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. Reference Oza, Rosales and Bush2013). 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 Reference Moláček and Bush2013b ). 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 Reference Moláček and Bush2013b ; Couchman et al. Reference Couchman, Turton and Bush2019). The time-averaged horizontal force balance may then be written as (Moláček & Bush Reference Moláček and Bush2013b )

(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. Reference Oza, Rosales and Bush2013). 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. Reference Oza, Rosales and Bush2013) of pilot-wave hydrodynamics, which has proven to be adequate in rationalising the bulk of observed walker behaviours (Oza et al. Reference Oza, Harris, Rosales and Bush2014a ; Turton et al. Reference Turton, Couchman and Bush2018). Nevertheless, it is known to have shortcomings when non-resonant effects arise (Primkulov et al. Reference Primkulov, Evans, Been and Bush2025), specifically when the walker bouncing phase varies owing to variability in the local wave form (Harris et al. Reference Harris, Moukhtar, Fort, Couder and Bush2013; Durey et al. Reference Durey, Milewski and Wang2020), which is particularly prevalent at high memory (Moláček & Bush Reference Moláček and Bush2013b ; Wind-Willassen et al. Reference Wind-Willassen, Moláček, Harris and Bush2013). 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. Reference Protière, Boudaoud and Couder2006; Wind-Willassen et al. Reference Wind-Willassen, Moláček, Harris and Bush2013) 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. Reference Tadrist, Shim, Gilet and Schlagheck2018; Couchman et al. Reference Couchman, Turton and Bush2019), 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. Reference Oza, Rosales and Bush2013).

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 Reference Moláček and Bush2013b )

(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. Reference Liu, Durey and Bush2023), 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. Reference Couchman, Turton and Bush2019), 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 Reference Moláček and Bush2013b )

(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 Reference Abramowitz and Stegun1964, 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.

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. Reference Oza, Harris, Rosales and Bush2014a ; Labousse et al. Reference Labousse, Oza, Perrard and Bush2016a ; Liu et al. Reference Liu, Durey and Bush2023). 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. Reference Oza, Harris, Rosales and Bush2014a ), 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. Reference Fort, Eddi, Boudaoud, Moukhtar and Couder2010; Harris & Bush Reference Harris and Bush2014; Oza et al. Reference Oza, Harris, Rosales and Bush2014a ). 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. Reference Oza, Harris, Rosales and Bush2014a ; Liu et al. Reference Liu, Durey and Bush2023). Following Oza et al. (Reference Oza, Harris, Rosales and Bush2014a ), 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 Reference Harris and Bush2014; Oza et al. Reference Oza, Wind-Willassen, Harris, Rosales and Bush2014b ). 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.