The energetics of pilot-wave hydrodynamics

Abstract

A millimetric droplet may bounce and self-propel across the surface of a vertically vibrating liquid bath, guided by the slope of its accompanying Faraday wave field. The ‘walker’, consisting of a droplet dressed in a quasi-monochromatic wave form, is a spatially extended object that exhibits many phenomena previously thought exclusive to the quantum realm. While the walker dynamics can be remarkably complex, steady and periodic states arise in which the energy added by the bath vibration necessarily balances that dissipated by viscous effects. The system energetics may then be characterised in terms of the exchange between the bouncing droplet and its guiding or ‘pilot’ wave. We here characterise this energy exchange by means of a theoretical investigation into the dynamics of the pilot-wave system when time-averaged over one bouncing period. Specifically, we derive simple formulae characterising the dependence of the droplet’s gravitational potential energy and wave energy on the droplet speed. Doing so makes clear the partitioning between the gravitational, wave and kinetic energies of walking droplets in a number of steady, periodic and statistically steady dynamical states. We demonstrate that this partitioning depends exclusively on the ratio of the droplet speed to its speed limit.

1. Introduction

The hydrodynamic pilot-wave system (Bush 2010), discovered in 2005 by Yves Couder and Emmanuel Fort (Couder et al. 2005a ) has garnered considerable interest owing to its relation to quantum systems (Couder et al. 2005b ), and has provided the basis for the field of hydrodynamic quantum analogues (Bush & Oza 2021). The system consists of a millimetric droplet bouncing on the surface of a vibrating liquid bath. The vibrational acceleration is always less than the Faraday threshold (Benjamin & Ursell 1954; Kumar & Tuckerman 1994; Kumar 1996), above which Faraday waves arise even in the absence of the droplet, so the wave field navigated by the droplet is always entirely of its own making. The control parameter of the system is the bath’s vibrational acceleration, which prescribes the longevity of the waves generated by the droplet impacts, and so the ‘memory’ of the system (Eddi et al. 2011). Increasing the memory progressively towards the Faraday threshold prompts the stationary bouncing droplet to destabilise into a ‘walker’, a macroscopic realisation of wave–particle duality consisting of a self-propelling droplet dressed in a quasi-monochromatic pilot-wave form (Bush 2015; Bush & Oza 2021). As the memory is further increased, the walking speed and spatial extent of the wave field increase, while the droplet moves downwards on its pilot wave, towards a region with higher slope (see figure 1).

Figure 1. Stroboscopic pilot-wave fields, $h({\boldsymbol{x}},t)$ , generated by $(a)$ a stationary bouncer and $(b,c)$ walkers for vibrational forcing (a) $\gamma = \gamma _W$ (the walking threshold), (b) $\gamma /\gamma _F = 0.9$ and (c) $\gamma /\gamma _F = 0.97$ . Upper panels: overhead view of the stroboscopic wave field accompanying the droplet, with regions of elevation and depression highlighted in red and blue, respectively. Lower panels: the wave profile (in μm) along the line of droplet motion. The droplet speed and wave energy increase with the vibrational forcing, with the wave field spanning the plane in the high-memory limit. As the droplet moves faster, it moves away from the wave crest towards a region with higher slope, inducing a corresponding decrease in the local wave elevation. The plots are generated by solving the stroboscopic pilot-wave model (2.1) for walking at a constant speed (see (4.3)) with physical parameters $f = 80$ Hz, $\rho = 950 \mbox { kg m}^{-3}$ , $\nu = 20.9$ cSt, $\sigma = 0.206 \mbox { N m}^{-1}$ , $\mathcal {H} = 4$ mm, $R =$ 0.4 mm, $\sin \varPhi = 0.2$ , and $T_d = 0.0174$ s (see table 1). The walking threshold is $\gamma _W/\gamma _F = 0.782$ , the Faraday wavelength is $\lambda _F = 4.75$ mm and the maximum steady walking speed is $c = 13.3 \mbox { mm s}^{-1}$ .

The hydrodynamic pilot-wave system is driven and dissipative (Kutz et al. 2022; Rahman & Kutz 2023); specifically, energy is fed into the system through the bath vibration and dissipated through the action of fluid viscosity. This dissipation ultimately leads to a warming of the bath and an associated drift of the Faraday threshold (Douady 1990; Harris et al. 2013; Ellegaard & Levinsen 2020). Nevertheless, a number of steady, periodic and statistically steady dynamical states may be achieved in which the driving and dissipation effectively balance over the time scales of interest (seconds or minutes), so that the global energetics need not be considered. Examples include steady walking states and orbital motion when the droplet moves in response to an external force field (Fort et al. 2010; Harris & Bush 2014; Perrard et al. 2014b ). The existence and stability of the system’s various dynamical states change with the vibrational forcing, which in turn prescribes the spatial extent and longevity of the wave field. In the high-memory limit, the pilot-wave dynamics often appears chaotic (Tambasco et al. 2016), with the droplet motion intermittently switching between weakly unstable periodic states (Harris & Bush 2014; Perrard et al. 2014a ). A number of connections have been made between the energy of different dynamical states and their relative stability. For example, stable steady walking, orbiting and periodic states are all energetically favourable relative to unstable stationary bouncing (Durey & Milewski 2017; Durey et al. 2018; Liu et al. 2023). Energetic considerations also arise in bound states formed from multiple droplets. For example, the wave energy of promenading pairs (Arbelaiz et al. 2018), consisting of two coupled droplets walking in tandem, is less than that of two free walkers (Borghesi et al. 2014; Durey & Milewski 2017). Finally, rings of bouncing droplets destabilise and execute a number of discrete rearrangements in such a way as to minimise their net gravitational potential energy (Couchman & Bush 2020; Thomson et al. 2020b ).

Despite the growing number of connections between energy and stability, a global account of the energy evolution of the pilot-wave system has not been forthcoming. One recent advancement, however, was made by Liu et al. (2023), who demonstrated that for droplets executing circular orbits, the wave energy and the droplet gravitational potential energies are both prescribed by the droplet speed. Through systematic analysis of the system energetics, we here demonstrate the generality of this result by deriving a speed-dependent energy diminution factor relating the wave energy and drop gravitational potential energy of dynamic states to those of the static bouncing state. Our results are valid for a number of canonical settings, including steady and quasi-steady dynamics, periodic and slow oscillatory dynamics and statistically steady states.

In § 2, we introduce the pilot-wave system and determine evolution equations for the wave and droplet energies. We utilise these results in § 3 to determine the dependence of energy on the droplet’s speed for canonical dynamical and statistically steady states. A number of illustrative numerical examples are presented in § 4. In § 5, we enumerate insights into pilot-wave hydrodynamics provided by our energetic analysis.

Table 1. Table of parameters defining the stroboscopic pilot-wave model (Moláček & Bush 2013a , b ; Oza et al. 2013).

2. Pilot-wave hydrodynamics

We consider a vertically vibrating liquid with periodic acceleration $\gamma \sin (2\pi f t)$ at time $t$ . All relevant parameters are defined in table 1. For $\gamma \gt \gamma _F$ , the fluid rest state destabilises to a field of subharmonic standing or Faraday waves, with wavelength $\lambda _F = 2\pi /k_F$ and period $T_F = 2/f$ . Below the Faraday threshold, $\gamma \lt \gamma _F$ , any waves generated at the free surface, for example by a bouncing droplet, are subcritical, decaying over a time scale, $T_M$ , that increases with proximity to the Faraday threshold. For the weakly viscous liquids used in experiments (Wind-Willassen et al. 2013), the Faraday wavenumber, $k_F$ , approximately satisfies the gravity–capillary dispersion relation, $(\pi f)^2 = (g k_F + \sigma k_F^3/\rho)\tanh (k_F \mathcal {H})$ (Benjamin & Ursell 1954).

2.1. Governing equations

We model the evolution of a resonant walker bouncing with period $T_F$ , in perfect synchrony with its subharmonic pilot wave, $\eta ({\boldsymbol{x}},t)$ . We thus assume that each drop impact arises at the same wave amplitude and phase, the basis of the stroboscopic approximation (Oza et al. 2013). The droplet is propelled by the wave force $\boldsymbol{F\!}_w(t) = -F(t) \nabla \eta ({\boldsymbol{x}}_{{\kern-1.5pt}p},t)$ and resisted by drag, where ${\boldsymbol{x}}_{{\kern-1.5pt}p}(t)$ denotes the droplet’s horizontal position at time $t$ and $F(t)$ is the periodic vertical force acting on the droplet. Notably, the vertical force averaged over one bouncing period, $T_F$ , is precisely equal to the droplet weight, namely $\overline {F(t)} = mg$ . As the time scale of the droplet’s horizontal motion greatly exceeds that of its vertical motion, we model the wave field as $\eta ({\boldsymbol{x}},t) = \cos (\pi f t) \bar \eta ({\boldsymbol{x}},t)$ , representing the superposition of a fast subharmonic oscillation modulated by a slowly varying spatial profile (Moláček & Bush 2013b ). Time-averaging the droplet wave force over one bouncing period thus yields $\overline {\boldsymbol{F\!}_w(t)} = -mg\nabla h({\boldsymbol{x}}_{{\kern-1.5pt}p},t)$ , where $h({\boldsymbol{x}},t) = \mathscr {C}\, \bar \eta ({\boldsymbol{x}},t)$ is the stroboscopic wave field and $\mathscr {C} = \overline {F(t)\cos (\pi f t)}/mg$ denotes the phase of droplet impact relative to the wave-field oscillation (Moláček & Bush 2013b ; Couchman et al. 2019). The time-averaged horizontal force balance may then be written as (Moláček & Bush 2013b )

(2.1a) \begin{equation} m \ddot{\boldsymbol{x}}_{{\kern-1.5pt}p} + D \dot{\boldsymbol{x}}_{{\kern-1.5pt}p} = -mg \nabla h ({\boldsymbol{x}}_{{\kern-1.5pt}p}, t) + \boldsymbol{F}, \end{equation}

where $\boldsymbol{F}$ is an applied force and dots denote derivatives with respect to time.

We model the stroboscopic pilot wave by the integral

(2.1b) \begin{equation} h({\boldsymbol{x}},t) = \frac {A}{T_F} \int _{-\infty }^t \mathscr {H}\ (|{\boldsymbol{x}} - {\boldsymbol{x}}_{{\kern-1.5pt}p}(s)|)\mathrm {e}^{-(t - s)/T_M}{\, \mathrm {d}s}, \end{equation}

which represents a superposition of axisymmetric quasi-monochromatic waves of amplitude $A$ centred along the droplet’s path and decaying over the memory time scale, $T_M$ (Oza et al. 2013). Experimentally, this represents the wave form observed when the system is strobed at the Faraday frequency, and images captured at the phase of droplet impact, so that the droplet appears to surf along its pilot wave. The system (2.1) represents the stroboscopic model (Oza et al. 2013) of pilot-wave hydrodynamics, which has proven to be adequate in rationalising the bulk of observed walker behaviours (Oza et al. 2014a ; Turton et al. 2018). Nevertheless, it is known to have shortcomings when non-resonant effects arise (Primkulov et al. 2025), specifically when the walker bouncing phase varies owing to variability in the local wave form (Harris et al. 2013; Durey et al. 2020), which is particularly prevalent at high memory (Moláček & Bush 2013b ; Wind-Willassen et al. 2013). Consequently, the results of our study are unlikely to be applicable in experiments close to the Faraday threshold, for which exotic and chaotic bouncing modes (Protière et al. 2006; Wind-Willassen et al. 2013) are expected to significantly alter the pilot-wave energetics.

For the sake of simplicity, we proceed by focusing on the case of resonant walkers, and so neglect temporal variations in the droplet impact phase. We may thus treat the wave amplitude, $A$ , as a constant. We further assume that the wave kernel, $\mathscr {H}\ (r)$ , is a differentiable function, with $\mathscr {H}\ (0) = 1$ and $\mathscr {H}\ (r) \lt 1$ for all $r \gt 0$ . Furthermore, we posit that the spectrum is localised about $k_F$ , giving rise to a quasi-monochromatic form with wavelength $\lambda _F$ and exponential spatial decay in the farfield over the length scale $l_d$ . Motivated by the results of prior studies (Tadrist et al. 2018; Couchman et al. 2019), we assume that $l_d(\gamma)$ increases with the vibrational forcing, and diverges in the high-memory limit.

2.2. Droplet energy evolution

We proceed by describing the evolution of the droplet’s kinetic and potential energy, where we express our results in terms of the local wave height, $H(t) = h({\boldsymbol{x}}_{{\kern-1.5pt}p},t)$ . We first use the chain rule to write $\dot H = \partial _t h({\boldsymbol{x}}_{{\kern-1.5pt}p},t) + \dot{\boldsymbol{x}}_{{\kern-1.5pt}p} \cdot \nabla h({\boldsymbol{x}}_{{\kern-1.5pt}p},t)$ , whereupon substitution of (2.1) yields

(2.2) \begin{equation} \dot H = \frac {A}{T_F} - \frac {H}{T_M} + \frac {1}{mg}\dot{\boldsymbol{x}}_{{\kern-1.5pt}p} \cdot \Big (\boldsymbol{F} - m \ddot {\boldsymbol{x}}_{{\kern-1.5pt}p} - D \dot {\boldsymbol{x}}_{{\kern-1.5pt}p}\Big). \end{equation}

We then rearrange to find that the droplet’s energy evolves according to

(2.3) \begin{equation} \frac {\mathrm {d} {}}{\mathrm {d} {t}}\left (\frac {1}{2}m |\dot {\boldsymbol{x}}_{{\kern-1.5pt}p}|^2 + mgH\right) = \dot{\boldsymbol{x}}_{{\kern-1.5pt}p} \cdot \boldsymbol{F} + \frac {mgA}{T_F}\left (\gamma _D(|\dot {\boldsymbol{x}}_{{\kern-1.5pt}p}|) - \frac {H}{H_B}\right), \end{equation}

where $H_B = AT_M/T_F$ is the amplitude of the stroboscopic wave field for a bouncer,

(2.4) \begin{equation} \gamma _D(v) = 1 - \frac {v^2}{c^2} \end{equation}

is a speed-dependent diminution factor and $c = \sqrt {mgA/DT_F}$ is the maximum steady walking speed of a free walker, as arises in the high-memory limit (Oza et al. 2013).

For a droplet moving in response to the gradient of an applied potential, $\boldsymbol{F} = -\nabla V({\boldsymbol{x}}_{{\kern-1.5pt}p})$ , we may recast (2.3) as the work equation, as prescribes the rate of change of the droplet’s total energy:

(2.5) \begin{equation} \dot E_p = \frac {mgA}{T_F}\left (\gamma _D(|\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|) - \frac {H}{H_B}\right), \quad \mathrm {where}\quad E_p = \frac {1}{2}m|\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|^2 + V({\boldsymbol{x}}_{{\kern-1.5pt}p}) + mg H \end{equation}

is the sum of the droplet’s dimensionless kinetic and potential energies. Evidently, the droplet energy increases when $H/H_B \lt \gamma _D(|\dot {\boldsymbol{x}}_{{\kern-1.5pt}p}|)$ , remains constant when $H/H_B = \gamma _D(|\dot {\boldsymbol{x}}_{{\kern-1.5pt}p}|)$ and decreases otherwise. The implications of (2.5) for the system’s energy budget are discussed in Appendix A.

2.3. Wave-field energy evolution

For near-critical vibrational forcing, the evolution of the Faraday wave field, $\eta ({\boldsymbol{x}},t)$ , may be regarded as quasi-inviscid, characterised by an exchange between kinetic energy and a combination of gravitational potential and surface energies. We thus define the gravitational potential and surface energies for small-amplitude waves (i.e. $|\nabla \eta | \ll 1$ ) as (Moláček & Bush 2013b )

(2.6) \begin{equation} \mathscr {E}(t) = \iint _{\mathbb {R}^2} \frac {\rho g}{2} \eta ^2({\boldsymbol{x}},t) \, \mathrm {d}{\boldsymbol{x}} + \iint _{\mathbb {R}^2} \frac {\sigma }{2}|\nabla \eta |^2\, \mathrm {d}{\boldsymbol{x}}. \end{equation}

Owing to the assumed far-field exponential decay of the wave field, the integrals in (2.6) converge, which is not the case for the widely adopted and mathematically convenient choice $\mathscr {H}\ (r) = \mathrm {J}_0(k_F r)$ (Liu et al. 2023), an account of which is given in Appendix B.

By substituting the relationship $\eta ({\boldsymbol{x}},t) = \cos (\pi f t) h({\boldsymbol{x}},t)/\mathscr {C}$ into (2.6), we deduce that the energy of the wave field when time-averaged over one Faraday period, denoted $\overline {\mathscr {E}(t)}$ , is related to the energy of the stroboscopic pilot wave,

(2.7) \begin{equation} E(t) = \iint _{\mathbb {R}^2} \frac {\rho g}{2} h^2({\boldsymbol{x}},t) \, \mathrm {d}{\boldsymbol{x}} + \iint _{\mathbb {R}^2} \frac {\sigma }{2}|\nabla h|^2\, \mathrm {d}{\boldsymbol{x}}, \end{equation}

via $\overline {\mathscr {E}} = \frac {1}{2}E/\mathscr {C}\,^2$ . As $\mathscr {C}$ typically lies in the range $0.1 \lesssim \mathscr {C}\lesssim 0.2$ for walkers (Couchman et al. 2019), the time-averaged wave-field energy is at least one order of magnitude larger than the energy of the stroboscopic pilot wave. Nevertheless, we leverage this proportionality relationship to derive a simple formula governing the evolution for $E$ , and thus $\overline {\mathscr {E}}$ .

To proceed, we recast the stroboscopic pilot wave as

(2.8a) \begin{equation} h({\boldsymbol{x}},t) = \sum _{n = -\infty }^\infty \int _0^\infty k \hat {h}_n(k,t)\varPhi _n({\boldsymbol{x}},k)\, \mathrm {d}k, \end{equation}

where $\varPhi _n({\boldsymbol{x}},k) = \mathrm {J}_n(kr)\mathrm {e}^{\mathrm {i} n \theta }$ denotes an orthogonal set of functions defined in plane polar coordinates, ${\boldsymbol{x}} = r(\cos \theta,\sin \theta)$ , $\mathrm {J}_n$ is the Bessel function of the first kind of order $n$ , $k$ is the wavenumber and $\mathrm {i}$ denotes the imaginary unit. In terms of this basis decomposition, the stroboscopic wave energy defined in (2.7) is (Moláček & Bush 2013b )

(2.8b) \begin{equation} E(t) = \pi \sum _{n = -\infty }^\infty \int _0^\infty k \big (\rho g + \sigma k^2\big)|\hat {h}_n(k,t)|^2 \, \mathrm {d}k. \end{equation}

Furthermore, an application of Graf’s addition theorem (Abramowitz & Stegun 1964, 9.1.79) to (2.1b ) determines that each wave mode evolves according to

(2.9) \begin{equation} \hat {h}_n(k,t) = \frac {\hat {h}_B(k)}{T_M}\int _{ -\infty }^t \varPhi _n^*({\boldsymbol{x}}_{{\kern-1.5pt}p}(s), k)\mathrm {e}^{-(t-s)/T_M}{\, \mathrm {d}s}, \end{equation}

where $*$ denotes complex conjugation and $\hat {h}_B(k) = \int _0^\infty r h_B(r) \mathrm {J}_0(kr)\, \mathrm {d}r$ is the Hankel transform of the bouncer wave field, defined as $h_B(r) = H_B\mathscr {H}\ (r)$ with $H_B = AT_M/T_F$ .

We now exploit the fact that the spectrum of the bouncer wave field is sharply peaked around $k_F$ , with $\hat {h}_B(k)$ being appreciable only when $|k - k_F|l_d = O(1)$ , to determine approximate expressions for $H(t)$ and $E(t)$ valid in the high-memory limit, for which $l_d k_F \gg 1$ . To facilitate this analysis, we first substitute the leading-order Taylor expansion $\varPhi _n^*({\boldsymbol{x}}_{{\kern-1.5pt}p}, k) = \varPhi _n^*({\boldsymbol{x}}_{{\kern-1.5pt}p}, k_F) + O(\Delta k)$ (where $\Delta k = k/k_F - 1$ ) into (2.9), giving

(2.10) \begin{equation} \frac {\hat {h}_n(k,t)}{\hat {h}_B(k)} = \frac {1}{T_M}\int _{-\infty }^t \varPhi _n^*({\boldsymbol{x}}_{{\kern-1.5pt}p}(s), k_F)\mathrm {e}^{-(t-s)/T_M}{\, \mathrm {d}s} + O(\Delta k) = \frac {\hat {h}_n(k_F,t)}{\hat {h}_B(k_F)} + O(\Delta k), \end{equation}

or, equivalently,

(2.11) \begin{equation} \frac {\hat {h}_n(k,t)}{\hat {h}_n(k_F,t)} = \frac {\hat {h}_B(k)}{\hat {h}_B(k_F)} + O(\Delta k). \end{equation}

This relationship demonstrates that the spectrum close to $k_F$ is proportional to that of the bouncer wave field, which we exploit below to approximate the local wave height, $H(t)$ , and stroboscopic wave energy, $E(t)$ .

To approximate the local wave height, $H(t) = h({\boldsymbol{x}}_{{\kern-1.5pt}p},t)$ , we substitute $\varPhi _n({\boldsymbol{x}}_{{\kern-1.5pt}p}, k) = \varPhi _n({\boldsymbol{x}}_{{\kern-1.5pt}p}, k_F) + O(\Delta k)$ and (2.11) into (2.8a ), and then exploit the fact that the integrand is sharply peaked about $k_F$ for $l_d k_F \gg 1$ , giving

(2.12) \begin{equation} H(t) = \sum _{n=-\infty }^\infty \frac {\hat {h}_n(k_F,t)}{\hat {h}_B(k_F)} \varPhi _n({\boldsymbol{x}}_{{\kern-1.5pt}p}, k_F)\int _0^\infty k \hat {h}_B(k)\, \mathrm {d}k + O\big ((k_F l_d)^{-1}\big). \end{equation}

By noting that $h_B(0) = \int _0^\infty k \hat {h}_B(k)\, \mathrm {d}k$ , where $h_B(0) = H_B$ is the amplitude of the bouncer wave field, we deduce that

(2.13a) \begin{equation} H(t) = H_B \sum _{n = -\infty }^\infty \frac {\hat {h}_n(k_F, t)}{\hat {h}_B(k_F)}\varPhi _n({\boldsymbol{x}}_{{\kern-1.5pt}p}(t), k_F) + O\big ((k_F l_d)^{-1}\big), \end{equation}

thereby approximating the local wave height in terms of the system’s wave modes. Likewise, we substitute (2.11) into (2.8b ), giving

(2.13b) \begin{equation} E(t) = \pi \sum _{n=-\infty }^\infty \frac {|\hat {h}_n(k_F,t)|^2}{\hat {h}_B^2(k_F)}\int _0^\infty k(\rho g + \sigma k^2) \hat {h}_B^2(k)\, \mathrm {d}k + O\big ((k_F l_d)^{-1}\big), \end{equation}

where we have used that $\hat {h}_B(k)$ is real. This expression may be written more concisely as

(2.13c) \begin{equation} E(t) = E_B\sum _{n=-\infty }^\infty \frac {|\hat {h}_n(k_F,t)|^2}{\hat {h}_B^2(k_F)} + O\big ((k_F l_d)^{-1}\big), \end{equation}

where

(2.14) \begin{equation} E_B = \pi \int _0^\infty k (\rho g + \sigma k^2)\hat {h}_B^2(k)\, \mathrm {d}k \end{equation}

is the energy of the bouncer wave field. We thus deduce from (2.13) that $H$ and $E$ deviate from their respective values for a stationary bouncing droplet, denoted $H_B$ and $E_B$ , according to a weighted sum of the wave modes with wavenumber $k_F$ . These approximations form the foundation for our forthcoming analysis of the energy evolution of the stroboscopic wave field.

To determine the evolution of $E(t)$ , we first differentiate (2.9) and (2.13c ) with respect to time. By neglecting higher-order connections of size $O ((k_F l_d)^{-1})$ in the high-memory limit, corresponding to $k_F l_d \gg 1$ , we deduce that

(2.15) \begin{equation} \frac {\partial {\hat {h}_n}}{\partial {t}} = \frac {\hat {h}_B(k)}{T_M} \varPhi _n^*({\boldsymbol{x}}_{{\kern-1.5pt}p}(t), k) - \frac {\hat {h}_n}{T_M} \quad \mathrm {and}\quad \frac {\mathrm {d} {E}}{\mathrm {d} {t}} = E_B \sum _{n = -\infty }^\infty \frac {2\mathrm {Re}[\partial _t \hat {h}_n^*(k_F,t) \hat {h}_n(k_F,t)]}{\hat {h}_B^2(k_F)}, \end{equation}

respectively, where $\mathrm {Re}$ denotes the real part. By eliminating $\partial _t \hat {h}_n^*(k_F,t)$ from the second equation and using (2.13c ), we obtain

(2.16) \begin{equation} \frac {T_M}{2}\frac {\mathrm {d} {E}}{\mathrm {d} {t}} = E_B \sum _{n = -\infty }^\infty \frac {\hat {h}_n(k_F, t)}{\hat {h}_B(k_F)} \varPhi _n({\boldsymbol{x}}_{{\kern-1.5pt}p}(t), k_F) - E. \end{equation}

Finally, we use the approximate definition for $H(t)$ given in (2.13a ) to deduce the leading-order evolution equation for the wave energy:

(2.17) \begin{equation} \frac {\mathrm {d} {E}}{\mathrm {d} {t}} = \frac {2E_B}{T_M}\left (\frac {H(t)}{H_B} - \frac {E(t)}{E_B}\right). \end{equation}

The wave energy increases when the ratio of the wave height relative to a bouncer exceeds the corresponding ratio for wave energy, and decreases otherwise. Equation (2.17) represents a powerful formula for elucidating the evolution of wave energy in pilot-wave hydrodynamics, the implications of which for the system’s energy budget are discussed in Appendix A. We proceed by considering its application in a series of canonical dynamical regimes.

3. Energy evolution for canonical pilot-wave phenomena

Although the range of dynamics observed in experiments and predicted by the stroboscopic pilot-wave model (2.1) is vast (Bush 2015; Bush & Oza 2021), the qualitative behaviour may often be characterised by a small number of canonical regimes. To address the evolution of energy in these canonical regimes, we analyse the evolution formulae for the droplet and wave energy, as detailed in (2.5) and (2.17), respectively. Our results shed light on the dependence of energy on the droplet speed in a number of dynamical states.

The first canonical regime describes steady pilot-wave dynamics, for which the droplet and wave energies are conserved (Kutz et al. 2022; Liu et al. 2023; Rahman & Kutz 2023), as is the case for walking and orbiting at a constant speed. We thus deduce from (2.5) and (2.17) the respective relationships

(3.1a) \begin{equation} H/H_B = \gamma _D(|\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|) \quad \mbox {and}\quad E/E_B = H/H_B, \end{equation}

giving rise to the following expressions for the droplet and wave energies:

(3.1b) \begin{equation} E_p = \frac {1}{2}m |\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|^2 + V({\boldsymbol{x}}_{{\kern-1.5pt}p}) + mgH_B \gamma _D(|\dot {\boldsymbol{x}}_{{\kern-1.5pt}p}|) \quad \mathrm {and}\quad E = E_B \gamma _D(|\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|). \end{equation}

Notably, the wave and droplet gravitational potential energies are both maximised for a stationary droplet, and decrease as the droplet walks faster and its kinetic energy increases. We thus deduce that the wave energy of a walker is always less than the wave energy of the unstable bouncer at the same vibrational forcing, providing a theoretical rationale for the observations made by Durey & Milewski (2017) based on their numerical computations.

The second canonical dynamical regime is periodic droplet motion, for which the average droplet and wave energies are conserved over one periodic cycle (Kutz et al. 2022; Rahman & Kutz 2023), as is the case for wobbling and drifting orbital states in a rotating frame (Harris & Bush 2014), and lemniscate and trefoil trajectories in a central force (Perrard et al. 2014b ). In order to determine the average wave height, $\langle H \rangle$ , and wave energy, $\langle E \rangle$ , over one periodic cycle, we time-average (2.5) and (2.17), giving

(3.2a) \begin{equation} \langle H \rangle / H_B = \gamma _D(\bar v) \quad \mbox {and}\quad \langle E \rangle / E_B = \langle H \rangle / H_B, \end{equation}

where $\bar {v} = \sqrt {\langle |\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|^2 \rangle }$ is the time-averaged droplet speed. Consequently,

(3.2b) \begin{equation} \langle E_p \rangle = \frac {1}{2}m\bar {v}^2 + \langle V({\boldsymbol{x}}_{{\kern-1.5pt}p}) \rangle + mg H_B \gamma _D(\bar v) \quad \mathrm {and}\quad \langle E \rangle = E_B\gamma _D(\bar v), \end{equation}

with the average droplet and wave energies both taking a form analogous to that of their steady counterparts given in (3.1).

The third canonical regime corresponds to dynamical states that modulate slowly in time. In particular, equation (3.1) approximates quasi-steady dynamics, for which the time scale over which the pilot-wave system evolves greatly exceeds the memory time, and thus one may neglect $\dot E_p$ and $\dot E$ in (2.5) and (2.17). This slow variation is typical of the low-memory regime and near-critical non-oscillatory perturbations about a steady state, such as the ‘jump up’ and ‘jump down’ instabilities observed in a rotating frame, wherein the droplet shifts to an orbit with larger or smaller orbital radius, respectively (Oza et al. 2014b ). Similarly, equation (3.2) approximates dynamics consisting of a fast oscillation augmented by slow drift and amplitude modulations. Specifically, the slow drift and amplitude modulations render the average droplet and wave energies over one fast cycle relatively small and thus one may neglect $\langle \dot E_p \rangle$ and $\langle \dot E \rangle$ when averaging (2.5) and (2.17), respectively. This regime characterises near-critical oscillatory perturbations about a steady state, such as the onset of wobbling of circular orbits arising when a droplet is constrained by an external force (Oza et al. 2014b ).

The final canonical regime characterises statistically steady states, as might arise for chaotic pilot-wave dynamics in the long-path-memory limit, including the intermittent switching between weakly unstable states when the droplet moves in response to a Coriolis or central force (Harris & Bush 2014; Perrard et al. 2014b ). Of particular note is the prevalence of wake-like statistics in the hydrodynamic pilot-wave system and their connections to quantum mechanics (Bush & Oza 2021). We assume only that the droplet speed, local wave height and wave energy approach statistically steady states, and that $E_p$ and $E$ are bounded. By taking long-time averages of (2.5) and (2.17), we deduce relationships that are algebraically identical to (3.2), but with angled brackets now denoting long-time averages (Rahman & Kutz 2023).

Equations (3.1) and (3.2) have important consequences for energy evolution in the hydrodynamic pilot-wave system. Notably, the droplet gravitational potential and wave energies both decrease when the droplet speed increases, providing a simple heuristic for distinguishing energetically favourable states. In particular, the energy of an unstable bouncer always exceeds that of a droplet moving at a constant speed (Durey & Milewski 2017; Liu et al. 2023). Finally, the bound $E \gt 0$ imposes a droplet speed limit: specifically, (3.1) implies that $v \lt c$ for all steady states, while (3.2) implies that $\bar {v} \lt c$ for all periodic states.

We proceed by demonstrating that the droplet must move at its speed limit in order to maintain a finite local wave height in the high-memory limit. We first note that the wave height of a bouncer, $H_B = A T_M/T_F$ , increases linearly with memory. It thus follows from (3.1) that the limit $\lim _{T_M\rightarrow \infty } H = H_0$ for steady droplet motion requires that

(3.3) \begin{equation} \lim _{T_M\rightarrow \infty } \frac {A T_M}{T_F}\gamma _D(v) = H_0, \end{equation}

where $v(T_M)$ is the droplet speed and $H_0$ is the limiting wave height, which is assumed to be finite and positive. As $\gamma _D(c) = 0$ (see (2.4)), this limit can only be satisfied when

(3.4) \begin{equation} v \sim c\left (1 - \frac {H_0 T_F}{2 A T_M }\right) \quad \mbox {as}\quad T_M\rightarrow \infty, \end{equation}

with a similar relationship holding for the average speed, $\bar {v}$ , of periodic states. Consequently, the speed bounds $v \lt c$ and $\bar {v} \lt c$ are necessarily achieved for all steady and periodic states that retain a finite local wave height in the vicinity of the Faraday threshold.

Conversely, the wave energy diverges as the Faraday threshold is approached, by virtue of waves being excited across the entire bath. Specifically, we show in Appendix C that the energy of the bouncer wave field satisfies the scaling $E_B \sim T_M^2 l_d$ in the high-memory limit, reflecting the diverging wave amplitude and spatial decay length, $l_d(\gamma)$ , of the bouncer wave field close the Faraday threshold. For steady states with a finite wave amplitude in the high-memory limit, we combine (3.3) with the relationship $E = \gamma _D(v)E_B$ to deduce that

(3.5) \begin{equation} E \sim \frac {H_0 T_F}{AT_M}E_B \quad \mbox {as}\quad T_M\rightarrow \infty, \end{equation}

and likewise for periodic states. Consequently, the wave energy dominates the droplet’s kinetic and gravitational potential energies at high memory, as is evident in the energy evolution and partitioning to be considered in § 4.

4. Numerical examples

We proceed to exemplify our theoretical developments (§ 3) through a series of numerical examples. Drawing upon prior theoretical and experimental investigations (Damiano et al. 2016; Couchman et al. 2019), we define the axisymmetric wave kernel

(4.1) \begin{equation} \mathscr {H}\,(r) = \mathrm {J}_0(k_F r)\left [(1- \mathcal {S}(k_F r)) + \frac {r}{l_d}\mathrm {K}_1(r/l_d)\mathcal {S}(k_F r)\right], \end{equation}

where $\mathrm {K}_1$ is the modified Bessel function of the second kind of order one. The smoothing function, $\mathcal {S}(x) = \mathrm {e}^{-x^{-2}}$ , ensures regularity of the wave kernel near the origin, with $\mathcal {S}(x) \rightarrow 0$ as $x\rightarrow 0$ and $\mathcal {S}(x)\rightarrow 1$ as $x\rightarrow \infty$ (Couchman et al. 2019). The exponential far-field spatial decay is characterised by the parameter $l_d = \frac {1}{2}\sqrt {T_M/\alpha }$ , which increases with proximity of the vibrational forcing to the Faraday threshold, with $\alpha$ defined in table 1 (Moláček & Bush 2013b ). We consider the regime for which stationary bouncing is unstable, namely $\gamma _W \lt \gamma \lt \gamma _F$ , where the walking threshold is (Oza et al. 2013)

(4.2) \begin{equation} \gamma _W = \gamma _F\left (1 - \frac {c k_F T_d}{\sqrt {2}}\right), \end{equation}

with $T_d$ defined in table 1. We first consider steady walking and orbital motion (§ 4.1), before considering the onset of orbital instability in a rotating frame (§ 4.2). We emphasise that the validity of our theoretical developments is not restricted to the choice of wave kernel considered here.

4.1. Steady droplet motion

For steady droplet motion at speed $v \gt 0$ , the relationship $H = H_B\gamma _D(v)$ and (2.1b ) determine that the corresponding memory time, $T_M$ , satisfies

(4.3) \begin{equation} 1 - \frac {v^2}{c^2} = \frac {1}{T_M} \int _0^\infty \mathscr {H}\,\big (\delta (t)\big)\mathrm {e}^{-t/T_M}\, \mathrm {d}t, \end{equation}

where $\delta (t)$ denotes the droplet displacement over time $t$ . For rectilinear walking, $\delta (t) = vt$ , and for orbital motion with radius $r_0$ , $\delta (t) = 2r_0 |\sin (v t/2r_0)|$ . Notably, we parameterise orbital motion by its radius, as is the case for a droplet moving in response to a Coriolis or central force (Oza et al. 2014a ; Labousse et al. 2016a ; Liu et al. 2023). We solve (4.3) numerically, and then compute the energy, $E$ (defined in (2.7)), of the accompanying stroboscopic pilot wave using Fourier transforms on a large, doubly periodic domain of size $48\lambda _F \times 48 \lambda _F$ with $512 \times 512$ grid points. The numerical results are insensitive to increasing the size of the computational domain and to refining the spatial discretisation.

The rectilinear walking speed, $v_0$ , increases monotonically with the vibrational forcing, approaching the speed limit, $c$ , at the Faraday threshold (see figure 2 $a$ ). As the droplet walks faster, the local wave height, $H$ , and wave energy, $E$ , both decrease relative to that of a stationary bouncer, denoted $H_B$ and $E_B$ , respectively (see figure 2 $b,\!c$ ). The theoretical prediction for the local wave height, $H/H_B = \gamma _D(v_0)$ (see (3.1)), where $\gamma _D(v) = 1-v^2/c^2$ is the speed-dependent diminution factor, holds exactly for all $\gamma _W \lt \gamma \lt \gamma _F$ . Despite the theoretical prediction for wave energy, $E/E_B = \gamma _D(v_0)$ (see (3.1)), being formally valid only in the high-memory limit (for which the error in the approximation is of size $O((k_F l_d)^{-1})$ ), our numerical results demonstrate the efficacy of this prediction both within and beyond the high-memory regime. Indeed, the discrepancy between the theoretical and numerical solutions is visually indistinguishable in figure 2 $(b,\!c)$ , with the relative error being less than 1 % for all values of $\gamma$ .

Figure 2. The dependence of energy on the steady rectilinear walking speed, $v_0$ , computed using (4.3). $(a)$ The free-walking speed increases with the vibrational forcing for $\gamma _W \lt \gamma \lt \gamma _F$ , approaching the speed limit, $c$ , at the Faraday threshold. $(b, c)$ The magnitude of the local wave height, $H$ (red squares), and stroboscopic wave energy, $E$ (green diamonds), relative to that of a bouncer ( $H_B$ and $E_B$ , respectively) as a function of $(b)$ vibrational acceleration, $\gamma$ , and $(c)$ free-walking speed, $v_0$ . The numerical results coincide with the theoretical predictions $H/H_B = \gamma _D(v_0)$ and $E/E_B = \gamma _D(v_0)$ for steady droplet motion (see § 3) indicated by the black curve, where $\gamma _D(v) = 1-v^2/c^2$ is the speed-dependent diminution factor (see (2.4)). The physical parameter values are the same as in figure 1.

Figure 3. The dependence of energy on the steady orbital speed, $v$ , computed using (4.3) for $\gamma / \gamma _F$ taking values 0.9 (blue), 0.95 (green) and 0.98 (red). $(a)$ The orbital speed oscillates over half the Faraday wavelength as the orbital radius, $r_0$ , is increased, approaching the rectilinear walking speed, $v_0$ , for large radii. The variations in the orbital speed are most pronounced close to the Faraday threshold. $(b, c)$ The magnitude of the local wave height, $H$ (squares), and stroboscopic wave energy, $E$ (diamonds), relative to that of a bouncer ( $H_B$ and $E_B$ , respectively) as a function of $(b)$ orbital radius, $r_0$ , and $(c)$ orbital speed, $v$ . The numerical results coincide with the theoretical predictions $H/H_B = \gamma _D(v)$ and $E/E_B = \gamma _D(v)$ for steady droplet motion (see § 3) indicated by the black curve, where $\gamma _D(v) = 1-v^2/c^2$ is the speed-dependent diminution factor (see (2.4)). The physical parameter values are the same as in figure 1.

A similar physical picture emerges for orbital states. As the orbital radius, $r_0$ , is increased, the orbital speed, $v$ , varies with orbital radius, oscillating over half the Faraday wavelength, before approaching the rectilinear walking speed, $v_0$ , in the large-radius limit (see figure 3 $a$ ). This speed modulation is a consequence of the increased influence of the droplet’s wake on its motion at high orbital memory (Oza et al. 2014a ), with similar variations in the local wave height, $H$ , and wave energy, $E$ , evident as the orbital radius is increased (see figure 3 $b$ ). Despite this more complex dependence on orbital speed, the theoretical predictions prescribed by the speed-dependent diminution factor, $\gamma _D$ , are in excellent agreement with the numerical results across a wide range of memory and orbital radius. Indeed, the dependences of normalised local wave height, $H/H_B$ , and wave energy, $E/E_B$ , on orbital speed collapse onto the curve $\gamma _D(v)$ for all values of $\gamma$ (see figure 3 $c$ ), underscoring the universal dependence of energy on speed in orbital pilot-wave dynamics.

Of particular interest is the partitioning of energy in pilot-wave hydrodynamics, which we quantify here for steady walking and orbiting states. To do so, we define the total stroboscopic energy as

(4.4) \begin{equation} \mathcal {E} = \frac {1}{2}m|\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|^2 + mgH + E, \end{equation}

representing the sum of kinetic, gravitational potential and wave energies. For steady walking and orbiting, we apply the approximations given by (3.1) (as verified in figures 2 and 3) to deduce from (4.4) the approximate relationship

(4.5) \begin{equation} \mathcal {E} = \frac {1}{2}m|\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|^2 + \gamma _D(|\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|)\big (mg H_B + E_B\big), \end{equation}

which makes clear the dependence on the speed-dependent diminution factor, $\gamma _D$ , and the relative sizes of the local wave height, $H_B$ , and wave energy, $E_B$ , for stationary bouncing. In particular, we see from (4.5) that the contribution of kinetic energy to the total stroboscopic energy, $\mathcal {E}$ , increases with droplet speed, while the contributions of wave and gravitational potential energies both decrease.

Figure 4. The dependence of the energy partitioning on vibrational forcing for stationary bouncing (dashed curves) and steady walking at the free-walking speed, $v_0$ (solid curves). $(a)$ The contribution to the total energy, $\mathcal {E}$ (grey, see (4.4)), in terms of the stroboscopic wave energy, $E$ (red), droplet gravitational potential energy, $mg H$ (blue), and droplet kinetic energy, $\frac {1}{2}m v_0^2$ (black). The total energy is normalised by that of a bouncer at the walking threshold, $\mathcal {E}(\gamma _W)$ . The wave energy diverges in the high-memory limit for both bouncing and walking. Notably, the droplet gravitational potential energy diverges at high memory for a bouncer, yet decreases towards a finite value (comparable to the kinetic energy) for a walker. $(b)$ The relative contributions of each type of energy to the total energy, with the wave energy dominating in the high-memory limit. The physical parameter values are the same as in figure 1.

Figure 5. The dependence of the energy partitioning on the orbital radius for steady orbiting at $\gamma /\gamma _F = 0.95$ . $(a)$ The contributions to the total energy, $\mathcal {E}$ (grey, see (4.4)), from the stroboscopic wave energy, $E$ (red), droplet gravitational potential energy, $mg H$ (blue), and droplet kinetic energy, $\frac {1}{2}m v^2$ (black). The total energy is normalised by that of a bouncer at the walking threshold, $\mathcal {E}(\gamma _W)$ . $(b)$ Energy partition of $\mathcal {E}$ , with minima in the wave and gravitational potential energies corresponding to maxima in the kinetic energy. The physical parameter values are the same as in figure 1.

In figure 4, we present the dependence of the total stroboscopic energy on the vibrational forcing for stationary bouncing and steady walking. For both bouncing and walking, the wave energy diverges in the high-memory limit, where the wave field spans the entire plane and so is the main contributor to the total energy (see Appendix C). For steady walking, the kinetic energy increases with the vibrational forcing (consistent with the walking speed curve in figure 2 $a$ ), while the gravitational potential energy decreases, approaching a finite value in the high-memory limit. The adjustment of the droplet from its wave crest to a region with higher slope is also evident in the wave-field plots in figure 1. Finally, we note that although the gravitational potential energy of a bouncer diverges as the Faraday threshold is approached, this contribution is subdominant to the wave energy, with $mgH_B \ll E_B$ at high memory (see (4.5)).

Similar trends for orbital motion are evident in figure 5. At high memory, the wave energy dominates the droplet kinetic and gravitational potential energies. We note that modulations in the droplet speed with the orbital radius lead to peaks in the partitioning of kinetic energy, and corresponding troughs in the partitioning of wave energy. While it is not easily discernible from figure 5 $(b)$ , the droplet gravitational potential energy oscillates in a manner similar to that of the wave energy, by virtue of the relationships $H/H_B = E/E_B = \gamma _D(v)$ . As the vibrational forcing is progressively increased, the peaks and troughs in figure 5 $(b)$ become more pronounced, and the wave energy dominates the total energy.

4.2. Onset of unstable orbital motion

To investigate the efficacy of (3.2) for slowly varying and periodic states, we consider the onset of orbital instability in a rotating frame. A droplet walking in a rotating frame moves in response to a Coriolis force, $\boldsymbol{F} = -2m \boldsymbol{\Omega } \times \dot{\boldsymbol{x}}_{{\kern-1.5pt}p}$ , where the rotation vector, $\boldsymbol{\Omega } = \Omega \boldsymbol{e}_z$ , is orientated vertically, perpendicular to the droplet’s plane of motion (Fort et al. 2010; Harris & Bush 2014; Oza et al. 2014a ). For a droplet moving at steady speed $v = r_0 \omega$ along a circular orbit of radius $r_0$ , we deduce from pilot-wave system (2.1) the radial and tangential force balances:

(4.6a) \begin{align} -m r_0 \omega ^2 = -\frac {mgA}{T_F} \int _0^\infty \mathscr {H}\, \, '\left (2 r_0 \sin \left (\frac {\omega t}{2}\right)\right)\sin \left (\frac {\omega t}{2}\right)\mathrm {e}^{-t/T_M}\, \mathrm {d}t + 2m\varOmega r_0 \omega, \end{align}

(4.6b) \begin{align} D r_0 \omega = -\frac {mgA}{T_F}\int _0^\infty \mathscr {H}\, \, '\left (2 r_0 \sin \left (\frac {\omega t}{2}\right)\right)\cos \left (\frac {\omega t}{2}\right)\mathrm {e}^{-t/T_M}\, \mathrm {d}t,\qquad \end{align}

where $\mathscr {H}\, \, '(r)$ denotes the derivative of $\mathscr {H}\,\,$ with respect to $r$ (with an odd extension so as to be defined over the real line). Notably, one may apply integration by parts to the tangential force balance (4.6b ) to deduce the equation for local wave height (4.3) (Oza et al. 2014a ; Liu et al. 2023). Following Oza et al. (2014a ), we parameterise the circular orbits by their orbital radius for a fixed vibrational forcing: specifically, we fix $r_0$ and use (4.6b ) (or (4.3)) to solve for the orbital frequency, $\omega$ , and then use (4.6a ) to determine the corresponding bath rotation rate.

We investigate the evolution of the local wave height, $H(t)$ , and stroboscopic wave energy, $E(t)$ , for an oscillatory instability whose corresponding perturbation grows towards a 2-wobble (Harris & Bush 2014; Oza et al. 2014b ). We thus anticipate that (3.2) (with $V = 0$ ) will hold sufficiently close to the instability threshold, for which the instability time scale greatly exceeds the orbital time scale. This behaviour is evidenced in figure 6, for which the local wave height and wave energy when averaged over one speed cycle (denoted $\langle H \rangle /H_B$ and $\langle E \rangle /E_B$ , respectively) agree closely with the theoretical predictions made by (3.2) during both the unstable transient and the periodic 2-wobble. Notably, the theoretical prediction for $\langle E \rangle / E_B$ slightly exceeds the accompanying numerical result; this discrepancy is a direct consequence of the high-memory approximation made when deriving (2.17), and so will be less significant closer to the Faraday threshold.

We further test the efficacy of our theoretical predictions by comparing in figure 7 the evolution of $\mathrm {d}E/\mathrm {d}t$ in the simulations with the theoretical prediction prescribed by (2.17). This investigation is based on the same simulation as in figure 6, but with the focus shifted onto the instantaneous wave energy, rather than that time-averaged over each speed cycle. For the numerical simulations, $\mathrm {d}E/\mathrm {d}t$ is computed using a second-order finite-difference approximation of $E(t)$ , as defined by (2.7). For the theoretical prediction, the numerically computed values of $E(t)$ and $H(t)$ are substituted into the right-hand side of (2.17). Akin to figure 6 $(d)$ , the theoretical prediction for $\mathrm {d}E/\mathrm {d}t$ is typically slightly larger than its numerical counterpart, as is most apparent in the unstable transient (see figure 7 $a$ ). Nevertheless, there is excellent agreement when the droplet executes the periodic 2-wobble, as demonstrated in figure 7 $(b)$ for the final two orbital periods. Our findings verify that (2.17) may be reliably used to investigate the evolution of energy in pilot-wave hydrodynamics, forming a platform on which to base future investigations of unsteady systems.

Figure 6. Wobbling motion of an inertial orbit in a frame rotating at $\varOmega = -2.78 \mbox { rad s}^{-1}$ and vibrating with $\gamma /\gamma _F = 0.96$ . $(a)$ Droplet trajectory following a small perturbation from an anticlockwise circular orbit of radius $r_0/\lambda _F = 0.8$ . After an initial transient, the trajectory approaches a 2-wobble (grey curve, with the final orbital period denoted in blue). $(b)$ Evolution of the droplet speed (black curve). The red diamonds denote the maxima of each speed cycle, over which the average speed, $\bar v = \langle |\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|^2 \rangle ^{1/2}$ , is computed (see § 3). $(c,d)$ The black curves denote the evolution of $(c)$ the local wave height, $H/H_B$ , and $(d)$ the wave energy, $E/E_B$ , each normalised by that of a stationary bouncer at the same memory. Time is scaled by the orbital period, $T_O = 2\pi /\omega$ , of the initial circular orbit. The red circles denote the average values of $H/H_B$ and $E/E_B$ over each speed cycle, and the blue curve denotes the corresponding theoretical prediction given by $\langle H \rangle /H_B = \langle E \rangle /E_B = \gamma _D(\bar {v})$ (see (3.2)). The physical parameter values are the same as in figure 1.

Figure 7. The rate of change of wave energy for a droplet executing the same wobbling trajectory as in figure 6. $(a)$ The black curve denotes $\mathrm {d}E/\mathrm {d}t$ computed from the simulation, and the blue curve denotes $\mathrm {d}E/\mathrm {d}t$ as computed from the theoretical prediction (2.17). Time is scaled by the orbital period, $T_O$ , of the initial unstable orbit, and wave energy is normalised by that of a stationary bouncer, $E_B$ , at the same memory. $(b)$ Zoom-in of $(a)$ underscores the efficacy of the theoretical prediction when the droplet executes a 2-wobble.

5. Discussion

Our study has yielded several new insights into the energetics of pilot-wave hydrodynamics. The walking droplet system is damped and driven, with energy input from the bath vibration and energy dissipated through the action of viscosity. We have paid particular attention to the cases in which these energy inputs and outputs balance, specifically steady and periodic dynamical states, slowly varying states and statistically steady states. For such states, our study illustrates how the partitioning of energy between droplet kinetic energy, droplet gravitational energy and wave energy depends on the system memory. It further illustrates how this partitioning evolves with time for unsteady states. Specifically, the rate of change of droplet energy (2.5) is prescribed by the local wave amplitude relative to that of a bouncer and the diminution factor, $\gamma _D(|\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|)$ , which decreases with the droplet’s speed. The rate of change of wave energy (2.17) also depends on the instantaneous wave energy relative to that of a bouncer, and thus the diminution factor, $\gamma _D$ . Notably, the walker’s wave-induced added mass, $m\gamma _B(|\dot{\boldsymbol{x}}_{{\kern-1.5pt}p}|)$ , also decreases with its speed (Bush et al. 2014). A comparison between $\gamma _B$ and the energy diminution factor, $\gamma _D$ , is presented in Appendix D.

Our theoretical developments may be extended in a straightforward fashion in order to treat the energetics of $N$ interacting droplets with constant impact phase, as outlined in Appendix E. In particular, insight into multiple-droplet systems (e.g. lattices or rings) may be gained through consideration of the wave energy, $E(t)$ , which evolves according to (E7). For quasi-steady droplet motion, the correspondence

(5.1) \begin{equation} \frac {E(t)}{E_B} = \frac {N\,\overline{\!H}(t)}{H_B} \end{equation}

holds, where $\,\overline{\!H}$ is the average height of the wave field beneath the droplets, which incorporates wave-mediated inter-droplet spatio-temporal correlations (see (E8)). Consequently, the evolution of the mean lattice wave height may be used as a proxy for that of wave energy for slowly evolving lattices. The decrease in average droplet height between discrete rearrangements of unstable ring structures noted by Couchman & Bush (2020) was thus accompanied by a decrease in the energy of the global pilot-wave field, and so reflects the tendency of multiple-droplet systems to minimise their total energy. We note, however, that (5.1) does not account for dynamic or spatial variations in the droplets’ impact phase, and so is not necessarily applicable to irregular or unsteady lattice structures.

It is worth emphasising the generality of our results, but also noting their limitations. In our theoretical developments, we have assumed only that the axisymmetric wave kernel oscillates over the Faraday wavelength and decays exponentially at infinity, with the spatial decay length increasing with proximity to the Faraday threshold. Our results thus apply not only to the particular wave kernel considered in our numerical examples (§ 4) but to any such pilot-wave form. As a caveat, we note that our results have been deduced from the stroboscopic model, which is based on the assumption of resonance between the droplet and its guiding wave. The role of impact phase is somewhat obscured in the stroboscopic formulation, but becomes clear when one recalls that all of our energetic formulations have been made in terms of the stroboscopic wave height, $h({\boldsymbol{x}},t)$ , rather than the temporally oscillating wave field, $\eta ({\boldsymbol{x}},t) = h({\boldsymbol{x}},t) \cos (\pi ft)/\mathscr {C}$ . Notably, the energy of the stroboscopic wave field, $E$ , is related to the energy of the temporally oscillating wave field averaged over one Faraday period, $\overline {\mathscr {E}}$ , via $\overline {\mathscr {E}} = ({1}/{2})E/\mathscr {C}\ ^2$ , so the two energies are proportional. As the memory is increased, the phase of drop impact relative to the wave oscillation, $\mathscr {C}$ , also changes, and with it the relative magnitudes of the wave and droplet energies. This dependence is not captured with the stroboscopic model, which assumes that the impact phase is independent of memory. Nevertheless, our results may be applied given the value of $\mathscr {C}$ at a particular memory, as may be measured experimentally.

While the stroboscopic model has been successful in rationalising a number of phenomena in pilot-wave hydrodynamics, it is known to have shortcomings in situations where the droplet’s impact phase varies owing to its interaction with a time-varying wave field, as may arise at high memory (Wind-Willassen et al. 2013) and in closed settings (Harris et al. 2013; Durey et al. 2020; Abraham et al. 2024). Relaxation of the stroboscopic approximation requires consideration of the droplet’s vertical dynamics (Moláček & Bush 2013a ; Couchman et al. 2019; Primkulov et al. 2025). Consideration of the energetics in this more general setting will yield insight into the myriad complex periodic, aperiodic and chaotic walking states arising at high memory (Moláček & Bush 2013b ; Wind-Willassen et al. 2013) and will be the subject of future work.