Time-reversed water waves generated from an instantaneous time mirror

Here we first summarize the theory of Bacot et al [10] To realize an ITM, a medium whose wave velocity is easy to control is necessary. Water proves to be a good candidate since water waves can be very dispersive. Specifically, the wave speed ${c}_{0}=\sqrt{{gh}}$ in shallow water [26] shows a dependence on the water depth h and the gravitational acceleration g. The wave velocity can then be easily changed by accelerating the water container, which in turn changes the effective gravitational acceleration inside the water domain. Meanwhile, a water wave pattern travels at a relatively low velocity and can be observed with naked eyes. Due to these reasons, an ITM can be created for the gravity-capillary waves by shaking a water tank in a short period of time.

During the perturbation, the wave velocity increases from an initial c₀ to $c^{\prime}$ and quickly reduces back to c₀, forming an impulse in the speed-time diagram. This process can be separated into three stages, including a short high-speed period corresponding to the perturbation and two long low-speed periods before and after that. The time boundaries joining these three stages can be characterized as two temporal interfaces, where the wave velocity changes abruptly. Just like waves reflect and refract at a spatial interface, they also undergo reflection and refraction at these temporal interfaces. According to the d'Alembert wave equation and the continuity condition, the reflective coefficient $R=(c^{\prime} -{c}_{0})/2c^{\prime}$ and the refractive coefficient $T=(c^{\prime} +{c}_{0})/2c^{\prime}$ should apply at the first interface [10]. Likewise, the temporal reflective coefficient $R^{\prime} =({c}_{0}-c^{\prime} )/2{c}_{0}$ and the refractive coefficient $T^{\prime} =({c}_{0}+c^{\prime} )/2{c}_{0}$ are valid when the wave velocity changes from $c^{\prime}$ back to c₀ at the second interface [10]. For example, after passing the first temporal interface at t = 0, the wave field changes from ϕ( r , t) to R ϕ( r , − t) + T ϕ( r , t) due to temporal reflection and refraction, with r being the position vector of a spatial point, while the wave vector remains unchanged.

As there exists two temporal interfaces, the waves undergo two temporal reflections and two refractions across each perturbation period. Consequently, the final reversed waveform is the superposition of several components. For convenience, set the first interface at t = 0 and the second at t = τ. The waveform evolves as

$$\begin{eqnarray*}\begin{array}{rcl}\phi \left({\boldsymbol{r}},t\right) & \to & R\phi \left({\boldsymbol{r}},-t\right)+T\phi \left({\boldsymbol{r}},t\right)\\ & \to & {RR}^{\prime} \phi \left({\boldsymbol{r}},t-2\tau \right)+{RT}^{\prime} \phi \left({\boldsymbol{r}},-t\right)+{TR}^{\prime} \phi \left({\boldsymbol{r}},2\tau -t\right)+{TT}^{\prime} \phi \left({\boldsymbol{r}},t\right),\end{array}\end{eqnarray*}$$

where each arrow represents the change of the waveform at a temporal interface. For simplicity, let the perturbation period τ be short enough such that the peak in the speed-time diagram can be approximated as an impulse described by a Dirac function. The reversed wave field ψ can then be written as [10],

$$\begin{eqnarray}&&\psi \left({\boldsymbol{r}},t\right)=\alpha \displaystyle \frac{\partial \phi }{\partial t}\left({\boldsymbol{r}},-t\right),\end{eqnarray} \tag{ 1 }$$

in which α is a coefficient indicating the amplitude of the perturbation. The minus sign before t on the right side suggests a backward propagation, that the time is 'reversed'. Therefore, the reversed wave pattern is the time derivative of the original waveform. There should exist two other packets that propagate forwards in the wave field, which are −α∂ϕ( r , t)/∂t and ϕ( r , t), but they cannot refocus back to the initial pattern.

When the perturbation period is infinitesimally short, the reversed wave pattern becomes the time derivative of the original one. However, equation (1) actually describes the time domain relationship between the waveforms ψ( r , t) and ϕ( r , t). Since wave shapes, especially those of water waves, are naturally mapped to the spatial domain when being observed or recorded, it is essential that the relationship between the shapes of the original and the reversed wave patterns be further interrogated.

As the experiment begins at t = −T, it is assumed that a wave pattern ϕ( r , −T) starts to emerge from the flat water surface, giving ∂ϕ( r , t)/∂t∣_t=−T  = 0. Within a short period Δ t, the field becomes ϕ( r , − T + Δt) and the wave pattern propagates outwards. After the perturbation at t = 0, the reversed waveform ψ( r , t) emerges, and is proportional to the time derivative of ϕ( r , − t). At the end of the experiment when t = T, ψ( r , T) = ∂ϕ( r , t)/∂t∣_t=−T  = 0 is expected. At this moment, the reversal water wave field completely subsides and becomes flat again provided that there are no boundary reflections. The final reversed pattern should appear at t = T − Δt as ψ( r , T − Δt). Integrating the d'Alembert wave equation yields,

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial \phi \left(r,t\right)}{\partial t}\right|}_{t=-T+{\rm{\Delta }}t}={\left.\displaystyle \frac{\partial \phi \left(r,t\right)}{\partial t}\right|}_{t=-T}+{c}^{2}{\rm{\Delta }}t{{\rm{\nabla }}}^{2}\phi \left(r,-T\right),\end{eqnarray} \tag{ 2 }$$

where Δt is again a short period. According to equation (1), ψ( r , T − Δt) is proportional to the time derivative of ϕ( r , − T + Δt). By further considering ∂ϕ( r , t)/∂t∣_t=−T  = 0 that is obtained earlier, one can conclude that,

$$\begin{eqnarray}&&\psi \left(r,T-{\rm{\Delta }}t\right)=\alpha \displaystyle \frac{\partial \phi \left(r,-T+{\rm{\Delta }}t\right)}{\partial t}=\alpha {c}^{2}{\rm{\Delta }}t{{\rm{\nabla }}}^{2}\phi \left(r,-T\right).\end{eqnarray} \tag{ 3 }$$

Since Δt is short, a Laplacian of the initial pattern should be a good measure of the final reversed pattern ψ( r , T − Δt).

An ITM is set up in the experiment. The 3D morphology of the water wave field is measured by a laser array device during the whole process. To verify the results of the theoretical analysis, the wave profiles at the beginning and at the end of the experiment are reconstructed for comparison.

As shown in figures 1(a) and (b), a cuboid water tank (20(L) × 20(W) × 10(H) cm³) is supported by four wooden planks, which are fixed at the corners of a vibrating table with strong double-sided tapes. Water drops are dripped into the tank from 10-cm above the water surface to produce ripples that spread out. At a certain moment, the vibrating table is shaken at an amplitude of −0.25 cm within a 100-ms period, allowing the effective gravitational acceleration change from 10 to 24.1 m· s⁻² and then back to 10 m· s⁻², which immediately results in a wave packet that propagates backwards and refocuses at the source location. This sudden perturbation serves as an instantaneous time mirror and produces the time-reversed wave pattern. At the same time, the whole process is recorded with an iPhone X working in the slow-motion mode.

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Throughout the experiment, the 3D shape of the water wave field is determined through displacement measurements. A 635-nm, 100-mW laser source emits light that passes through 400 (20 × 20) circular apertures on an aluminum baffle, forming an array of paralleled laser beams going upwards, see figure 1(c). To prevent the laser source itself from vibrating, it is isolated from other parts of the system. The square array of beams penetrates the tank and can be refracted at the water surface.

In the absence of water waves, the laser beams can produce 400 red points evenly distributed in a square array on a light screen (which is a piece of A4-size paper), see figure 1(d). As the waves propagate, the water surface becomes wavy, deflecting the laser beams. Due to the varying slope of the water surface, the red points on the light screen keep moving around their origins. Therefore, by measuring the offsets of the points, local slopes of the water surface can be determined according to the Snell's Law. By integrating the slope as a function of time, the global morphology of the water surface is obtained. In each experiment, a GNU software Tracker helps to extract the in-plane displacements of the 400 laser points. The local shapes of the water wave field are in turn figured out at several moments before and after the time mirror is activated. Using this method, an effective area of 4 × 4 cm² of the water surface can be measured in a single experiment.

In such an experimental protocol, it is critical to examine how different physics, e.g., the fluid viscosity and surface tension, play their roles, and whether the theoretical analysis given in the above applies here. To do this, some important dimensionless numbers [26] are evaluated. At the moment when the waves begin to propagate, the Reynolds number is Re = 6741.6, the Bond number is Bo = 55.6, and the Weber number We = 25.0. In determining these numbers, the following parameters are used: water density ρ = 10³kg· m⁻³, dynamic viscosity μ = 8.9 × 10⁻⁴ Pa·s, surface tension γ = 7.2 × 10⁻² N· m⁻¹, the characteristic length l = 0.02 m (approximately evaluated as the width of the wave packet), and the characteristic velocity U = 0.3 m· s⁻¹ (corresponding to the maximum particle velocity at the water surface). Therefore, inertia and gravity are dominating the wave motions at the water surface, while the influence of viscosity and surface tension can be reasonably neglected. Consequently, the theoretical analysis given in the above, which is based on the d'Alembert equation, is valid here.

Figure 2 shows several snapshots at equally distributed time intervals during the focusing process of the reversed waves, including the recorded laser point arrays, the reconstructed 3D water surface, and the numerical simulation results based on the d'Alembert wave equation where the wave speed c is a piecewise function described as

$$\begin{eqnarray}c\left(t\right)=\left\{\begin{array}{cc}2\alpha {c}_{0}/\tau , & -\tau /2\lt t\lt \tau /2\\ {c}_{0}, & \mathrm{otherwise}\end{array}\right..\end{eqnarray} \tag{ 4 }$$

and the parameters α = 0.25 s and τ = 100 ms are selected.

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As the water drop contacts the surface at t = −0.510 s, behaviors of the reversal waves at t = 0.344, 0.352, 0.360, and 0.369 s are presented in columns 1–4 in figures 2(a) and (b), respectively. In figure 2(a), the dense circle of points indicates a ripple, which propagates inwards with time (from left to right), focusing towards the source. From the displacements of the red points, the 3D morphologies of the water surface are reconstructed and given in figure 2(b). It is clear to observe that, initially the center of the ripple is not disturbed by waves. As the refocused waves propagate inwards, a wave packet is raised at the center. In order to validate the reconstructed patterns, a similar refocusing process is simulated using a commercial software Mathematica (v12.0, Wolfram). In the simulations, a standard Gaussian wave packet is used to model the pattern generated by a water droplet. Corresponding results in figure 2(c) show good consistency with the experimentally reconstructed patterns. Especially, in the last snapshots of figures 2(b) and (c), the water wave pattern automatically restores itself to the original form of a water droplet.

The wave behaviors at the very beginning of the spreading process and the very end of the refocusing process deserve attention, since equation (3) actually relates wave patterns at these two moments. To compare them explicitly, the corresponding reconstructed 3D morphologies are first given in figures 3(a) and (b), respectively. It is obvious that the ripple actually does not exist in the initial Gaussian wave pattern at t = −0.510 s, but is observed in the final reversed pattern at t = 0.505 s. It is mentioned that, the final reversed pattern is observed slightly before t = T = 0.510 s, just as we predicted theoretically in equation (3). One dimensional (1D) profiles along x = y are then plotted in figures 3(c) and (d) for comparison, where the two waveforms show distinct differences. To verify equation (3), the Laplacian of the initial wave profile is also presented in figure 3(d). Despite a small difference, the two wave profiles in figure 3(d) coincide well with each other, verifying the Laplacian relationship that we obtained. The difference might be ascribed to the reflections of the  − α∂ϕ( r , t)/∂t and ϕ( r , t) packets from the tank boundaries. In fact, after t = T = 0.510 s, we were only able to observe complicated grid-like patterns, which are obviously due to the reflected sequences.

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The results in the above only reveals the situation in a homogeneous medium. It is demonstrated by Fink that receiver-based time reversal can also be realized in a complex medium [25], which is very useful for wave focusing and imaging in the presence of medium scattering. For water waves studied here, the problem of medium inhomogeneity is also important because the wave velocity is dependent on the water depth h. Specifically, if the water tank has an uneven bottom, or is slightly tilted, the wave velocity should vary from place to place, and the pattern does not spread out at a uniform speed. In this case, the d'Alembert wave equation becomes

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}\phi }{\partial {t}^{2}}={c}^{2}\left({\boldsymbol{r}}\right){{\rm{\nabla }}}^{2}\phi ,\end{eqnarray} \tag{ 5 }$$

which is still linear with respect to t and satisfies time reversal symmetry. Therefore, ϕ( r , − t) should be a valid solution to this wave equation. By taking a partial time derivative of the equation, one obtains

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}}{\partial {\left(-t\right)}^{2}}\left(-\alpha \displaystyle \frac{\partial \phi }{\partial t}\right)={c}^{2}\left({\boldsymbol{r}}\right){{\rm{\nabla }}}^{2}\left(-\alpha \displaystyle \frac{\partial \phi }{\partial t}\right),\end{eqnarray} \tag{ 6 }$$

Since the continuity condition at t = 0 is valid whether or not the wave velocity is homogeneous, ψ( r , t) = α∂ϕ( r , − t)/∂t always satisfies equation (6) at any time after the perturbation. In other words, no matter how the wave velocity is distributed, the instantaneous time mirror always works, while the reversed wave profile is the time derivative of the original in the time domain and the Laplacian relationship also holds in the spatial domain.

This conclusion is demonstrated through a numerical simulation with the results given in figure 4. Here, the wave velocity c is designed as c² = ∣θ∣/2π + 0.3, where θ is the azimuthal angle. As a result, the waves spread faster on the left side (the  − x direction) than on the right side. By setting the initial wave pattern as a symmetric flower, the unevenly distributed c gradually breaks the symmetric pattern—the left half of the flower gets larger than the right half. After the ITM is activated at t = 0, a reversed wave pattern emerges and begins to propagate backwards, finally restoring the flower pattern to its initial symmetric shape. Therefore, the ITM is effective in an inhomogeneous medium.

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Activation of the ITM is known to change the total energy [10], which makes the temporal reflections and refractions distinctly different from the spatial ones. For example, in spatial reflection and refraction, the total energy is conserved when the wave pattern impinges onto a boundary. But here, energy of the gravity-capillary wave field is inconsistent before and after the ITM works. It should be of interest to discuss why this happens and how different parameters influence the total energy change. Firstly, the potential energy is conserved during the perturbation, since the shape of the water wave field remains unchanged at the perturbation moment. The kinetic energy, however, is changed due to the contribution of the work done by the vibrating table. For simplicity, the wave motions can be considered as a collection of harmonic oscillators, so the kinetic energy density is proportional to (∂ϕ/∂t)². From the continuity condition, the difference of the total energy before and after the perturbation can be obtained by integrating the kinetic energy density over the entire field. The ratio between the kinetic energy increment Δ E and the initial kinetic energy E₀ is

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}E}{{E}_{0}}\propto \displaystyle \frac{{\alpha }^{2}{c}^{2}}{{\sigma }^{2}},\end{eqnarray} \tag{ 7 }$$

in which σ is the width of the wave packet. From the results shown in figures 5(a) and (b), it is quite unexpected that the ITM requires more energy input when the waveform gets narrower. This is actually true due to the fact that, if σ approaches infinity, the water surface becomes completely flat and a perturbation cannot change the shape of the wave field, and no energy could be transferred to the system. In a spatial reflection, the energy remains unchanged but the momentum changes; in a temporal reflection, here the momentum remains unchanged but the energy changes; these reveal beautiful symmetry in physics.

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