2.1 Experimental set-up

The bath and droplet are composed of pure 20 cSt silicone oil with density $\unicode[STIX]{x1D70C}=950~\text{kg}~\text{m}^{-3}$ and surface tension $\unicode[STIX]{x1D70E}=20.6~\text{dynes}~\text{cm}^{-1}$ . The circular bath of diameter 16 cm is surrounded by a shallow border of width 9 mm and depth 1.5 mm. This shallow region serves to damp the waves generated by the oscillating meniscus at the edge of the container, thereby minimizing wave reflection and so limiting the range of influence of the outer boundary (Eddi et al. Reference Eddi, Fort, Moisy and Couder2009; Harris & Bush Reference Harris and Bush2014). The bath is driven sinusoidally with frequency $f=80$  Hz and peak acceleration $\unicode[STIX]{x1D6FE}$ by an electromagnetic shaker guided by a linear air bearing which provides vertical vibration that is spatially uniform to within 0.1 % (Harris & Bush Reference Harris and Bush2015). The forcing is maintained at a constant acceleration amplitude to within $\pm 0.002g$ (where $g$ is the gravitational acceleration) using closed-loop feedback control. Above the Faraday threshold, $\unicode[STIX]{x1D6FE}_{F}$ , subharmonic standing waves form spontaneously on the free surface of the vibrating bath (Faraday Reference Faraday1831) with a typical wavelength determined by the standard water-wave dispersion relation (Benjamin & Ursell Reference Benjamin and Ursell1954), here $\unicode[STIX]{x1D706}_{F}=4.75$  mm.

The Faraday threshold is measured every 20 min as follows. We first set $\unicode[STIX]{x1D6FE}$ slightly above $\unicode[STIX]{x1D6FE}_{F}$ , then decrease it by $0.005g$ every minute. When the waves disappear, as visualized readily by a reflected light source, we take the Faraday threshold as the last recorded value of the acceleration. This procedure yields an uncertainty in $\unicode[STIX]{x1D6FE}_{F}$ of $\pm 0.005g$ . The sum of the relative errors of $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FE}_{F}$ gives a maximum uncertainty in $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}$ of 0.002. In the experiments closest to the Faraday threshold, at $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.998\pm 0.002$ , we continuously checked that the bath was quiescent in the absence of a drop, that the system was below the Faraday threshold at all times and throughout the bath. We henceforth refer to the experimental control parameter $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}$ as the memory. (We note that early theoretical formulations of the problem (Eddi et al. Reference Eddi, Sultan, Moukhtar, Fort, Rossi and Couder2011; Moláček & Bush Reference Moláček and Bush2013b ) defined the characteristic decay time of the waves as the memory time $T_{M}(\unicode[STIX]{x1D6FE})=T_{d}/(1-\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F})$ , where $T_{d}\sim 1/\unicode[STIX]{x1D708}{k_{F}}^{2}$ is the viscous decay time of the waves in the absence of forcing. These models approximate the waves as linear Faraday waves damped exclusively by the action of viscosity. When $\unicode[STIX]{x1D6FE}\rightarrow \unicode[STIX]{x1D6FE}_{F}$ (and $T_{M}$ diverges), this approximation breaks down and nonlinear wave effects become significant. The memory time $T_{M}$ appears nowhere in our theoretical formulation.) The threshold $\unicode[STIX]{x1D6FE}_{F}$ is always in the range $\unicode[STIX]{x1D6FE}_{F}=(4.18-4.23)g$ , with the variation resulting from the sensitivity of the fluid viscosity to temperature (Bechhoefer et al. Reference Bechhoefer, Ego, Manneville and Johnson1995; Harris & Bush Reference Harris and Bush2014).

We note an important point for future experimental investigations of the walking droplet system. Curved trajectories in the absence of barriers may be taken as an indication of air currents, non-uniform vibration, operating above the Faraday threshold, or interactions with the cell boundaries. Thus, a simple but critical test is to send the walker across the cell in the absence of barriers (i.e. with the slit barriers removed) and check that it executes rectilinear motion. We verified this to be the case in our system for all values of forcing considered.

Figure 1.

(a) Regime diagram indicating the dependence of the drop bouncing or walking state on the forcing amplitude $\unicode[STIX]{x1D6FE}/g$ and the vibration number $\unicode[STIX]{x1D6FA}=2\unicode[STIX]{x03C0}f\sqrt{\unicode[STIX]{x1D70C}D^{3}/8\unicode[STIX]{x1D70E}}$ (Wind-Willassen et al. Reference Wind-Willassen, Moláček, Harris and Bush2013; Bush Reference Bush2015). In the $(m,n)$ bouncing or walking mode, the drop bounces $n$ times in $m$ driving periods. The range of walkers considered in this work, of diameter $D=0.67$  mm, is indicated by the white line as arising entirely within the chaotic walking regime. (b) Visualization of the wave field of a walker as it passes through the single slit (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2017.790). (c,d) Trajectories of a walking droplet of diameter $D=0.80$  mm passing through the slit. For both panels, the impact parameter and experimental parameters are the same, the forcing amplitude is $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.95$ and the free speed of the walker is $u_{o}=12~\text{mm}~\text{s}^{-1}$ . (c) Three trajectories deduced without isolating the bath from ambient air currents. (d) Eight trajectories deduced after isolating the bath from air currents with a lid.

To demonstrate the influence of air currents in the diffraction experiments, we performed an experiment with a relatively large walker ( $D=0.80$  mm) with and without the lid (figure 1 c,d). Enhanced divergence of trajectories is clearly present when the system is not isolated from air currents. For smaller droplets, $D<0.80$  mm, the walker in the absence of the lid is deviated even more strongly by air currents and often reflected back from the barrier, so that it rarely passes through the aperture. The sensitivity to ambient air currents is consistent with the fact that walking droplets are neutrally stable to lateral perturbations (Oza, Rosales & Bush Reference Oza, Rosales and Bush2013): their direction of motion can be readily altered. It is now evident that isolation from air currents is a necessary requirement for repeatable experiments involving walkers interacting with complex bounding geometries.

For the fluid and driving frequency considered, bouncing droplets may become walkers if their diameter $D$ is within the range 0.49–0.95 mm, leaving a significant range of parameter space to explore. One of the key features of Couder & Fort (Reference Couder and Fort2006)’s experiments was the reported independence of the deflection angle and the impact parameter. For the bulk of the parameter regime considered in a preliminary series of experiments, this independence was not observed and quantum-like diffraction was not recovered. This independence only arose for a limited range of parameter space, including a drop of diameter $D=0.67$  mm. We thus focused on drops of this size, which have chaotic bouncing dynamics (figure 1 a): their natural vertical motion is not periodic even in the absence of perturbations such as those resulting from interaction with boundaries. Larger and smaller drops than $D=0.67$  mm exhibit qualitatively similar behaviour when interacting with slits, but notable differences. Larger drops are generally deflected less and smaller drops only pass through the slit for a limited range of impact parameters $|y_{i}|<y_{max}$ . Therefore, unless otherwise stated, silicone oil droplets of diameter $D=0.67$  mm are used in our study.

The drops are created using a piezoelectric droplet generator with repeatability in diameter of 1 % (Harris et al. Reference Harris, Liu and Bush2015). The droplets are deposited gently onto the bath by letting them slide down a curved surface wetted with the same silicone oil (Gilet & Bush Reference Gilet and Bush2012). After a droplet is deposited on the vibrating bath, the container is sealed with a transparent lid to isolate the system from ambient air currents while still allowing for clear visualization and tracking of the droplet position from above. Care is taken to avoid vibrational excitation of the lid: its fundamental mode of vibration has a frequency ${\sim}1000$  Hz, significantly higher than the bath vibration frequency. The walker is filmed at 10 frames per second and tracked using a custom particle-tracking algorithm implemented in MATLAB. Despite their chaotic bouncing dynamics, these walking droplets execute rectilinear motion with a constant horizontal speed  $u_{0}$ , provided they are sufficiently far from boundaries.

Figure 2.

Experimental set-ups for single- and double-slit geometries. (a) Side view of the submerged slit geometry. The depth above the barriers is $h_{1}=0.42$  mm. (b) Plan view. The droplet launcher (white) directs the drop towards the slit of width $L$ and breadth $b$ with an impact parameter $y_{i}$ . The droplet is deflected by an angle $\unicode[STIX]{x1D6FC}$ . The solid blue curve is a sample trajectory. The dotted square indicates the cell size used in the original single-slit experiments of Couder & Fort (Reference Couder and Fort2006). Trajectory reproducibility and the reloading process are apparent in supplementary movie 2. (c) Side view of the double-slit geometry. (d) Plan view. The solid blue curve is a sample trajectory. Here, $d$ is the separation between the slit centrelines.

If the container is to be sealed, the walker needs to be automatically and continuously guided towards the slits. For this purpose, we designed a droplet launcher, lying 1.2 mm beneath the free surface (figure 2 a). It consists of a straight channel that centres the droplet in the launcher and a diverging channel that gradually reduces the confinement so that the walker is not deflected as it exits. The impact parameter $y_{i}$ can then be varied by shifting the launcher. A sample trajectory is shown in figure 2(a). After passing through the slit, the trajectory of the walker is deviated by an angle $\unicode[STIX]{x1D6FC}$ , measured as the angle of the straightest consecutive 30 points (corresponding to a time interval of approximately 3 s) along the trajectory (identified as the segment with the maximum $r^{2}$ value of a fitted line). The measured angle is only weakly sensitive to the choice of the number of points, with an estimated maximum measurement error of $\pm 1^{\circ }$ . After passing through the slit, the walker follows the outer boundary, re-enters the launcher and the process is repeated (see supplementary movie 2). At extremely high memory, $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.998$ , the reloading process can take several minutes because of the strong interaction of the walker with the boundaries. We note that we could reduce the reloading time to 1–2 min by temporarily decreasing the acceleration to $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}\simeq 0.99$ during the reloading process.

In the single-slit experiment, two barriers of breadth $b=6$  mm are bolted to an aluminium base plate, the opening between them constituting the slit of width $L=14.7~\text{mm}=3.1\unicode[STIX]{x1D706}_{F}$ (figure 2 a). The slit geometry is thus as close as possible to that of Couder & Fort (Reference Couder and Fort2006), who used slit widths of $2.1\unicode[STIX]{x1D706}_{F}$ and $3.1\unicode[STIX]{x1D706}_{F}$ . We set the fluid depth above the barriers to be $h_{1}=0.42\pm 0.02$  mm, comparable to that used in previous experiments on walking droplets confined to a circular corral (Harris et al. Reference Harris, Moukhtar, Fort, Couder and Bush2013). The bath depth is $h_{o}=7.4$  mm. The wave field generated by a walker in a typical passage of the slit is shown in figure 1(b).

In the double-slit experiment, three barriers of breadth $b=6$  mm are bolted to the aluminium base plate. The double-slit geometry is composed of two slits of width $L=14.7~\text{mm}=3.1\unicode[STIX]{x1D706}_{F}$ whose centrelines are separated by $d=20~\text{mm}=4.21\unicode[STIX]{x1D706}_{F}$ (figure 2 b). The two slits are chosen with width equal to the single-slit geometry in order to directly compare the results. The bath depth is $h_{0}=8.4$  mm and the depth above the barriers again $h_{1}=0.42\pm 0.02$  mm. The wave field generated by a walker in a typical passage through the double-slit arrangement is shown in supplementary movie 3.

2.2 Experimental results

We begin by assessing whether the pilot-wave dynamics is sufficient to produce chaotic trajectories in the single-slit experiment. We first span the possible range of impact parameters for $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.985$ and obtain the pattern shown in figure 3(a). The incident ‘beam’ is clearly splayed by the spatial confinement. As the droplet approaches the slit, it drifts slightly towards the nearest barrier before receiving a lateral kick as it closes in. This pushes the droplet over the centreline, where it receives another weaker kick as it approaches the opposite barrier. This process acts to focus the incident walker towards the centre of the slit, crossing the centreline in the process. There is only a narrow range of impact parameters ( $|y_{i}|<0.1L$ ) where the droplet is weakly deflected. Otherwise, the deflection angle tends to one of two preferred angles, approximately $\pm 55^{\circ }$ , relative to the normal. The maximum observed deflection angle is $60^{\circ }$ . It should be noted also that the entire range of impact parameters is not accessible in this experiment: for $|y_{i}|>0.35L$ , the walker does not pass through the slit, but is instead reflected back towards the launcher. When the walker does pass through the slit, the deflection angle depends continuously on the impact parameter, as evidenced in figure 3(b). By fitting a continuous curve to these data, we can compute the corresponding probability density function by assuming a uniform density of impact parameters (figure 3 c). We proceed by examining the role of memory on the system behaviour.

Figure 3.

The interaction of a walking drop of diameter $D=0.67$  mm with a single slit; the free speed is $u_{0}=6.7\pm 0.1~\text{mm}~\text{s}^{-1}$ and the forcing is $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.985\pm 0.002$ . (a)  $N_{e}=171$ experimental trajectories. (b) Dependence of the deflection angle $\unicode[STIX]{x1D6FC}$ on the impact parameter $y_{i}$ . (c) The probability density function corresponds to the fitted curve in (b) assuming uniform density of impact parameters over the accessible range $|y_{i}|<0.35L$ . (d–f) Analogous theoretical results obtained from $N_{t}=600$ trajectories. The probability density function is obtained with [ $\sqrt{N_{t}}$ ] bins, where $[\cdot ]$ denotes the nearest integer.

In figure 4, we illustrate the dependence on the forcing amplitude of trajectories with a fixed impact parameter. Up to approximately $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.990$ , the behaviour is as previously described: the impact parameter uniquely determines the deflection angle (figure 4 a,b), which tends to increase with memory (figure 4 e). As the Faraday threshold is approached, the behaviour changes dramatically. At $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.995$ (figure 4 c), the walker is still deflected by the slit, but the deflection angle is no longer uniquely prescribed by the impact parameter. We henceforth refer to the associated trajectories as chaotic to indicate their unpredictable nature. When $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.998$ (figure 4 d), deflection angles in the range $-60^{\circ }<\unicode[STIX]{x1D6FC}<60^{\circ }$ are obtained. The splaying of the trajectories after the slit is preceded by a damped lateral oscillation in the trajectory and visible disturbances to the vertical motion of the droplet. While the form of the incident trajectories is relatively insensitive to the forcing amplitude, the outgoing trajectories evidently change dramatically as the Faraday threshold is approached. The reproducibility of trajectories and their dependence on memory is examined in the Appendix.

Figure 4.

Trajectories of a walking droplet with diameter $D=0.67$  mm passing through the slit with fixed impact parameter $y_{i}=+0.17L$ , and with respective forcing and free speed: (a)  $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.985$ , $u_{0}=6.4~\text{mm}~\text{s}^{-1}$ , (b)  $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.990$ , $u_{0}=6.6~\text{mm}~\text{s}^{-1}$ , (c)  $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.995$ , $u_{0}=6.8~\text{mm}~\text{s}^{-1}$ and (d)  $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.998$ , $u_{0}=6.9~\text{mm}~\text{s}^{-1}$ . (e) Dependence of the deflection angle $\unicode[STIX]{x1D6FC}$ on the forcing $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}$ for $y_{i}=+0.17L$ .

At the highest memory considered ( $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.998$ ), the impact parameter is varied uniformly over the accessible range to simulate a plane wave incident on the slit. Again, we see that the incident ‘beam’ is deflected as a result of the spatial confinement of the guiding wave of the walker (figure 5 a). However, there is no longer a simple relationship between the impact parameter and the deflection angle (figure 5 b). A limiting deflection angle still exists: deflection angles greater than $65^{\circ }$ are never observed. Once again, not all impact parameters are accessible: for $|y_{i}|>0.35L$ , the walker does not pass through the slit.

Figure 5.

(a) Trajectories of a walking droplet of diameter $D=0.67$  mm passing through the slit with forcing $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.998\pm 0.002$ and free speed $u_{0}=6.8\pm 0.2~\text{mm}~\text{s}^{-1}$ . The total number of independent trajectories is $N_{e}=235$ . (b) Dependence of the deflection angle $\unicode[STIX]{x1D6FC}$ on the impact parameter $y_{i}/L$ . In this chaotic regime, $\unicode[STIX]{x1D6FC}$ is evidently independent of $y_{i}/L$ . (c) Probability density function obtained with [ $\sqrt{N_{e}}$ ] bins. The curve is the far-field intensity pattern of a deflected plane wave with wavelength $\unicode[STIX]{x1D706}_{F}$ impinging on a slit of width $L$ , corresponding to $I(\unicode[STIX]{x1D6FC})$ in (1.1).

The statistical behaviour of the walkers in this chaotic regime is shown in figure 5(c). While some preferred angles do emerge, including a weak central peak, there is a dominance of large deflection angles, $|\unicode[STIX]{x1D6FC}|\sim 60^{\circ }$ . Thus, in both the chaotic and non-chaotic regimes, we observe a preference for large deflection angles near $55{-}60^{\circ }$ . We repeated the experiments with a reduced slit breadth ( $b=2$  mm) and observed very similar behaviour; in particular, the same favoured angle dominates the statistical distribution and a limiting deflection angle again exists. We also explored the behaviour for different values of $h_{1}$ and found that in the range $h_{1}=0.04{-}1.0$  mm, the trajectories and angle distributions are qualitatively similar to those reported in figure 5. We note that insensitivity to barrier depth was also reported by Pucci et al. (Reference Pucci, Sáenz, Faria and Bush2016) in their study of walkers reflecting off a planar barrier.

Figure 6.

(a) The green lines indicate the observed trajectories of a walker of diameter $D=0.67$  mm passing a single barrier or ‘edge’ (green) of breadth $b=2$  mm. Here, $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.990\pm 0.002$ and $u_{o}=6.8\pm 0.2~\text{mm}~\text{s}^{-1}$ . The grey background represents the results from figure 5(a). (b) Theoretical trajectories deduced for a droplet walking past an edge at $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.995$ . (c) Computed deflection angle as a function of the impact parameter. Trajectories far from the edge (with impact parameter $y_{i}>10\,\unicode[STIX]{x1D706}_{F}$ ) are essentially unaffected by the presence of the barrier, while trajectories sufficiently close to the edge are drawn towards $\unicode[STIX]{x1D6FC}\approx 60^{\circ }$ . (d) Probability density function for a single slit of width $20\,\unicode[STIX]{x1D706}_{F}$ inferred by symmetrizing the computed trajectories past an edge up to $y_{i}=10\,\unicode[STIX]{x1D706}_{F}$ ( $N_{t}=570$ data points and [ $\sqrt{N_{t}}$ ] bins).

To gain further insight, we consider the case of diffraction from an edge of breadth $b=2$  mm (figure 6 a). If released sufficiently far from the barrier edge, the walkers continue along a straight path. However, if they approach within several $\unicode[STIX]{x1D706}_{F}$ , they are deflected towards a preferred angle $\unicode[STIX]{x1D6FC}$ . The emergence of the same preferred angle $\unicode[STIX]{x1D6FC}\sim 55{-}60^{\circ }$ for the edge geometry suggests that this angle is a generic feature of walker–wall interactions, including reflection (Pucci et al. Reference Pucci, Sáenz, Faria and Bush2016).

To complete our experimental investigation, we explore the behaviour of walkers in the double-slit geometry at extremely high memory, $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.998$ (figure 7), where the statistical behaviour for the single slit is relatively rich. We see again that the incident ‘beam’ is deflected as a result of the spatial confinement of the wave field of the walker (figure 7 a). Here, again, the behaviour and statistics are dominated by large deflection angles $\unicode[STIX]{x1D6FC}\sim 60^{\circ }$ and not all of the impact parameters are accessible: for $|y_{i}|>0.4(L+d/2)$ , the walker does not pass through the slit. For the largest impact parameters, $y_{i}=\pm 0.4(L+d/2)$ , the droplet is strongly attracted by the second slit after passing through the first; consequently, it reflects off the wall and may even loop around its own wave field (Labousse et al. Reference Labousse, Perrard, Couder and Fort2016b ) (see figure 7 a). Our experimental data are not perfectly symmetric because this anomalous behaviour arises for a narrow range of impact parameters explored only for one of the two slits.

Chaotic trajectories emerge predominantly for relatively large impact parameters, $|y_{i}|\geqslant 0.3(L+d/2)$ (figure 7 b), in contrast to the single-slit geometry, where chaotic trajectories arise for most impact parameters (figure 5 b). Once again, the wall effect dominates and most trajectories tend to the angles ${\sim}60^{\circ }$ , substantially reducing the chaotic behaviour and evidently suppressing the central peak (figure 7 c). The difference between these trajectories and those obtained with the single slit is highlighted in figure 8. Our experiments underscore the fact that, while the drop passes through one slit, it is influenced by both.

Figure 7.

Double-slit experiments (a–c) and numerical simulations (d–f). (a) Trajectories of a walking droplet passing through the slits; the vibrational forcing is $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.998\pm 0.002$ and the free speed is $u_{o}=6.8\pm 0.2~\text{mm}~\text{s}^{-1}$ ; the total number of independent trajectories is $N_{e}=266$ . (b) Dependence of the deflection angle $\unicode[STIX]{x1D6FC}$ on the impact parameter  $y_{i}$ . (c) The probability density function, as obtained with [ $\sqrt{N_{e}}$ ] bins. The weak asymmetry results in part from slight lateral asymmetries in the distribution of impact parameters. The curve is the far-field intensity pattern of a deflected plane wave with wavelength $\unicode[STIX]{x1D706}_{F}$ impinging on two slits of width $L$ whose centrelines are separated by $d$ , corresponding to (1.2). (d) Numerical simulations of a walking droplet passing through the slits with forcing $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.995$ ; the number of trajectories is $N_{t}=416$ . (e) The computed dependence of $\unicode[STIX]{x1D6FC}$ on $y_{i}$ . (f) The computed probability density function with [ $\sqrt{N_{t}}$ ] bins.

Figure 8.

Comparison between observed (a) single-slit and (b) double-slit trajectories, with forcing $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.998\pm 0.002$ and free walking speed $u_{o}=6.8\pm 0.2~\text{mm}~\text{s}^{-1}$ . The marked difference between the two indicates the influence of the second slit.

3.1 Theoretical model

We briefly review the theoretical model of Faria (Reference Faria2017), which will be used here to investigate the dynamics of walkers passing through single and double slits. The model builds upon the quasipotential theory of Milewski et al. (Reference Milewski, Galeano-Rios, Nachbin and Bush2015) by making the simplifying assumption that the waves are nearly monochromatic; therefore, only waves with the Faraday frequency need to be considered. This approximation yields simplified equations for the surface waves which consist essentially of damped dispersive wave equations, given by

Here $\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D719}$ denote the free-surface displacement and the velocity potential on the free surface, $\boldsymbol{x}=(x,y)$ are the coordinates in the plane of the bath, $\unicode[STIX]{x1D735}_{\bot }=(\unicode[STIX]{x2202}_{x},\unicode[STIX]{x2202}_{y})$ is the surface gradient operator and $g(t)=g_{0}(1+\unicode[STIX]{x1D6FE}\cos (2\unicode[STIX]{x03C0}ft-\unicode[STIX]{x1D711}))$ . The parameter $\unicode[STIX]{x1D708}^{\ast }$ is an effective kinematic viscosity, chosen to capture the correct instability threshold $\unicode[STIX]{x1D6FE}_{F}$ . Finally, $b(\boldsymbol{x})$ is an effective depth, chosen to ensure that the dispersion relation for Faraday waves agrees precisely with that given by the quasipotential theory in regions of constant depth (see Faria (Reference Faria2017) for further details). The source term $P_{D}$ represents the effect of the drop on the surface and $\boldsymbol{x}_{p}$ denotes the horizontal position of the drop, which evolves according to the trajectory equation of Moláček & Bush (Reference Moláček and Bush2013b ),

The parameters $m,R_{0},\unicode[STIX]{x1D707}_{air}$ 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 bath. In the context of our simplified model, according to which the vertical drop motion is periodic, it can be shown that $F(t)=mg_{0}\sum _{n=0}^{\infty }\unicode[STIX]{x1D6FF}((t-nT_{F})/T_{F})$ , where the $n$ denote prior impacts.

We note that all information concerning the bottom topography enters the model through the term $b(\boldsymbol{x})$ , which is taken to be piecewise constant for the slit geometry considered here. Thus, the main effect of the topography on the waves is to change the local wave speed, allowing for reflection and transmission of waves respectively from and across the boundaries. Because the effective depth is chosen such that the Faraday waves are correctly modelled in both the deep and the shallow regions, the model adequately captures the reflection and transmission properties of walkers interacting with submerged barriers (Pucci et al. Reference Pucci, Sáenz, Faria and Bush2016).

A limitation of the model worth mentioning is its oversimplified treatment of the vertical dynamics of the drop. By assuming that the drop bounces periodically with a contact time infinitesimally small relative to the Faraday period, the vertical dynamics is completely eliminated. Although such an assumption is reasonable for drops bouncing in the periodic $(2,1)^{2}$ mode (Moláček & Bush (Reference Moláček and Bush2013a ,Reference Moláček and Bush b ); see figure 1 a), it is likely to have shortcomings in the chaotic walking regime considered here (Wind-Willassen et al. Reference Wind-Willassen, Moláček, Harris and Bush2013).

3.2 Theoretical results

Simulations of single-slit, edge and double-slit deflection are considered for a parameter regime similar to that explored experimentally. For all of the following simulations, we consider a drop of diameter 0.67 mm and fix the bouncing phase $\unicode[STIX]{x1D711}/(2\unicode[STIX]{x03C0})=0.28$ so as to have a free walking speed consistent with that of the experiments ( ${\approx}6.8~\text{mm}~\text{s}^{-1}$ ). The coefficient of tangential restitution is taken to be $c_{4}=0.17$ , as was suggested by Moláček & Bush (Reference Moláček and Bush2013b ). A resolution of $512\times 512$ Fourier modes is used on a horizontal domain of size $64\times 64$ Faraday wavelengths.

Overall, the simulations of the single-slit geometry show good agreement in terms of both dynamics and statistics (figure 3). A slight difference can be noted for trajectories with small impact parameters. The transition to large deflection angles, $\pm 55^{\circ }$ , is smoother in the simulations, where a larger number of weakly deflected trajectories results in a central peak in the statistical distribution of deflection angles (figure 3 e,f). Notably, this central peak is significantly smaller than those corresponding to the larger wall-induced angles, $\pm 55^{\circ }$ . Another difference is that in the simulations the entire range of impact parameters $y_{i}$ is accessible, so that the walker always crosses the slit (figure 3 d,e). We note that some differences between experimental and theoretical trajectories were also found in our study of reflection from a planar wall (Pucci et al. Reference Pucci, Sáenz, Faria and Bush2016). In both cases, the attraction of the drop to the barrier is less pronounced in the theoretical model than in experiments.

Figure 9.

(a,c,e) Computed trajectories of a drop of diameter $D=0.67$  mm passing through single slits of width $L=4\unicode[STIX]{x1D706}_{F}$ , $10\unicode[STIX]{x1D706}_{F}$ and $20\unicode[STIX]{x1D706}_{F}$ (number of trajectories $N_{t}=548$ , 544, 570 respectively). (b,d,f) The resulting probability density function of the deflection angle $\unicode[STIX]{x1D6FC}$ with $[\sqrt{N_{t}}]$ bins. All simulations are at $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.99$ . We see that for sufficiently wide slits ( $L\gtrsim 10\unicode[STIX]{x1D706}_{F}$ ), a central peak emerges as trajectories passing near the centreline are only weakly affected by the presence of the boundaries.

As previously noted, most of the trajectories through the single and double slits settle onto an approximately constant angle which appears to be caused by a wall effect. In figure 6(b,c), we observe that, as in the experiment reported in figure 6(a), such is also the case for motion past an edge: most trajectories with impact parameters smaller than $4\unicode[STIX]{x1D706}_{F}$ deflect to an angle of approximately $60^{\circ }\pm 5^{\circ }$ . In order to investigate the effect of the slit width on the trajectories, we show in figure 9 three different geometries corresponding to single slits of width $L=4\unicode[STIX]{x1D706}_{F},10\unicode[STIX]{x1D706}_{F},20\unicode[STIX]{x1D706}_{F}$ . The emergence of a central peak becomes clear in figure 9(f) and results from trajectories with impact parameters, $y_{i}<10\unicode[STIX]{x1D706}_{F}$ , too far from either edge to feel the wall effect. By knowing the deflection angle as a function of the impact parameter for trajectories past an edge (figure 6 c), we may infer, for sufficiently wide slits, the resulting probability density function of the deflection angle $\unicode[STIX]{x1D6FC}$ . Such a construction is shown in figure 6(d), where we plot the predicted probability density function for drops passing a slit of the same breadth and width, $L=20\unicode[STIX]{x1D706}_{F}$ . The consistency between figures 6(d) and 9(f) underscores the fact that the walker–wall interaction dominates the dynamics in the single-slit geometry.

Finally, we use the model to explore the double-slit geometry at a very high memory of $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{F}=0.995$ (see figure 7). We obtain satisfactory agreement with the experiments reported in § 2.2 and note two salient effects. First, while the droplet passes through only one slit, its trajectory is influenced by the presence of the second slit. This is readily apparent from the lack of symmetry of trajectories emerging from either slit (figure 8). Second, the distribution of trajectories is again dominated by the wall effect and does not resemble wave diffraction.