2.1 Surface waves model

We are interested in the surface waves of a bath undergoing a vertical sinusoidal oscillation. In the bath’s frame of reference, the problem is equivalent to considering a time-dependent gravity: $g=g_{0}(1+\unicode[STIX]{x1D6E4}\cos (\unicode[STIX]{x1D6FA}t))$ , where $g_{0}$ is the gravitational acceleration, $\unicode[STIX]{x1D6FA}$ denotes the angular frequency of oscillation and $\unicode[STIX]{x1D6E4}$ the amplitude. For a fixed $\unicode[STIX]{x1D6FA}$ , there exists a critical amplitude of oscillation, $\unicode[STIX]{x1D6E4}_{F}$ , above which the surface becomes unstable. In the nearly inviscid limit of interest to this paper, the instability is always subharmonic, with frequency $\unicode[STIX]{x1D714}_{0}=\unicode[STIX]{x1D6FA}/2$ (Benjamin & Ursell Reference Benjamin and Ursell1954; Kumar Reference Kumar1996). The bouncing drop experiments described in the introduction are typically carried out near (and below) the stability boundary, so that the flat surface of the bath is stable, but subharmonic modes excited within it decay very slowly. Furthermore, the drops are small enough that the waves they generate may be treated as linear.

Instead of beginning with the linearized Navier–Stokes equations, we take as a starting point the quasi-potential theory developed in Milewski et al. (Reference Milewski, Galeano-Rios, Nachbin and Bush2015), which when extended to a bath of finite depth is given by

Here $\unicode[STIX]{x1D702},\unicode[STIX]{x1D719}$ , denote the free-surface displacement and velocity potential. The parameter $\unicode[STIX]{x1D708}^{\ast }$ is an effective kinematic viscosity, chosen to capture the correct stability threshold $\unicode[STIX]{x1D6E4}_{F}$ , and $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70C}$ denote surface tension and density. Finally, $\boldsymbol{n}$ is a vector normal to the bottom profile, $h(\boldsymbol{x})$ the depth of the undisturbed surface, and $\unicode[STIX]{x1D6E5}_{\bot }$ the horizontal Laplacian (i.e. $\unicode[STIX]{x1D6E5}_{\bot }=\unicode[STIX]{x2202}_{xx}+\unicode[STIX]{x2202}_{yy}$ ).

We consider a bath having a deep and a shallow region, separated by a discontinuous jump, so $h(\boldsymbol{x})$ is given by:

where $h_{0}>h_{1}$ . The key feature that we aim to exploit is that the sinusoidal oscillations of the bath favour the resonating temporal frequencies, causing the waves to be nearly monochromatic. Thus we need not model all waves correctly, but only those for which the frequency is approximately $\unicode[STIX]{x1D6FA}/2$ ; other waves are less important owing to their faster temporal decay.

The main difficulty in solving (2.1)–(2.4) is that $\unicode[STIX]{x1D719}_{z}$ in (2.4) couples the surface evolution to Laplace’s equation in the three-dimensional domain, making the problem computationally expensive. When $h(\boldsymbol{x})=\text{const.}$ , it can be shown that (see e.g. Whitham Reference Whitham1974)

where ${\mathcal{F}}$ denotes the two-dimensional Fourier transform in $x$ and $y$ and $\boldsymbol{k}=(k_{x},k_{y})$ is the wavenumber vector. This allows for the surface evolution to be represented, in Fourier space, using only data on the surface, reducing a three-dimensional problem to two dimensions. When $h(\boldsymbol{x})$ is not constant, however, no simple explicit representation of $\unicode[STIX]{x1D719}_{z}(\boldsymbol{x},0,t)$ in terms of $\unicode[STIX]{x1D719}(\boldsymbol{x},0,t)$ exists . Our goal is to approximate $\unicode[STIX]{x1D719}_{z}$ in (2.4) using data on the surface only, and to do so in a way that correctly models waves with frequency $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D6FA}/2$ .

Motivated by shallow-water theory, we seek an approximation of the form

and choose $b(\boldsymbol{x})$ to exactly match the dispersion relation of Faraday waves in both the shallow and deep regions. Given $\unicode[STIX]{x1D6FA}$ and $h_{0},h_{1}$ , we must thus determine the most unstable wavenumber in each region, denoted by $k_{F_{0}},k_{F_{1}}$ . Following appendix A of Milewski et al. (Reference Milewski, Galeano-Rios, Nachbin and Bush2015), the Faraday wavenumber in each region of constant depth is computed by finding the $k$ which minimizes $\unicode[STIX]{x1D6E4}$ in the following expression:

where $\unicode[STIX]{x1D714}_{I}^{2}(k)=k\tanh (kh)(g_{0}+\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}k^{2})$ is the ideal/inviscid dispersion relation, $\unicode[STIX]{x1D6FE}=2\unicode[STIX]{x1D708}^{\ast }k^{2}$ is the dissipation rate and $\unicode[STIX]{x1D714}_{g}=g_{0}k\tanh (kh)$ . Notice that this reduces to the well-known result of Benjamin & Ursell (Reference Benjamin and Ursell1954) when $\unicode[STIX]{x1D6FE}\rightarrow 0$ , and in that limit $k_{F_{0}},k_{F_{1}}$ simply satisfy

In order for (2.7) to exactly match the dispersion relation of Faraday waves in the regions of constant depth, we must choose

This leads to the following wave model:

where $b$ is given by (2.10).

It is worth noting that in the long-wave limit (i.e. $kh\rightarrow 0$ ), we have that $b=h$ and formally the linear shallow-water theory is recovered. Our approximation, however, does not rely on $k_{F_{0}}h_{0}\ll 1$ , which is not typically true for the bouncing drop experiments, but on the assumption that the wavefield is nearly monochromatic, with all the energy in modes of temporal frequency near $\unicode[STIX]{x1D6FA}/2$ . For drops bouncing periodically over a vibrating bath, parametric resonance ensures that such an approximation is justifiable, as later shown in the quantitative comparison performed in figure 1. In the next section we modify the wave model given by (2.11)–(2.12) in order to account for a drop bouncing on its surface.

Figure 1.

(a) Comparison between our approximation ( $y>0$ ), and the quasi-potential theory ( $y<0$ ) of Milewski et al. (Reference Milewski, Galeano-Rios, Nachbin and Bush2015) for a walker using exactly the same parameters. (b,c) Show the $x$ and $y$ cross-sections of the wavefield. The red line represents the quasi-potential theory, and the black line our model. Here $R=0.38~\text{mm}$ , $\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6E4}_{F}=0.9$ , and the drop is located at the origin.

2.2 Drop dynamics and wave/drop coupling

The wave–drop coupling is identical to that of Moláček & Bush (Reference Moláček and Bush2013b ) and Milewski et al. (Reference Milewski, Galeano-Rios, Nachbin and Bush2015), where the drop is treated as an excess pressure on the dynamic surface condition. This leads to the following system:

The new term $P_{D}$ represents the effect of the drop, and $\boldsymbol{x}_{p}$ denotes the drop’s horizontal position, which evolves according to

The parameters $m,R_{0},\unicode[STIX]{x1D707}_{\mathit{air}}$ and $c_{4}$ denote the drop mass, drop radius, air viscosity and coefficient of tangential restitution, respectively (Moláček & Bush Reference Moláček and Bush2013b ). The function $F(t)$ represents the reaction force exerted on the drop by the fluid.

Calculating $F(t)$ requires solving for the vertical dynamics of the particle, which, as shown in Moláček & Bush (Reference Moláček and Bush2013a ) can be well approximated by a logarithmic spring during contact. Although it is possible to couple the wave model here presented to the log-spring model of Moláček & Bush (Reference Moláček and Bush2013a ), for the sake of simplicity we assume that the drop bounces periodically, with period $T_{F}=4\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FA}$ . It can then be shown (see Moláček & Bush Reference Moláček and Bush2013b ) that $\int _{t}^{t+T_{F}}F(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}=mg_{0}T_{F}$ . Assuming that the contact time is small enough (relative to the bouncing period) that $F$ may be treated as an impulse force, and taking the impacts to happen at $t=nT_{F}$ , we obtain $F(t)=mg_{0}\sum _{n=0}^{\infty }\unicode[STIX]{x1D6FF}(t-nT_{F}/T_{F})$ , where $\unicode[STIX]{x1D6FF}$ is the Dirac delta function. This eliminates the need to solve for the vertical motion of the drop; however, by considering the impact to be instantaneous, the model ignores the evolution of the waves during contact, which may be important for certain types of interaction given that the contact time can be as large as one-fourth of the period. Finally the excess pressure $P_{D}$ may be modelled as $P_{D}(\boldsymbol{x}-\boldsymbol{x}_{p},t)=F(t)I(\Vert \boldsymbol{x}-\boldsymbol{x}_{p}\Vert )$ , where $I$ describes the spatial distribution of the excess pressure (and has units of $1/\text{length}^{2}$ ).

It is now convenient to non-dimensionalize the equations using a representative length scale, $\unicode[STIX]{x1D706}_{F}$ , and time scale $T_{F}$ . Using tildes for dimensionless variables, we obtain

where $G=g_{0}T_{F}^{2}/\unicode[STIX]{x1D706}_{F}$ , $Re=\unicode[STIX]{x1D706}_{F}^{2}/(t_{0}\unicode[STIX]{x1D708})$ , $Bo=\unicode[STIX]{x1D70E}T_{f}^{2}/(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D706}_{F}^{3})$ , $M=m/(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D706}_{F}^{3})$ , $\tilde{b}=b/\unicode[STIX]{x1D706}_{F}$ and $\unicode[STIX]{x1D711}$ denotes the impact phase. Similarly (2.15) becomes

where $C_{i}=c_{4}\sqrt{\unicode[STIX]{x1D70C}R_{0}/\unicode[STIX]{x1D70E}}gT_{F}$ and $C_{\mathit{air}}=6\unicode[STIX]{x03C0}R_{0}\unicode[STIX]{x1D707}_{\mathit{air}}T_{F}/m$ .

The last simplification comes from considering a point impact: $\tilde{I} (\tilde{\boldsymbol{x}}-\tilde{\boldsymbol{x}}_{p})=\unicode[STIX]{x1D6FF}(\tilde{\boldsymbol{x}}-\tilde{\boldsymbol{x}}_{p})$ . Equations (2.16)–(2.18) represent the final form of our model, and the rest of this paper is devoted to their study. Henceforth, we drop the tilde notation, but all variables are assumed dimensionless unless otherwise stated.

3.1 Constant depth

We consider first walkers in a uniform bath as a benchmark for the model. As expected, in the absence of barriers the drops move at a constant speed that is independent of the initial conditions. For any fixed set of physical parameters, the final free-space velocity will depend only on the impact phase $\unicode[STIX]{x1D711}$ . Because we have chosen not to consider the details of the vertical dynamics, $\unicode[STIX]{x1D711}$ is extraneous to the model and may therefore be treated as a free parameter. As in previous strobed models that average out the vertical dynamics (e.g. Oza et al. Reference Oza, Rosales and Bush2013; Labousse Reference Labousse2014), $\unicode[STIX]{x1D711}$ is chosen so that the theoretical walking speed approximately agrees with the experimentally reported value. A summary of the average experimental walking speed, as well as the inferred impact phase used for all configurations reported in this paper can be found in table 2.

Table 2.

Experimentally measured speed, $v$ , at different memories for the two drop sizes considered in this paper. The phase $\unicode[STIX]{x1D711}$ is obtained by requiring the numerically observed speed to match the experimental value, and is in the range consistent with experimental observation (Moláček & Bush Reference Moláček and Bush2013a ).

The first comparison we perform is between our model and the equations that are intended to approximate, specifically (2.1)–(2.4). We show in figure 1 the wavefield generated by a walker using our nearly monochromatic approximation, and compare it to the wavefield obtained when using the quasi-potential theory of Milewski et al. (Reference Milewski, Galeano-Rios, Nachbin and Bush2015). Although the match is not perfect, the overall wave features appear to be correctly captured. This indicates that, at least in places of constant depth, where it is easy to solve for $\unicode[STIX]{x1D719}_{z}(\boldsymbol{x},0,t)$ exactly using (2.6), our monochromatic approximation $\unicode[STIX]{x1D719}_{z}(\boldsymbol{x},0,t)\approx \unicode[STIX]{x1D735}\boldsymbol{\cdot }(b\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}(\boldsymbol{x},0,t))$ performs reasonably well. In the next sections, when considering a variable bottom topography, it becomes very difficult to solve (2.1)–(2.4); therefore, in § 3.2 we instead compare our model to experimental observations of Harris (Reference Harris2015) and Pucci et al. (Reference Pucci, Sáenz, Faria and Bush2016b ).

Figure 2.

Wave amplitude (contour plot) and trajectories (white line) for wall reflection. The black line at $x=50$ divides the deep ( $x<50$ ) and shallow ( $x>50$ ) regions. The leftmost and rightmost images are taken 60 bounces before and after the turning point, respectively. The centre images depict the wavefield exactly at the turning point. The red line represents the corresponding experimental data (Pucci et al. Reference Pucci, Sáenz, Faria and Bush2016b ). (a) $\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6E4}_{F}=0.90$ , (b) $\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6E4}_{F}=0.99$ .

3.2 Reflection from planar boundary

We consider now one of the simplest types of walker–boundary interactions: reflection from a planar submerged obstacles, and compare two of our trajectories to the experimental data of Pucci et al. (Reference Pucci, Sáenz, Faria and Bush2016b ). The simulation consists of sending a walker towards a shallower region at a given incident angle and then recording both the trajectory and the wavefield for several hundred impacts. The walker–boundary interaction can be divided into three distinct stages, depicted in figure 2 for two different values of $\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6E4}_{F}$ . First, at sufficient distance from the shallow region ( $x>50~\text{mm}$ in figure 2), the walker does not feel the presence of the step, and therefore moves in a straight line. Then, as it approaches the barrier, its reflected waves start to affect the dynamics, eventually causing the walker to turn around. The drop continues to interact with the submerged step through its wavefield as it moves away from it: the trajectory appears to curve towards the step, becoming asymptotic to a rectilinear motion. At high memory (figure 2 b), we also see the formation of interference fringes caused by the reflected waves. Finally, we observe that although the model approximately captures the correct reflection angle (experimental and numerical lines are nearly parallel in figure 2), the details of the trajectory near the wall are slightly different from the experiments, where the experimental trajectories turn more sharply. Interestingly, the reflection is non-specular (i.e. incident and reflected angles are not equal), and there is a small preferred range of reflected angles. A detailed study of the walker’s non-specular reflection, as well as a more through comparison between the theory here developed and experiments, is reported in Pucci et al. (Reference Pucci, Sáenz, Faria and Bush2016b ). We also see in figure 2 that, as expected, the waves have a much smaller amplitude in the shallow region ( $x>50$ ) than in the deep region ( $x<50$ ). However, neither $\unicode[STIX]{x1D702}_{x}$ nor $\unicode[STIX]{x1D702}$ appears to be zero at the deep–shallow interface. This inference, consistent with the schlieren imaging of Eddi et al. (Reference Eddi, Fort, Moisy and Couder2009) and Damiano et al. (Reference Damiano, Brun, Harris, Galeano-Rios and Bush2016), 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.

Behaviour of the wavefield at the interface between the shallow (0.42 mm) and the deep (6.09 mm) regions. In (a) we plot the $x$ cross-section of the wavefield of a walker impinging perpendicularly upon a step, as a function of time. The walker’s trajectory is given by $(x(t),0)$ ), and is shown in red. In (b) we plot both $\unicode[STIX]{x1D702}$ and its normal derivative, $\unicode[STIX]{x1D702}_{x}$ . In (c) we show the relation between $\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D702}_{x}$ at $(x,y)=(10,0)$ through the reflection process.

In order to study in more detail what happens to $\unicode[STIX]{x1D702}$ at the deep–shallow interface, we consider in figure 3 a walker impinging perpendicularly upon the step. The drop’s trajectory remains orthogonal to the step at all times, allowing for an easier study of the waves by focusing on the $x$ cross-section of the wavefield. We show in figure 3(a) the $x$ cross-section of the wavefield as a function of time (i.e. $\unicode[STIX]{x1D702}(x,0,t)$ ), where $t=0$ has been chosen as the time where the drop turns away from the shallow region. We see that within about 1.5 wavelengths into the shallow region the wave amplitude has already decayed to nearly zero. The behaviour of $\unicode[STIX]{x1D702}$ above the step, indicated by the black line in the figure, suggests that neither $\unicode[STIX]{x1D702}=0$ nor $\unicode[STIX]{x1D702}_{x}=0$ holds at the deep–shallow interface. This is better seen in figure 3(b), where we plot $\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D702}_{x}$ at $(x,y)=(10,0)$ (i.e. above the step) as a function of time. We see that, as the drop approaches the wall, the wave height above the step (blue curve in figure 3 b) oscillates in time, indicating the presence of a transmitted wave.

Figure 4.

Interaction between a walker and a submerged step as a function of the depth above the step. In (a) we show a few trajectories for depths ranging from very shallow ( $h_{1}\approx 0.01\unicode[STIX]{x1D706}_{F}$ ) to very deep ( $h_{1}\approx 0.9\unicode[STIX]{x1D706}_{F}$ ). In (b) we show the final angle, measured relative to the $x$ -axis, as a function of the step’s depth. The vertical solid line separates the drops which were reflected from the drops which were transmitted. The dashed horizontal lines indicate the asymptotic angles for either very shallow or very deep regions.

The relationship between $\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D702}_{x}$ at the deep–shallow interface, plotted in figure 3(c), appears to be rather complex. If the variable depth in the fluid were to be modelled by means of an effective boundary condition of the Robin type, i.e. $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}+\unicode[STIX]{x1D6FD}\unicode[STIX]{x2202}\unicode[STIX]{x1D702}/\unicode[STIX]{x2202}n=\unicode[STIX]{x1D705}$ , then figure 3(c) should show (at least approximately) a straight line; for the depth ratio considered in figure 3, this is not the case. Thus, in the context of our model, an effective boundary condition of the Robin type cannot be used in lieu of the spatially varying coefficient in the partial differential equation (PDE) without sacrificing quantitative agreement, and a more complex boundary condition must be sought. Of course, it may very well be that in certain limits (e.g. as $h_{1}\rightarrow 0$ ) the relation between $\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D702}_{x}$ becomes simple, and an effective boundary of the Robin type becomes appropriate.

We also use the model to investigate the effect of the height above the step on the trajectory. In figure 4 we consider a walker coming from a deep region ( $h_{0}=6.09~\text{mm}$ ) and impinging upon a step at an angle of $45^{\circ }$ . We vary the depth above the step between 0.05 mm and 6.09 mm by small increments, and then calculate the final angle after the walker has interacted with the step. We see that the final angle appears to asymptote to a constant for either very shallow or very deep steps. The drops is unable to feel the bottom (i.e. the trajectories do not significantly deviate from a straight line) for depths larger than ${\approx}0.6\unicode[STIX]{x1D706}_{F}$ . For depths less than ${\approx}0.2\unicode[STIX]{x1D706}_{F}$ , we also see that the reflection angle depends only weakly on the depth. This observation, consistent with experiments (G. Pucci, private communication, 2016), suggests that as long as the depth above the obstacles is small enough, the drop’s interaction with boundaries will be largely independent of the depth in the shallow region.

3.3 Diffraction by a slit

We now briefly consider the case of a walker launched towards a slit. If the drop behaved like a classical macroscopic particles, we would expect its trajectory to remain unaffected by the passage through a slit. In Couder & Fort (Reference Couder and Fort2006), it was shown that this is not the case: passage through a slit deflects the drop owing to the interaction of its wavefield and the barrier. It was found that walkers behaved (on average) much like a plane wave diffracted through a slit. Subsequent experiments conducted by Andersen et al. (Reference Andersen, Madsen, Reichelt, Ahl, Lautrup, Ellegaard, Levinsen and Bohr2015), Harris (Reference Harris2015) and Batelaan et al. (Reference Batelaan, Jones, Huang and Bach2016), were unable to reproduce the precise diffraction pattern observed in Couder & Fort (Reference Couder and Fort2006). Nevertheless, walker deflection was apparent in all of them. Harris (Reference Harris2015) found that, in the absence of external air currents, the single-slit trajectories were dominated by two preferred angles similar to those emerging in the reflection experiments. We proceed by showing that our numerical simulations are consistent with the findings reported in Harris (Reference Harris2015).

Figure 5.

Single-slit diffraction. In (a) we show the particle trajectories, colour coded according to the impact parameter. In (b) we plot the histogram of the deflection angle, measured with respect to the $y$ -axis.

The bath topography consists of a deep region of depth $h_{0}=6.09~\text{mm}$ , and shallow region, in the shape of a slit, of depth $h_{1}=0.42~\text{mm}$ (see figure 5). The width and breadth of the slit are respectively 15 mm and 5 mm wide. Letting $x_{i}$ be the distance of the initial position of the drop from the centre of the slit, we took a uniform sample of trajectories with impact parameters ranging from 0 mm to 7.5 mm by increments of 0.025 mm (300 cases in total). Trajectories with $x_{i}>6.375~\text{mm}$ reflected from the barrier, and were thus discarded. All simulations started at $y=-75~\text{mm}$ , which ensures that a steady walking state is achieved prior to interaction with the slit. We set $\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6E4}_{F}=0.99$ , and considered a drop of radius 0.335 mm, corresponding to the parameters examined by Harris (Reference Harris2015).

In figure 5(a) we show a few sample trajectories of our numerical simulations. The slit appears to focus trajectories inside it: its width plays an important role in the dynamics. The observed pattern does not appear to be of the classical wave diffraction type; in fact, most trajectories settle to a small range of angles $\unicode[STIX]{x1D703}$ after passing through the slit, which is reminiscent of the planar wall reflection and consistent with the findings reported in Harris (Reference Harris2015). Plotting a histogram of the final angle of each trajectory (figure 5 b), indicates a strong preference for $55^{\circ }<\unicode[STIX]{x1D703}<65^{\circ }$ , which is similar to the preferred angle observed in Harris (Reference Harris2015). The number of bins in figure 5(b) was chosen to be approximately $\sqrt{N}$ , where $N$ is the number of trajectories considered. The main qualitative feature of the histogram, i.e. its lateral peaks, remains unaffected by variations in bin size. A thorough comparison between our model and recent single- and double-slit experiments will be reported in Pucci et al. (Reference Pucci, Harris, Faria and Bush2016a ).