
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Figure 6
Set of parameters ( $e$ , $L$ ) for which a cobra stick wave can be observed (red region). This region is limited by two conditions given in Eqs. (13) and (15): the limit set by gravity $L_{\max} = {(36 E e^{4} / ρ g)}^{1 / 5}$ and the breaking limit $L_{\min} = 1.5 \times 3 \sqrt{E / σ^{*}} e$ . Solid blue dots: sets of parameters for which the cobra-stick wave is observed. Open blue dots: sets of parameters for which the cobra-stick wave could not be observed (the sticks are too small and therefore break). (i) Schematics of the shape of a stick in the lattice with the most probable breaking region.
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