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Surface topography measurements of the bouncing droplet experiment

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Adam P. Damiano
Adam P. Damiano
Adam P. Damiano
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Pt Brun at Princeton University
Pt Brun
Daniel M. Harris at Brown University
Daniel M. Harris
Carlos Galeano Rios at Minerva University
Carlos Galeano Rios
  • Minerva University
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Abstract and Figures

A free-surface synthetic Schlieren (Moisy et al. in Exp Fluids 46:1021–1036, 2009; Eddi et al. in J Fluid Mech 674:433–463, 2011) technique has been implemented in order to measure the surface topography generated by a droplet bouncing on a vibrating fluid bath. This method was used to capture the wave fields of bouncers, walkers, and walkers interacting with boundaries. These wave profiles are compared with existing theoretical models and simulations and will prove valuable in guiding their future development. Specifically, the method provides insight into what type of boundary conditions apply to the wave field when a bouncing droplet approaches a submerged obstacle.
Layout of our experimental setup (Harris and Bush 2015) comprising an electromagnetic shaker (1) interfaced to the bath via a rod (2) and a linear air bearing (3). The air bearing carriage is fixed to an optical table (4) via a leveled platform (5). The bath acceleration is monitored via two accelerometers (6) and is surmounted by a lid (7). The bath (8) comprises a pattern printed on paper inserted below a transparent lower plate so that the pattern may be used as an imaging background
Layout of our experimental setup (Harris and Bush 2015) comprising an electromagnetic shaker (1) interfaced to the bath via a rod (2) and a linear air bearing (3). The air bearing carriage is fixed to an optical table (4) via a leveled platform (5). The bath acceleration is monitored via two accelerometers (6) and is surmounted by a lid (7). The bath (8) comprises a pattern printed on paper inserted below a transparent lower plate so that the pattern may be used as an imaging background
… 
The FSSS method (Moisy et al. 2009): a A sloped interface causes an apparent displacement of the points on a background image, u(x, y). The refraction angle is denoted by ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document} and the interface slope by ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}. The background image is a random dot pattern (b) and the apparent displacement of the pattern creates a gradient field (c) that is used to reconstruct the surface topography (d)
The FSSS method (Moisy et al. 2009): a A sloped interface causes an apparent displacement of the points on a background image, u(x, y). The refraction angle is denoted by ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document} and the interface slope by ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}. The background image is a random dot pattern (b) and the apparent displacement of the pattern creates a gradient field (c) that is used to reconstruct the surface topography (d)
… 
Regime diagram [see Molácek and Bush (2013b) for its derivation] displaying the drop’s bouncing or walking mode as a function of the vibration number ω0/σ/ρR3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0/\sqrt{\sigma /\rho R^3}$$\end{document} and driving acceleration γ/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /g$$\end{document}. A drop in the (m,n)i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m, n)^i$$\end{document} mode bounces n times in m forcing periods, with the integer i ordering multiple (m, n) states according to their total mechanical energy, with i = 1 being the lowest. Here, we have ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\,{\mathrm{Hz}}$$\end{document}, ν=20cSt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =20\,{\mathrm{cSt}}$$\end{document} and fixed drop radius R = 0.38 mm. The red symbols indicate the experiments performed in our study
Regime diagram [see Molácek and Bush (2013b) for its derivation] displaying the drop’s bouncing or walking mode as a function of the vibration number ω0/σ/ρR3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0/\sqrt{\sigma /\rho R^3}$$\end{document} and driving acceleration γ/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /g$$\end{document}. A drop in the (m,n)i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m, n)^i$$\end{document} mode bounces n times in m forcing periods, with the integer i ordering multiple (m, n) states according to their total mechanical energy, with i = 1 being the lowest. Here, we have ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\,{\mathrm{Hz}}$$\end{document}, ν=20cSt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =20\,{\mathrm{cSt}}$$\end{document} and fixed drop radius R = 0.38 mm. The red symbols indicate the experiments performed in our study
… 
Temporal evolution of the wave field excited by a period-doubled bouncing droplet of radius 0.38 mm. The acceleration of the 20 cSt oil bath is γ=0.77γF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0.77\,\gamma _\mathrm{F}$$\end{document} and ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\mathrm{Hz}$$\end{document} (see the red star in Fig. 3). The wave field is shown at phases a 0°, b 150°, c 300°, d 450° and e 600°. Phase 0° corresponds to the bath at its maximum vertical displacement. Black lines show a projection of the crosssection of the wave field passing through the droplet’s center. The gray cylinders indicate the location of the droplet’s projected shadow, where the wave field cannot be determined.
Temporal evolution of the wave field excited by a period-doubled bouncing droplet of radius 0.38 mm. The acceleration of the 20 cSt oil bath is γ=0.77γF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0.77\,\gamma _\mathrm{F}$$\end{document} and ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\mathrm{Hz}$$\end{document} (see the red star in Fig. 3). The wave field is shown at phases a 0°, b 150°, c 300°, d 450° and e 600°. Phase 0° corresponds to the bath at its maximum vertical displacement. Black lines show a projection of the crosssection of the wave field passing through the droplet’s center. The gray cylinders indicate the location of the droplet’s projected shadow, where the wave field cannot be determined.
… 
Dependence of the wave profile of a bouncing droplet on the dimensionless forcing accelerations γ/γF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /\gamma _\mathrm{F}$$\end{document} where γF denotes the Faraday threshold. The drop radius is 0.38 mm, while ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\,\mathrm{Hz}$$\end{document} and ν=20cSt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =20\,\mathrm{cSt}$$\end{document}. Each experiment is represented by a dot in Fig. 3. The wave profile is captured when the bath is at its minimum vertical displacement. The dark vertical band in the vicinity of r = 0 represents the droplets projected shadow
+4
Dependence of the wave profile of a bouncing droplet on the dimensionless forcing accelerations γ/γF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /\gamma _\mathrm{F}$$\end{document} where γF denotes the Faraday threshold. The drop radius is 0.38 mm, while ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\,\mathrm{Hz}$$\end{document} and ν=20cSt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =20\,\mathrm{cSt}$$\end{document}. Each experiment is represented by a dot in Fig. 3. The wave profile is captured when the bath is at its minimum vertical displacement. The dark vertical band in the vicinity of r = 0 represents the droplets projected shadow
… 
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1 3
Exp Fluids (2016) 57:163
DOI 10.1007/s00348-016-2251-4
RESEARCH ARTICLE
Surface topography measurements of the bouncing droplet
experiment
Adam P. Damiano1 · P.‑T. Brun1 · Daniel M. Harris1 · Carlos A. Galeano‑Rios2,3 ·
John W. M. Bush1
Received: 15 June 2016 / Revised: 6 September 2016 / Accepted: 10 September 2016 / Published online: 11 October 2016
© Springer-Verlag Berlin Heidelberg 2016
unstable to Faraday waves. Couder et al. (2005) and Protière
et al. (2006) discovered that in certain experimental regimes,
the droplets may self-propel along the surface of the bath due
to interactions with their own wave fields. These walking
droplets, henceforth walkers, are spatially extended objects
that exhibit several phenomena reminiscent of quantum sys-
tems (Couder and Fort 2006; Eddi et al. 2009, 2011; Bush
2010, 2015a, b; Fort et al. 2010; Harris et al. 2013; Perrard
et al. 2014a, b; Oza et al. 2014; Harris and Bush 2014).
While the droplet, of typical radius 0.4 mm, is read-
ily discerned by eye, the waves excited by the droplet,
of typical amplitude of 1–20 μm, are relatively diffi-
cult to observe and quantify. Various theoretical models
have been developed to describe the waves created by a
bouncing droplet (Eddi et al. 2011; Molácek and Bush
2013a, b; Oza et al. 2013; Labousse 2014; Milewski
et al. 2015; Gilet 2016; Blanchette 2016), on the basis
of which much headway has been made in rationalizing
the behavior of the walkers in a variety of settings [see
Bush (2015a, b) for reviews]. Nevertheless, theoretical
developments would benefit from quantitative measure-
ments of the wave field. In particular, walker-boundary
interactions as arise in a number of key quantum ana-
logues (Couder and Fort 2006; Eddi et al. 2009; Harris
et al. 2013; Harris 2015) remain poorly characterized
and understood. Specifically, some theoretical models
of walkers near boundaries apply a zero-wave-amplitude
boundary condition (Gilet 2016; Blanchette 2016) while
others apply a zero slope boundary condition (Duber-
trand et al. 2016).
We here report the results of an experimental effort to
measure the surface topography in the walking drop system
using the surface synthetic Schlieren technique originally
developed by Moisy et al. (2009), as was applied by Eddi
et al. (2009, 2011). Specifically, we utilize the refracted
Abstract A free-surface synthetic Schlieren (Moisy et al.
in Exp Fluids 46:1021–1036, 2009; Eddi et al. in J Fluid
Mech 674:433–463, 2011) technique has been implemented
in order to measure the surface topography generated by a
droplet bouncing on a vibrating fluid bath. This method
was used to capture the wave fields of bouncers, walkers,
and walkers interacting with boundaries. These wave pro-
files are compared with existing theoretical models and
simulations and will prove valuable in guiding their future
development. Specifically, the method provides insight into
what type of boundary conditions apply to the wave field
when a bouncing droplet approaches a submerged obstacle.
1 Introduction
A millimetric drop placed onto a vibrating liquid bath can
bounce indefinitely on the fluid surface due to a thin film of
air that prevents coalescence and is replenished with each
bounce (Walker 1978; Couder et al. 2005; Terwagne et al.
2007; Vandewalle et al. 2006). The drop dynamics depends
critically on the forcing acceleration of the bath γ relative
to the critical threshold γF, at which the interface becomes
Electronic supplementary material The online version of this
article (doi:10.1007/s00348-016-2251-4) contains supplementary
material, which is available to authorized users.
* P.-T. Brun
pierrethomas.brun@gmail.com
1 MIT, Cambridge, MA, USA
2 IMPA/National Institute of Pure and Applied Mathematics,
Est. D. Castorina, 110, Rio de Janeiro RJ 22460-320, Brazil
3 Department of Mathematical Sciences, University of Bath,
Bath BA2 7AY, UK
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... where b is given by equation (10). It is worth noting that in the long wave limit (i.e. ...
... When obstacles are present the depth above them is (unless otherwise indicated) h 1 = 0.42 mm, chosen to correspond roughly to the experiments of [32] and [19]. Computing k F using the method outlined in §2.1 yields k F0 ≈ 1.265mm −1 mm and k F1 ≈ 1.555mm −1 ; b(x) is then computed using (10). In order to compare our simulations to available experiments, two different drop sizes are considered: in subsections § 3.1-3.2 the drops have a radius of 0.39mm, while in § 3.3 they have a radius of 0.335mm. ...
... However, neither η x nor η appears to be zero at the deep-shallow interface. This inference, consistent with the schlieren imaging of [13] and [10], suggests that imposing an effective Dirichlet or Neumann boundary condition on the wavefield at places where the depth changes is likely to be inadequate. Figure 3: Behavior of the wavefield at the interface between the shallow (0.42mm) and the deep (6.09mm) regions. ...
A model for Faraday pilot waves over variable topography
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  • Jan 2017
Couder and Fort discovered that droplets walking on a vibrating bath possess certain features previously thought to be exclusive to quantum systems. These millimetric droplets synchronize with their Faraday wavefield, creating a macroscopic pilot-wave system. In this paper we exploit the fact that the waves generated are nearly monochromatic and propose a hydrodynamic model capable of quantitatively capturing the interaction between bouncing drops and a variable topography. We show that our reduced model is able to reproduce some important experiments involving the drop-topography interaction, such as non-specular reflection and single-slit diffraction.
View
... This technique is used to determine the global gradient map of a surface of fluid. The original FS-SS has been used to measure fluid surface-droplet dynamics with micrometer wave amplitudes (Damiano et al. 2016;Eddi et al. 2011) containing theoretical application to new understandings of quantum mechanics (Damiano et al. 2016; Dagan 2023) as well as more applied studies involving waves generated by both wind (Paquier et al. 2015) and mechanical (Moisy et al. 2009) means. ...
... This technique is used to determine the global gradient map of a surface of fluid. The original FS-SS has been used to measure fluid surface-droplet dynamics with micrometer wave amplitudes (Damiano et al. 2016;Eddi et al. 2011) containing theoretical application to new understandings of quantum mechanics (Damiano et al. 2016; Dagan 2023) as well as more applied studies involving waves generated by both wind (Paquier et al. 2015) and mechanical (Moisy et al. 2009) means. ...
Modified free-surface synthetic Schlieren method to adjust measurement sensitivity in high-strain waves
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This research revisits the free-surface synthetic Schlieren technique (FS-SS) (Moisy et al. in Exp Fluids 46(6):1021–1036, 2009) for the topographical measurement of the free surface of a liquid. In contrast to the aforementioned method, which utilizes the image refraction of a random dot pattern through a free surface to determine the surface gradients, this present study mathematically derives a method where the pattern may take an arbitrary three-dimensional shape. This is shown to be theoretically valid under a small pattern slope approximation. Our method is then verified against the previously mentioned, flat-pattern, free-surface synthetic Schlieren technique by resolving the free-surface elevation of plane waves across a channel, showing similar results in constant strain conditions, with improved results in variable strain conditions, particularly in cases where there is a very large difference in strain across the channel. The validation test cases investigated include a rectangular channel containing a transparent liquid with a random dot pattern placed below at a constant angle and a pattern placed on top of a cosine-shaped profile. Both of these setups are validated against the classical FS-SS technique involving a flat pattern. The new method involving an arbitrarily shaped pattern proposed here may increase the resolution in low-amplitude regions by increasing the surface–pattern distance below these regions and correspondingly reducing the sensitivity in high-strain regions by decreasing the surface–pattern distance. Geometries shown to produce advantageous results in waves that include both regions of very high strains and regions of very low amplitudes are explored, resolving the wave in both regions simultaneously. This shows promise in resolving multi-scale surface waves in highly viscous liquids, which may include very high-amplitude regions quickly followed by very low-amplitude regions due to damping effects.
View
... Of these optical techniques, the free-surface synthetic Schlieren (FS-SS) method developed by Moisy et al. (2009) provides a straightforward reconstruction of the liquid surface at relatively higher spatiotemporal resolution using inverse analysis of the gradient field, which is usually obtained by a digital image correlation (DIC) algorithm and linear transformation. The FS-SS method, based on analysis of the deformed random-dot background obtained after refraction through the water surface, has been applied in the measurement of oscillating systems to observe almost pure cross-waves (Moisy et al. 2012), surface three-wave resonant interactions (Abella & Soriano 2019) and the surface field induced by a bouncing droplet (Damiano et al. 2016). ...
... Some additional, unavoidable effects arising from the relative motion between the visual plane and random-dot pattern were discerned when the FS-SS method resolved the surface interface in the vibrating system (Damiano et al. 2016). Conventionally, the background pattern has been assumed to remain stationary in the derivation of the FS-SS method and in later implementations (Moisy et al. 2009). ...
Pattern evolution and modal decomposition of Faraday waves in a brimful cylinder
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This paper investigates the steady-state pattern evolution of symmetric Faraday waves excited in a brimful cylindrical container when driving parameters much exceed critical thresholds. In such liquid systems, parametric surface responses are typically considered as the resonant superposition of unstable standing waves. A modified free-surface synthetic Schlieren method is employed to obtain full three-dimensional spatial reconstructions of instantaneous surface patterns. Multi-azimuth structures and localized travelling waves during the small-elevation phases of the oscillation cycle give rise to modal decomposition in the form of ν\nu -basis modes. Two-step surface-fitting results provide insight into the spatiotemporal characteristics of dominant wave components and corresponding harmonics in the experimental observations. Arithmetic combination of modal indices and uniform frequency distributions reveal the nonlinear mechanisms behind pattern formation and the primary pathways of energy transfer. Taking the hypothetical surface manifestation of multiple azimuths as the modal solutions, a linear stability analysis of the inviscid system is utilised to calculate fundamental resonance tongues (FRTs) with non-overlapping bottoms, which correspond to subharmonic or harmonic ν\nu -basis modes induced by surface instability at the air–liquid interface. Close relationships between experimental observations and corresponding FRTs provide qualitative verification of dominant modes identified using surface-fitting results. This supports the validity and rationality of the applied ν\nu -basis modes.
View
... First, a filter H r (k, ω) is applied to the transverse profile in the Fourier space to filter the waves reflected by the boundaries of the tank. Then, Butter-worth filter function H bw (k, ω) is applied to suppress undesired frequencies at low and high wave numbers [40,41]. This filtered signal is then transformed back to a spatiotemporal signalη ⊥ (y, t) using inverse Fourier transform F −1 :η ⊥ (k, ω) = F[η ⊥ (y, t)], η ⊥ (y, t) = F −1 [H r (k, ω)H bw (k, ω)η ⊥ (k, ω)]. ...
Resonance of an object floating within a surface wavefield
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Full-text available
  • Jan 2025
We examine the interaction between floating cylindrical objects and surface waves in the gravity regime. Given that the impact of resonance phenomena associated with floating bodies, particularly at laboratory scales, remains underexplored, we focus on the influence of the floaters' resonance frequency on wave emission. We study the response of floating rigid cylinders to external mechanical perturbations. Using an optical reconstruction technique to measure wave fields, we study the natural resonance frequency of the floaters in geometry and their response to incoming waves. The results indicate that the resonance frequency is influenced by the interplay between the cylinder geometry and the solid-to-fluid density ratio. Experiments with incoming waves demonstrate that floaters diffract waves, creating complex patterns of diffusion and retro-diffusion while radiating secondary waves that interfere with the incident wavefield. Minimal wave generation is observed at resonance frequencies. These findings provide insights for elucidating the behavior of larger structures, such as sea ice floes, in natural wave fields.
View
... Thus, for µ = 0, we obtain a megastable structure with infinite limit cycles given by the zeros of the product of Bessel functions. wave form, it still does not accurately capture the exponential spatial decay observed in experiments [42]. To capture this whilst maintaining a smooth and differentiable wave field, we consider the wave form, W (x) = cos(x) sech(x/2l), which results in the following selfoscillator model ...
Megastable quantization in generalized pilot-wave hydrodynamics
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A classical particle in a harmonic potential gives rise to a continuous energy spectra, whereas the corresponding quantum particle exhibits countably infinite quantized energy levels. In recent years, classical non-Markovian wave-particle entities that materialize as walking droplets, have been shown to exhibit various hydrodynamic quantum analogs, including quantization in a harmonic potential via limit cycle orbits. However, the number of co-existing quantized states are limited to at most a few limit cycles. By considering a minimal generalized pilot-wave model of the system in the low-memory and low-dissipation regime, we obtain a classical harmonic oscillator perturbed by an oscillatory non-conservative force that displays a countably infinite coexisting limit-cycle states, also know as \emph{megastability}; thus forming a dynamical analog of infinite quantized states. Using averaging techniques, we derive an analytical approximation of the orbital radii, orbital frequency and Lyapunov energy function, of this megastable spectrum, and further show average energy conservation along these quantized states. Our formalism extends to a general class of self-excited oscillators and can be used to construct megastable spectrum with different energy-frequency relations.
View
... Thus, for µ = 0, we obtain a megastable structure with infinite limit cycles given by the zeros of the product of Bessel functions. wave form, it still does not accurately capture the exponential spatial decay observed in experiments [42]. To capture this whilst maintaining a smooth and differentiable wave field, we consider the wave form, W (x) = cos(x) sech(x/2l), which results in the following selfoscillator model ...
Megastable quantization in generalized pilot-wave hydrodynamics
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  • Oct 2024
A classical particle in a harmonic potential gives rise to a continuous energy spectra, whereas the corresponding quantum particle exhibits countably infinite quantized energy levels. In recent years, classical non-Markovian wave-particle entities that materialize as walking droplets, have been shown to exhibit various hydrodynamic quantum analogs, including quantization in a harmonic potential via limit cycle orbits. However, the number of co-existing quantized states are limited to at most a few limit cycles. By considering a minimal generalized pilot-wave model of the system in the low-memory and low-dissipation regime, we obtain a classical harmonic oscillator perturbed by an oscillatory non-conservative force that displays a countably infinite coexisting limit-cycle states, also known as megastability; thus forming a dynamical analog of infinite quantized states. Using averaging techniques, we derive an analytical approximation of the orbital radii, orbital frequency and Lyapunov energy function, of this megastable spectrum, and further show average energy conservation along these quantized states. Our formalism extends to a general class of self-excited oscillators and can be used to construct megastable spectrum with different energy-frequency relations.
View
... To be more specific, the present study conducts experiments on mechanically-induced meniscus water waves in square containers, where the hydrophilicities of the four lateral boundaries can be easily encoded 9 in order to explore the impact of different wetting conditions on the frequency-response characteristics of meniscus-wave pattern selection. To accurately measure the full-spatial 3-D surface topography of meniscus waves, a high-resolution optical surface reconstruction technique, called the free-surface synthetic Schlieren (FS-SS) method 29,30 , is employed, which has been widely used in experiments involving small-amplitude surface waves [31][32][33][34] . ...
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View
... First, the resonance between the droplet and its wave ensures that When the droplet's bouncing frequency matches that of the vibrating bath's most unstable Faraday mode, the drop destabilizes into a 'walker', and is self-propelled across the bath by its pilot wave. (c) Comparison between experimental measurements (top) and simulation (bottom) of the wave field accompanying a walker moving from left to right (Damiano et al., 2016). Color bar indicates the wave height in microns. ...
The State of Play in Hydrodynamic Quantum Analogs
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  • Nov 2023
We describe the manner in which the physical picture emerging from hydrodynamic quantum analogs (HQAs) may serve to resolve some of the longstanding difficulties of quantum mechanics. We enumerate some of the most significant intellectual cul-de-sacs of quantum mechanics, and the manner in which HQA suggests a route past them. Particular attention is given to enumerating the many guises of quantum nonlocality as it appears in the standard quantum interpretations. We illustrate how one might misinfer such nonlocality from the walking-droplet system if one had an incomplete description of the system dynamics, if the variables required for its complete description were hidden rather than in plain sight. We highlight recent work that illustrates how phenomena typically attributed to nonlocality in quantum systems may be rationalized in terms of classical, pilot-wave dynamics. Finally, we define the frontiers of the field of hydrodynamic quantum analogs, including attempts to achieve classical entanglement by demonstrating Bell violations in pilot-wave hydrodynamics, and attempts to develop a model of quantum dynamics informed by the walking-droplet system.
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Experimental investigation of walking drops: Wave field and interaction with slit structures
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  • Mar 2024
  • PRE
While bouncing walking silicone oil droplets (walkers) do show many quantumlike phenomena, the original, most intriguing, double-slit experiment with walkers has been shown to be an overinterpretation of data. Several experiments and numerical simulations have proven that for at least some parameter region there is no randomness. Still, true randomness was claimed to be observed in an experiment on chaotically bouncing walkers. Also, most of the available phase space has not been investigated. The main goal of this paper is therefore to look for true interference and chaos in the entire phase space. Recently, we made an extensive investigation of drops interacting with slits, but still in a limited range. However, the outcome was always deterministic and only incidentally mimicked the statistics of the corresponding quantum case. We also showed that the extra interference, already seen by others, in the double-slit case was caused by reflection of waves from the outlet of the unused slit after passage and thus was not a true double-slit effect. A new theoretical treatment of bouncing drop dynamics has since given analytic relations for the associated wave field, leading to a proposal for criteria for the possible occurrence of true interference in the double-slit experiment. Satisfying these criteria, requires working close to the onset of the Faraday instability, with two spatial conditions favoring slow walkers, and a temporal condition favoring fast walkers. The regions of high velocity, where the walkers bounce periodically, and of very low velocity, with chaotically bouncing walkers, have not been comprehensively investigated so far. We have therefore looked at these regions, probing the limits for interaction with slits. Furthermore, noting that a short transit time is essential to fulfill the criteria, experiments were done using double-slit barriers only 0.5 and 2 mm broad. Nowhere was true interference or a chaotic response found. As the theory has implications for many of the observed quantumlike phenomena involving walkers as, e.g., tunneling and interaction between drops, we have measured the spatial and temporal decay of the wave field. A comparison with the theory shows very good agreement but leads to a reformulation of the above-mentioned criteria.
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Scattering theory of walking droplets in the presence of obstacles
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We aim to describe a droplet bouncing on a vibrating bath. Due to Faraday instability a surface wave is created at each bounce and serves as a pilot wave of the droplet. This leads to so called walking droplets or walkers. Since the seminal experiment by {\it Couder et al} [Phys. Rev. Lett. {\bf 97}, 154101 (2006)] there have been many attempts to accurately reproduce the experimental results. Here we present a simple and highly versatile model inspired from quantum mechanics. We propose to describe the trajectories of a walker using a Green function approach. The Green function is related to Helmholtz equation with Neumann boundary conditions on the obstacle(s) and outgoing conditions at infinity. For a single slit geometry our model is exactly solvable and reproduces some general features observed experimentally. It stands for a promising candidate to account for the presence of any boundaries in the walkers'dynamics.
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Quantumlike statistics of deterministic wave-particle interactions in a circular cavity
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  • Apr 2016
  • PRE
A deterministic low-dimensional iterated map is proposed here to describe the interaction between a bouncing droplet and Faraday waves confined to a circular cavity. Its solutions are investigated theoretically and numerically. The horizontal trajectory of the droplet can be chaotic: it then corresponds to a random walk of average step size equal to half the Faraday wavelength. An analogy is made between the diffusion coefficient of this random walk and the action per unit mass h/m of a quantum particle. The statistics of droplet position and speed are shaped by the cavity eigenmodes, in remarkable agreement with the solution of Schrödinger equation for a quantum particle in a similar potential well.
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The new wave of pilot-wave theory
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  • Aug 2015
Small drops bouncing across a vibrating liquid bath display many features reminiscent of quantum systems.
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Faraday pilot-wave dynamics: Modelling and computation
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  • Sep 2015
  • J FLUID MECH
A millimetric droplet bouncing on the surface of a vibrating fluid bath can self-propel by virtue of a resonant interaction with its own wave field. This system represents the first known example of a pilot-wave system of the form envisaged by Louis de Broglie in his double-solution pilot-wave theory. We here develop a fluid model of pilot-wave hydrodynamics by coupling recent models of the droplet's bouncing dynamics with a more realistic model of weakly viscous quasi-potential wave generation and evolution. The resulting model is the first to capture a number of features reported in experiment, including the rapid transient wave generated during impact, the Doppler effect and walker–walker interactions.
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Investigation of a path-memory dynamics: a theoretical trial
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  • Dec 2014
Waves and particles are distinct objects at a macroscopic scale. The existence of walkers, drops bouncing on a vertically vibrated fluid bath is a surprising case of dual objects at our scale. The drop is self-propelled, piloted by the standing surface waves generated by its previous rebounds. These objects exhibit a rich dynamics relying on the concept of path memory. Indeed, the wave field results from the position of the past impacts left all along the walker trajectory. The memory is tunable at will by simply changing the vertical acceleration of the bath. A series of experiments have revealed the surprising dynamical behaviors of this dual drop-wave entity. In this PhD, we give a theoretical understanding of the temporal non local structure of walkers. We explore the dynamics of numerical walkers in a two-dimensional harmonic potential. We observe that the system only reaches a relatively limited set of stable attractors, quantized in both extension and mean angular momentum, in excellent agreement with the experimental results. We investigate how the different time scales are intertwined, which decouples the short-time acting propulsion from the build-up of coherent wave structures at much longer time scales. We analyze the non-local mechanism revealing the internal symmetries of the walker which drives the convergence of the dynamics to a set of low-dimensional eigenstates.
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A model for Faraday pilot waves over variable topography
Article
  • Jan 2017
Couder et al. ( Nature , vol. 437 (7056), 2005, p. 208) discovered that droplets walking on a vibrating bath possess certain features previously thought to be exclusive to quantum systems. These millimetric droplets synchronize with their Faraday wavefield, creating a macroscopic pilot-wave system. In this paper we exploit the fact that the waves generated are nearly monochromatic and propose a hydrodynamic model capable of quantitatively capturing the interaction between bouncing drops and a variable topography. We show that our reduced model is able to reproduce some important experiments involving the drop–topography interaction, such as non-specular reflection and single-slit diffraction.
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Non-specular reflection of walking droplets
Article
  • Oct 2016
Since their discovery by Yves Couder and Emmanuel Fort, droplets walking on a vibrating liquid bath have attracted considerable attention because they unexpectedly exhibit certain features reminiscent of quantum particles. While the behaviour of walking droplets in unbounded geometries has to a large extent been rationalized theoretically, no such rationale exists for their behaviour in the presence of boundaries, as arises in a number of key quantum analogue systems. We here present the results of a combined experimental and theoretical study of the interaction of walking droplets with a submerged planar barrier. Droplets exhibit non-specular reflection, with a small range of reflection angles that is only weakly dependent on the system parameters, including the angle of incidence. The observed behaviour is captured by simulations based on a theoretical model that treats the boundaries as regions of reduced wave speed, and rationalized in terms of momentum considerations.
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Modeling the vertical motion of drops bouncing on a bounded fluid reservoir
Article
  • Mar 2016
We present a first-principles model of drops bouncing on a liquid reservoir. We consider a nearly inviscid liquid reservoir and track the waves that develop in a bounded domain. Bouncing drops are modeled as vertical linear springs. We obtain an expression for the contact force between drop and liquid surface and a model where the only adjustable parameter is an effective viscosity used to describe the waves on the reservoir’s surface. With no adjustable parameters associated to the drop, we recover experimental bouncing times and restitution coefficients. We use our model to describe the effect of the Bond, Ohnesorge, and Weber numbers on drops bouncing on a stationary reservoir. We also use our model to describe drops bouncing on an oscillated reservoir, describing various bouncing modes and a walking threshold.
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