# Wave spectroscopy in a driven granular material

Michael Berhanu1<EMAIL_ADDRESS>Simon Merminod2
Gustavo Castilloh3  Eric Falcon1  1Université Paris Cité, CNRS, MSC, UMR 7057,
F-75013, Paris, France 2Department of Molecular and Cellular Biology, Harvard
University, Cambridge, MA 02138, USA 3Instituto de Ciencias de la Ingeniería,
Universidad O’Higgins, 2841959 Rancagua, Chile

###### Abstract

Driven granular media constitute model systems in out-of-equilibrium
statistical physics. By assimilating the motions of granular particles to
those of atoms, by analogy, one can obtain macroscopic equivalent of phase
transitions. Here, we study fluid-like and crystal-like two-dimensional states
in a driven granular material. In our experimental device, a tunable magnetic
field induces and controls remote interactions between the granular particles.
We use high-speed video recordings to analyse the velocity fluctuations of
individual particles in stationary regime. Using statistical averaging, we
find that the particles self-organize into collective excitations
characterized by dispersion relations in the frequency-wavenumber space. These
findings thus reveal that mechanical waves analogous to condensed matter
phonons propagate in driven granular media. When the magnetic coupling is
weak, the waves are longitudinal, as expected for a fluid-like phase. When the
coupling is stronger, both longitudinal and transverse waves propagate, which
is typically seen in solid-like phases. We model the dispersion relations
using the spatial distribution of particles and their interaction potential.
Finally, we infer the elastic parameters of the granular assembly from
equivalent sound velocities, thus realizing the spectroscopy of a granular
material.

##  1 Introduction

Granular systems are defined as assemblies of separate, macroscopic and
athermal particles Andreotti et al. (2013). Such systems occur in Nature under
a variety of forms: from the sand of beaches and dunes on Earth to small
astronomic bodies—interplanetary dust, meteoroids, and small asteroids—in
space. But despite the ubiquity of granular systems, a unified description of
the collective motions of their particles is still missing Mujica & Soto
(2016). A reason for this is the complexity due to energy dissipation through
particle-particle and particle-substrate friction and dissipative collisions.
Such energy dissipation also means that granular systems require continuous
inputs of energy to sustain their dynamics. Granular systems thus constitute
model systems to study matter out of the thermal equilibrium Mujica & Soto
(2016).

As it can be observed in everyday life, granular matter can be considered as a
solid, liquid or gas, depending on the particle density and the amount of
energy injected into the system, for instance mechanically. In molecular
systems, phase transitions result from the competition between thermal
agitation and the interaction between the particles. In granular systems, the
macroscopic counterpart to phase transitions Jaeger et al. (1996); Aranson &
Tsimring (2006); Andreotti et al. (2013) can be obtained by varying the number
of particles Falcon et al. (1999); Noirhomme et al. (2018, 2021), inducing
interactions between particles Aranson et al. (2000); Blair & Kudrolli (2003)
or tuning the confinement Schmidt & Löwen (1997). In particular, thin and
vibrated layers of grains Olafsen & Urbach (1998); Prevost et al. (2004);
Clerc et al. (2008); Castillo et al. (2012); Mujica & Soto (2016); Opsomer et
al. (2020) are of fundamental interest because they allow using particle
tracking techniques, yielding the dynamics of each grain individually and over
durations far exceeding those accessible through molecular dynamics
simulations. These studies bridge the gap between a description at the
particle scale and a global analysis of collective motions.

By analogy with the hydrodynamics description of molecular systems Hansen &
McDonald (2013), we characterize a driven granular system in a non-equilibrium
stationary state from the analysis of the particle velocity fluctuations in
the Fourier space. We previously introduced a model experimental system
Merminod et al. (2014, 2015); Castillo et al. (2020) to study the competition
between mechanical agitation and remote interactions in a quasi-two-
dimensional quasi-2D assembly of magnetized, millimeter-diameter spheres. For
a given cell filling and shaking strength, we observed a granular fluid-like
phase or a solid-like crystal phase depending on the strength of a tunable
dipolar repulsion Merminod et al. (2014). Here, we show that the spontaneous
collective excitations in this system can be described as a set of random
propagative waves characterized by dispersion relations. Thus, we obtain a
macroscopic equivalent of phonons, which correspond to the standard
decomposition of atomic vibrations under the form of normal modes in condensed
matter Ashcroft & Mermin (1976). Phonons propagate in both crystalline
lattices and amorphous matter. In our experiment, we take advantage of the
propagation of these waves to extract the elastic coefficients characterizing
the mechanical behaviour of the granular material. Therefore, our study
constitutes a sound spectroscopy, which gives access to the properties of the
material using a non-intrusive method, i.e., without submitting it to a
mechanical perturbation such as a shear. A similar approach using particle
tracking has been recently implemented in active matter for a system of self-
propelled colloidal rollers undergoing collective motion and which validated
the hydrodynamics description of active matter Geyer et al. (2018).

Collective excitations have also been studied in dusty plasmas, also called
complex plasmas, which are constituted of macroscopic, charged particles in
levitation and interacting through a screened Coulombian potential Fortov et
al. (2005); Morfill & Ivlev (2009). In 2D complex plasma crystals, phonon
spectra of longitudinal and transverse waves are generated by random particle
motions Nunomura et al. (2002); Couëdel et al. (2009, 2019). More recently,
numerical and theoretical works Golden et al. (2008, 2010); Khrapak et al.
(2018); Mistryukova et al. (2019) investigated the collective excitations of
2D assemblies of interacting dipoles ; however, these results are yet to be
experimentally validated. For dissipative granular gases, a few numerical
simulation works Vollmayr-Lee et al. (2011); Brito et al. (2013) computed
space-time spectra in dissipative granular gases, and showed that velocity
fluctuations organize into longitudinal waves. These works extracted effective
macroscopic coefficients —viscosity and diffusivity— which they compared with
the theoretical predictions from the fluctuating hydrodynamics theory of
driven granular media Sela & Goldhirsch (1998); van Noije et al. (1999); Sela
& Goldhirsch (2003); Gradenigo et al. (2011); Puglisi (2015). However, to our
knowledge, an experimental validation of these theoretical predictions is
still missing. In colloidal crystals, the mechanical constants of the medium
can be extracted from the fluctuations of particle displacements in response
to thermal agitation Sengupta et al. (2003); Zahn et al. (2003); von Grünberg
et al. (2004); Gasser et al. (2010), but this requires assuming thermal
equilibrium. A similar approach has been applied to granular particles with
electrostatic interactions Coupier (2006). In this article, the wave
spectroscopy that we carry out relies on the analysis of velocity fluctuations
in the spatial and temporal Fourier spaces; it thus necessitates high-speed
imaging with a frame rate of about $10^{3}$ Hz. This method does not require
thermal equilibrium and is thus advantageous for the out-of-equilibrium,
driven granular material studied here.

##  2 From granular gas to hexagonal crystal

Figure 1: Overview of our experimental system. (a) Top view schematics.
Millimetre-sized particles follow Brownian-like motions in quasi-two-
dimensions (black arrows) in the absence of magnetic field, and interact via
remote interactions when immersed in a transverse, operator-tuned magnetic
field, $\mathbf{B}$. (b) Side view schematics of a region of the experimental
cell. As the cell is vertically shaken (double grey arrow), the chrome steel
beads rebound on the rough bottom surface in random directions and velocities.
When the magnetic field immersing the cell is turned on (long red arrow), the
beads become induced magnetic dipoles (short red arrows) and repel each other
(orange arrows). The bead diameter is $a=1$ mm, the vertical gap is roughly
$1.5a$, and the root mean square roughness of the bottom surface is 20 $\mu$m.
(c) Dimensionless parameter $\varepsilon$, which quantifies the competition
between magnetic interactions and bead kinetic energy, increases with the
amplitude of the magnetic field $B=||\mathbf{B}||$. (d) Top-view photographs
show increasing order in the bead assembly as $\varepsilon$ increases and
dipolar forces strengthen. (e) The sixfold bond orientational parameter,
$|\Psi_{6}$—, monotonically increases as a function of $\varepsilon$. We
define four distinct regimes: “granular gas” (bead collisions occur,
$\varepsilon<10.7$), “dipolar fluid” (no collisions and low $|\Psi_{6}$—,
$10.7<\varepsilon<21.9$), “partially ordered phase” abbreviated as “partial
order” (no collisions and steeply increasing $|\Psi_{6}$—,
$21.9<\varepsilon<185$), “solid crystal” (no collisions and plateauing
$|\Psi_{6}$—, $\varepsilon>185$). Circled numbers correspond to the conditions
of the photographs in panel (d).

Our experimental system consists of a quasi-monolayer of millimetre-diameter
granular spheres that are vibrated on a rough plate and can be magnetized
through an external magnetic field (figure 1a-b). We used $N_{0}=2000$
spherical particles of diameter $a=1\,$mm and made of chromed steel, which is
a soft-ferromagnetic material. These particles form a monolayer inside a
horizontal, square cell of internal dimensions roughly 9 cm $\times$ 9 cm
$\times$ 1.5 mm. By vibrating the cell vertically with a frequency of 300 Hz
and an acceleration of $1.6g$, where $g$ is the gravitational acceleration,
the particles acquire kinetic energy and their motions projected onto the
horizontal plane are quasi-Brownian Olafsen & Urbach (1998); Losert et al.
(1999); Reis et al. (2007); Merminod et al. (2014). Using a high-speed camera
and tracking algorithms (see Materials and Methods) we compute the positions
and velocities of individual particles within a region of interest of area
$50.36\times 50.36~{}\text{mm}^{2}$ around the cell center.

Furthermore, we can turn on an external magnetic field, $\mathbf{B}$, which is
vertical—i.e., perpendicular to the bottom of the cell—so that each particle
becomes an induced magnetic dipole (figure 1b ). Thus, two particles located
in a same horizontal plane and whose centres are separated by a distance
$r_{i,j}$ interact through the repulsive potential Jackson (1999)

$U_{i,j}=\frac{4\pi}{\mu_{0}}\,B^{2}\frac{(a/2)^{6}}{r_{i,j}^{3}}\,,$ (1)

where $\mu_{0}$ is the vacuum permeability constant and $B=||\mathbf{B}||$.

Throughout this article, we use a single dimensionless parameter,
$\varepsilon$, to quantify the competition between kinetic energy and magnetic
repulsion in the particle assembly. Specifically, we define
$\varepsilon\equiv{E_{\textrm{m}}}/{E_{\textrm{K}}}$, where ${E_{\textrm{m}}}$
is the mean magnetic energy per particle (Eq. 4) and ${E_{\textrm{K}}}$ is the
mean kinetic energy per particle (Eq. 5). Parameter $\varepsilon$ is analogous
to an inverse temperature and constitutes our control parameter. We carried
out 31 experiments with $B$ between $[0,430]$ G, corresponding to values of
$\varepsilon$ between $[0,1500]$ (figure 1 c) ).

As we previously showed Merminod et al. (2014); Castillo et al. (2020),
increasing $\varepsilon$—through $B$—induces a gas-like to solid-like
transition in the granular quasi-monolayer (figure 1d). More specifically,
when $\varepsilon=0$, the combination of homogeneous mechanical forcing and
inherently dissipative collisions results in a granular gas state Olafsen &
Urbach (1998); Losert et al. (1999); Reis et al. (2007); Puglisi et al.
(2012); Mujica & Soto (2016). From $\varepsilon\sim 10$, we do not detect
collisions anymore and a fluid of interacting dipoles is formed Merminod et
al. (2014); Castillo et al. (2020). At the highest values of $\varepsilon$,
particles self-organize into a hexagonal, crystal-like structure. Notably, a
study with a similar experiment as our own Schockmel et al. (2013) showed that
this crystallization follows the Kosterlitz-Thouless-Halperin-Nelson-Young
(KTHNY) scenario Strandburg (1988); von Grünberg et al. (2007); Gasser et al.
(2010).

We define four distinct regimes of the granular assembly based on the degree
of orientational order and the collision rate in the system. As a measure
orientational order, we use the time- and ensemble-averaged sixfold bond-
orientational order parameter, or global hexagonal order parameter,
$|\Psi_{6}|$ (Eq. 7). $|\Psi_{6}|$ equals 1 for a perfectly hexagonal lattice
and is smaller otherwise. We find that $|\Psi_{6}|$ increases from roughly 0.4
to 0.9 across the full range of $\varepsilon$ (figure 1e). First, we define
the “granular gas” state for $0<\varepsilon<10.7$, where $|\Psi_{6}|$ is the
smallest, collisions occur—although at a decreasing rate as $\varepsilon$
increases—and the particle assembly behaves as a dissipative granular gas
Castillo et al. (2020) (Movies S1 and S5). Second, for
$10.7<\varepsilon<21.9$, the “dipolar fluid” regime consists in a disordered
fluid of strongly coupled dipoles, for which $|\Psi_{6}|$ is still near 0.4
but there are no more collisions (Movies S2 and S6). Magnetic energy then
significantly dominates kinetic energy, and the system behaves similarly to
dusty plasmas Donkó et al. (2008); Morfill & Ivlev (2009). Third, in the
“partial order” regime, for $21.9<\varepsilon<185$, the hexagonal order
parameter $|\Psi_{6}|$ increases from roughly 0.4 to 0.8, with the maximum
variation in $|\Psi_{6}|$ occurring near $\varepsilon=62$ (Movies S3 and S7).
Finally, for $\varepsilon>185$, we define the “solid crystal” state as
hexagonal order is high, with $|\Psi_{6}|>0.8$ (Movies S4 and S8). There,
while particles do not exchange neighbours, they vibrate around their local
magnetic energy minimum, which together define the hexagonal lattice, due to
the mechanical shaking. Note that defects are present in this crystal-like
structure, but they likely do not significantly impact the dynamical
properties which are the focus of this article. See the electronic
supplementary material Sections 5 and 6 for a more detailed characterization
of these four regimes using the pair correlation function and the mean-square
displacements.

Collective excitations correspond to a coherent behaviour of these vibrations
evidenced using a representation in the Fourier space. Such excitations have
been studied theoretically and numerically for 2D systems of interacting
dipoles Golden et al. (2008, 2010); Khrapak et al. (2018); Mistryukova et al.
(2019). In particular, Khrapak et al. Khrapak et al. (2018) studied in details
the dynamics and thermodynamics of such systems, depending on a parameter
equivalent to our $\varepsilon$, and for both strongly coupled dipolar fluid
and solid phases; their work provides a theoretical guide for our experimental
study.

##  3 Experimental spectra of velocity fluctuations

Figure 2:  Two-dimensional wave propagation. (a,c,e,g) Space-time spectra of
longitudinal excitations, ${{S_{\textrm{L}}}}(k,\omega)$, in each of the four
regimes. (b,d,f,h) Space-time spectra of transverse excitations,
${{S_{\textrm{T}}}}(k,\omega)$. (a–h) In each panel, we obtain an experimental
dispersion relation, ${{\omega_{\textrm{R,L}}}}(k)$ (respectively,
${{\omega_{\textrm{R,T}}}}(k)$, black bullets), and its width,
${{\Lambda_{\textrm{L}}}}(k)$ (resp., ${{\Lambda_{\textrm{T}}}}(k)$, grey
bullets) by fitting a double-Lorentzian function (Eq. 13) to the space-time
spectra, ${S_{\textrm{L}}}$ (resp., ${S_{\textrm{T}}}$). The solid white lines
at small $k$ show linear fits of slope ${C_{\textrm{G,L}}}$
$\equiv\partial{\omega_{\textrm{R,L}}}/\partial k|_{k\rightarrow 0}$ (resp.
${C_{\textrm{G,T}}}\equiv\partial{\omega_{\textrm{R,T}}}/\partial
k|_{k\rightarrow 0}$) The dashed white lines at large $k$ show linear fits of
slope ${C_{\textrm{P,L}}}$ $\equiv\partial{\omega_{\textrm{R,L}}}/\partial k$.
The red curves show fits of the experimental dispersion relations according to
Eq. 3, with a single adjustable parameter for each. Eq. 3 is derived from the
theoretical dispersion relation for a 2D array of point dipoles interacting
with each other Khrapak et al. (2018). (a,b) show that the granular gas has
properties typical of a fluid: propagation of acoustic longitudinal waves (a)
and absence of transverse waves (b). (c–h) show that dipolar forces between
the particles enable the propagation of both longitudinal and transverse
waves.

We now investigate the collective excitations propagating in the granular
medium in stationary regime. For that, as in studies of the hydrodynamic limit
of molecular liquids Hansen & McDonald (2013) and driven granular media
Gradenigo et al. (2011); Brito et al. (2013), we start by computing a current
density ${\mathbf{j}}$, which is the coarse grained velocity field associated
with the motions of individual particles, and whose space Fourier transform is
${\mathbf{\tilde{j}}}(\mathbf{k},t)=\sum_{i}^{N}\,\mathbf{v}_{i}(t)\,\mathrm{e}^{\imath\,\mathbf{k}\cdot\mathbf{r}_{i}(t)}$,
with $\mathbf{k}$ the 2D wave number, $\mathbf{r}_{i}$ the position of
particle $i$, and $\mathbf{v}_{i}$ the velocity of particle $i$. We also
assume isotropy, allowing us to consider the dynamics of the longitudinal
currents (i.e., verifying $\mathbf{k}\times\mathbf{r}_{i}=0$) and transverse
currents ($\mathbf{k}\cdot\mathbf{r}_{i}=0$) decoupled. Then, we identify the
collective modes and characterize their dynamics by computing the space-time
spectra (Eq. 11) of longitudinal and transverse excitations, respectively
${S_{\textrm{L}}}(k,\omega)$ and ${S_{\textrm{T}}}(k,\omega)$, where
$k=||\mathbf{k}||$. The wavenumber $k$ and the angular frequency $\omega$ are
the natural variables to study wave propagation.

Space-time spectra representative of the the four regimes we identified in
figure 1e—granular gas, dipolar fluid, partial order, and solid crystal—are
shown in figure 2. Under each condition, the ensemble of the energy maxima
across the range of $ka$ defines a curve called the dispersion relation. By
fitting each space-time spectra with a double Lorentzian function (Eq. 13), we
extract experimental dispersion relations, ${\omega_{\textrm{R,L}}}(k)$ and
${\omega_{\textrm{R,T}}}(k)$, and their widths, ${\Lambda_{\textrm{L}}}(k)$
and ${\Lambda_{\textrm{T}}}(k)$, respectively.

In the absence of magnetic field ($\varepsilon=0$) and for the longitudinal
spectrum, the energy is distributed around a non-vanishing dispersion relation
(figure 2a). For $ka<2$, we find that longitudinal waves have an acoustic
behaviour, i.e., ${\omega_{\textrm{R,L}}}(k)={C_{\textrm{G,L}}}\,k$, where
${C_{\textrm{G,L}}}$ is a longitudinal sound velocity. Thus we show
experimentally, that the velocity fluctuations in a driven granular medium
without remote interactions correspond to the propagation of sound waves
similar to those in a molecular gas. Such observation is predicted by the
hydrodynamic theory for homogeneously driven granular gas with dissipative
collisions van Noije et al. (1999); Gradenigo et al. (2011) and has been also
reported in event-driven numerical simulations Vollmayr-Lee et al. (2011);
Brito et al. (2013). For $ka>6$, the dispersion relation is fitted by
${\omega_{\textrm{R,L}}}(k)={C_{\textrm{P,L}}}\,k+\Omega$, where
${C_{\textrm{P,L}}}$ is a sound velocity and $\Omega$ is a constant, denoting
a dispersive propagation. In contrast, for the transverse spectrum, the energy
maximum is at $\omega=0$ at all $ka$, indicating the absence of transverse
waves, as expected in a fluid phase (figure 2b).

As the magnetic field is increased, the acoustic behaviour at low $k$ is lost,
i.e.,
${\omega_{\textrm{R,L}}}(k)={C_{\textrm{G,L}}}\,k+\omega_{\textrm{0,L}}$, with
$\omega_{\textrm{0,L}}$ a cut-off frequency and ${C_{\textrm{G,L}}}$ the
fitted group velocity. Moreover, the dispersion relation becomes non-
monotonic, thus showing regions of negative group velocity and even
oscillations when interactions are the strongest (figures 2c–h). In dispersion
relations, local minima due to regions of a negative group velocity are called
rotons. While rotons have recently been observed in a quantum gas Chomaz et
al. (2018), our results show that quantum effects are not necessary
ingredients for rotons—a finding also in agreement with studies attributing
the roton minimum to position correlations Kalman et al. (2010).

For longitudinal waves and moderate $\varepsilon$ (figure 2c,e), we observe
again a linear dispersion relation for $ka>6$, namely
${\omega_{\textrm{R,L}}}(k)={C_{\textrm{P,L}}}\,k+\Omega$, where
${C_{\textrm{P,L}}}$ is a sound velocity and $\Omega$ is a constant, which is
consistent with numerical simulations of interacting, thermally agitated
dipoles Golden et al. (2010); Khrapak et al. (2018). This branch corresponds
to lengthscales smaller than the typical distance between particles and we
attribute it to single-particle excitations Golden et al. (2010); Khrapak et
al. (2018). In contrast, in the solid phase, the dispersion relation is non-
monotonic even for large $ka$ (figure 2g).

Importantly, the presence of a magnetic field enables transverse waves to
propagate as well. We observe indeed dispersion relations for transverse
waves, ${\omega_{\textrm{R,T}}}(k)$, in all regimes but the granular gas
(figure 2d,f,h). This shows that, even with moderate magnetic interactions as
in the dipolar fluid regime, the system becomes rigid enough to transmit
transverse displacements as commonly observed in solids and in agreement with
numerical simulations of 2D arrays of interacting dipoles Golden et al.
(2010); Khrapak et al. (2018). Again, we report a non-acoustic behaviour of
transverse waves at low $k$:
${\omega_{\textrm{R,T}}}(k)={C_{\textrm{G,T}}}\,k+\omega_{\textrm{0,T}}$, due
to the presence of a cut-off frequency $\omega_{\textrm{0,T}}$.

We model our experimental dispersion relations based on a theory for wave
propagation in dipole arrays. The theoretical dispersion relations for a 2D
array of point dipoles within the harmonic potential approximation,
$\omega_{\textrm{Q,L}}(k)$ and $\omega_{\textrm{Q,T}}(k)$, can be analytically
computed using the quasilocalized charge approximation (QLCA) theory Golden et
al. (2008, 2010); Khrapak et al. (2018) (see Materials and Methods Eqs. 14 and
15). The QLCA theory involves a characteristic dipole oscillation frequency,

${\omega_{\textrm{D}}}=B\,\phi^{5/4}\,\sqrt{\dfrac{4\pi\,a}{\mu_{0}\,m}}{,}$
(2)

with $\phi$ the area fraction of dipoles. Note that ${\omega_{\textrm{D}}}$
increases linearly with the applied magnetic field $B$. Notably, in our
experiments, ${\omega_{\textrm{R,L}}}(k)$ and ${\omega_{\textrm{R,T}}}(k)$ do
not tend towards zero when $k$ does so. Instead, cut-off frequencies at $k=0$,
namely ${{\omega_{\textrm{0,L}}}}$ for longitudinal waves and
${{\omega_{\textrm{0,T}}}}$ for transverse waves, prevent acoustic wave
propagation at large scales. Therefore, we use the following modified version
of the QLCA theory results to model our experimental dispersion relations:

${\omega_{\textrm{R},\alpha}^{\textrm{th}}}^{2}(k)=\left[\dfrac{\omega_{\mathrm{A},\alpha}}{{\omega_{\textrm{D}}}}\,\omega_{\mathrm{Q},\alpha}(k)\right]^{2}+\omega_{\textrm{0},\alpha}^{2}\,,$
(3)

where the subscripts $\alpha$ should be substituted by “L” for longitudinal
waves, and “T” for transverse waves; and $\omega_{\mathrm{A},\alpha}$ are the
best-fit parameters that are our experimental estimates correcting the QLCA
characteristic frequency, ${\omega_{\textrm{D}}}$.

These semi-empirical dispersion relations describe increasingly well our
experimental dispersion relations as remote particle interactions strengthen
and order increases in the particle lattice. Indeed, in the solid crystal
regime, our model matches closely both longitudinal and transverse dispersion
relations (figure 2g,h), while in the partial order regime,
${\omega_{\textrm{R,T}}}$ is better fitted than ${\omega_{\textrm{R,L}}}$
(figure 2e,f), and neither ${\omega_{\textrm{R,T}}}$ or
${\omega_{\textrm{R,L}}}$ are well fitted in the dipolar fluid regime, where
orientational order is very low (figure 2c,d).

Figure 3:  Sound propagation. (a,b) Characteristic dipole frequency,
${\omega_{\textrm{D}}}$ (Eq. 2, solid curve), experimental cut-off frequency,
$\omega_{\textrm{0}}$ (squares), and adjusted frequency
${\omega_{\textrm{A}}}$ (circles) obtained by fitting
$\omega_{\textrm{R}}^{\textrm{th}}$ (Eq. 3) to each experimental dispersion
relation, ${\omega_{\textrm{R}}}(k)$, for longitudinal (A) and transverse (B)
waves. (c,d) Sound velocities as a function of $\varepsilon$ for longitudinal
(c) and transverse (d) waves: ${C_{\textrm{Q}}}$ (solid black curves) is
obtained from the QLCA theory (see Material and Methods Eqs. 17 and 18);
${C_{\textrm{A}}}$ (red circles) is obtained from the same two equations,
after ${\omega_{\textrm{D}}}$ was substituted by ${\omega_{\textrm{A}}}$;
${C_{\textrm{G}}}$ (blue squares) is the group velocity, taken as the slope of
the linear fits of the experimental dispersion relations at small $k$ (see
solid white lines in figure 2); and $3v_{\textrm{th}}$ (solid grey curve) is
three times the “thermal” velocity, that is, $3(E_{\textrm{K}}/m)^{1/2}$.
Insets in (c,d) show that ${C_{\textrm{A}}}$ is roughly proportional to the
magnetic field, $B$, when $B$ is strong. (e) The dimensionless decay rates at
large scales, ${\Lambda_{\textrm{L}}}/{\omega_{\textrm{R,L}}}$ and
${\Lambda_{\textrm{T}}}/{\omega_{\textrm{R,T}}}$ (averaged within $0<ka<1.68$
for each $\varepsilon$), show that waves are strongly damped at all
$\varepsilon$. (f) The coherence lengths,
${\mathcal{L}_{\mathrm{L}}}={C_{\textrm{G,L}}}/{\Lambda_{\textrm{L}}}$ and
${\mathcal{L}_{\mathrm{T}}}={C_{\textrm{G,T}}}/{\Lambda_{\textrm{T}}}$, reveal
that waves propagate only on distances of the order of the particle diameter,
$a=1\,$mm, at all $\varepsilon$.

##  4 Sound velocities

In this section, we discuss the relevance of our semi-empirical model for the
experimental dispersion relations, and we investigate their main
properties—namely sound velocities, excitation lifetimes, and excitation
coherence lengths.

We start by comparing the characteristic frequency from the QLCA theory,
${\omega_{\textrm{D}}}$, with our experimental estimates
${\omega_{\textrm{A,L}}}$ and ${\omega_{\textrm{A,T}}}$ in figures 3a,b. In
the granular gas regime, the comparison is not relevant because the dipolar
interactions between the particles are negligible. However, as magnetic
interaction increase between particles, the fitted frequencies become nearly
proportional to ${\omega_{\textrm{D}}}$, with ${\omega_{\textrm{A,L}}}\approx
0.75\,{\omega_{\textrm{D}}}$ and ${\omega_{\textrm{A,T}}}\approx
0.9\,{\omega_{\textrm{D}}}$ and thus close to the value
${\omega_{\textrm{D}}}$ predicted by the QLCA theory.

The presence of cut-off frequencies in our system is not trivial and calls for
an in-depth discussion. We find that the frequencies
${{\omega_{\textrm{0,L}}}}$ and ${{\omega_{\textrm{0,T}}}}$ become finite from
near the onset of the dipolar fluid regime and grow proportionally to
${\omega_{\textrm{D}}}$ (figures 3a,b). Usually, such cut-offs are associated
with an absence of translational invariance, which is typical of finite size
systems. Yet, here they vanish in the granular gas regime before they increase
proportionally to ${\omega_{\textrm{D}}}$. Thus, these cut-off frequencies
seem more likely to result from the dipolar interactions than from a
geometrical constraint such as in a wave-guide. Particle vertical
oscillations—unfortunately inaccessible with our 2D particle tracking—could
occur due to the magnetic forces, influence in-plane wave propagation, and
result in cut-off frequencies. Indeed, as long as the vertical gap in the
experimental cell is larger than the particle diameter, $a$, the configuration
of minimum magnetic potential energy for a local group of particles is where
particles are in contact alternatively with the bottom surface and the lid
Merminod et al. (2015); Merminod (2016), leading to vertical zigzag structures
Opsomer et al. (2020) (see electronic supplementary material Section 3). We
find the characteristic frequency associated with this energy minimum to be
roughly 80%–90% of ${\omega_{\textrm{D}}}$. We show with a simple
phenomenological model, that this coupling of horizontal motions with
localized vertical oscillations results in dispersion relations with an
acoustic branch for in-phase oscillations and an optical branch for antiphase
oscillations (see electronic supplementary material Section 4). The acoustic
branch of the dispersion relation intersects the origin, whereas for the
optical branch has a cut-off frequency. In an analogous geometrical
configuration, although using a repulsive electrostatic potential, acoustic
and optical branches of longitudinal and transverse oscillations have also
been obtained theoretically Dessup et al. (2015). However, the extension of
this work to 3D is challenging even in the crystal regime, because the
hexagonal order is incompatible with the up-down alternation and leads to
geometrical frustration Han et al. (2008). Experiments on quasi-2D dusty
plasmas have demonstrated that out-of-plane waves create an optical branch of
excitations Couëdel et al. (2009), which for high forcing can be coupled with
the longitudinal and transverses branches by mode coupling instability and
induce crystal melting Couëdel et al. (2010, 2019). Similarly, for a 2D system
of electrons trapped at the surface of liquid helium, the coupling of the
longitudinal and transverse waves with the capillary waves modifies the
dispersion relation of the optical phonons at low $k$ Grimes & Adams (1979);
Fisher et al. (1979); Marty et al. (1980); Gallet et al. (1982). The obtained
dispersion relation in $\omega^{2}$ corresponds to the sum of the squares of
the characteristic capillary frequency and of the 2D dispersion relation, as
here in Eq. 3. In our experiments, mechanical shaking appears to
preferentially force optical wave modes, i.e., the horizontal and vertical
oscillations are in antiphase for both the longitudinal and transverse
excitations, which explains the cut-off frequencies. An extension of the
theory by Khrapak et al. Khrapak et al. (2018) to 3D would thus be useful to
better understand wave propagation in confined layers.

We extract three types of sound velocities from the experimental dispersion
relations to characterize the propagation of the waves (Figs. 3c,d). Firstly,
the linear fits of the dispersion relations at small $k$ yield sound group
velocities ${C_{\textrm{G,L}}}$ and ${C_{\textrm{G,T}}}$, as explained in the
previous section. Secondly, we estimate theoretical sound velocities using the
QLCA theory Golden et al. (2010); Khrapak et al. (2018), ${C_{\textrm{Q,L}}}$
and ${C_{\textrm{Q,T}}}$ (see Materials and Methods Eqs. 17 and 18), which are
relevant for strong coupling (typically, $\varepsilon>10.7$). Thirdly, we
compute sound velocities from the formula defining ${C_{\textrm{Q,L}}}$ and
${C_{\textrm{Q,T}}}$, except that we replaced ${\omega_{\textrm{D}}}$ by
${\omega_{\textrm{A}}}$, to obtain ${C_{\textrm{A,L}}}$ and
${C_{\textrm{A,T}}}$. ${C_{\textrm{A,L}}}$ and ${C_{\textrm{A,T}}}$ provide a
good estimate of the sound velocity related to the structure of the 2D
granular assembly at scales of order the particle separation. We find that the
acoustic propagation of longitudinal waves, measured by ${C_{\textrm{G,L}}}$,
occurs at roughly 25 mm s-1, that is, nearly three times the thermal velocity
$3\,v_{\textrm{th}}=3\,\sqrt{{E_{\textrm{K}}}/m}$ and the predicted adiabatic
sound velocity for hard disks with dissipative collisions, which is
$2.04\,v_{\textrm{th}}$ Brito et al. (2013) (figure 3c). We note that both
${C_{\textrm{G,L}}}$ and ${C_{\textrm{G,T}}}$ remain much smaller than
${C_{\textrm{A,L}}}$ and ${C_{\textrm{A,T}}}$ when $\varepsilon>10.7$ due to
the cut-off frequencies. We also observe that for $\varepsilon\gtrsim 100$
(resp., $\varepsilon>10.7$), ${C_{\textrm{A,L}}}$ (${C_{\textrm{A,T}}}$) is
proportional to the applied magnetic field $B$, in agreement with the scaling
of ${\omega_{\textrm{D}}}$ (figure 3c,d insets) and are of the order of the
theoretical values ${C_{\textrm{Q,L}}}$ and ${C_{\textrm{Q,T}}}$ for a 2D
array of dipoles.

The widths of the experimental dispersion relations, $\Lambda_{\textrm{L}}(k)$
and $\Lambda_{\textrm{T}}(k)$, quantify the excitation lifetimes due to
dissipation and nonlinear effects in the interaction potential, and can be
considered as dimensionless decay rates when rescaled by an angular frequency.
The dissipation is caused by inelastic collisions between particles and
between the particle and the top and bottom walls. Even when the particle
collisions are suppressed, the level of dissipation remains significant, as
shown in one of our previous work Castillo et al. (2020). We show these decay
rates, obtained by fitting the spectra (Eq. 13) and averaged within
$0<k\,a<1.68$ for each $\varepsilon$, as functions of $\varepsilon$ for both
the longitudinal and transverse waves in figure 3E. We find that the decay
rates equal at most roughly 1, and are closer to 0.1 in the solid crystal
phase. Therefore, wave propagation is strongly damped because the excitations
disappear on a time scale similar to their period.

To make a similar analysis in space instead of time, we define coherence
lengths of the excitations, $\mathcal{L}_{\textrm{L}}$ and
$\mathcal{L}_{\textrm{T}}$, as the ratios
${C_{\textrm{G,L}}}/\Lambda_{\textrm{L}}$ and
${C_{\textrm{G,T}}}/\Lambda_{\textrm{T}}$, respectively. We find the coherence
lengths to be roughly between 1–2 mm, i.e., very close to the particle
diameter, $a=1$ mm (figure 3f). Thus, the waves propagate in a strong damping
regime and are not affected by the lateral boundaries of the cell, whose sides
are 90 mm-long. Such strong excitation damping likely is the reason why it is
difficult to detect the waves self-generated by mechanical agitation in
granular media. Indeed, strongly damped density waves have been reported in
only one experimental and numerical study Clerc et al. (2008) in the context
of the transient formation of solid-like clusters in a dense granular layer
without interactions and vertically shaken.

##  5 Elastic coefficients

Now, we non-intrusively estimate the elastic coefficients—the bulk and shear
moduli—of our quasi-2D, isotropic granular medium using two methods. In the
first method, we use an extension of the QLCA theory to compute the so-called
instantaneous bulk and shear moduli, $K_{\infty}$ and $G_{\infty}$, from the
structural arrangement of particles and the interaction potential Zwanzig &
Mountain (1965); Khrapak et al. (2018); Khrapak & Klumov (2018), according to
Eqs. 19 and 20. In the second method, we estimate the elastic coefficients
using the sound velocities deduced from the experimental dispersion relations.
Specifically, we use results from the hydrodynamics theory of granular
gases—which introduces effective elastic and dissipation coefficients at
scales larger than few particle separations—and from linear elasticity. In the
fluid phase, the hydrodynamics theory of granular gases predicts longitudinal
waves only and relates their velocity, $C_{\textrm{L}}$, to the bulk modulus,
$K$, through $C_{\textrm{L}}=\sqrt{K/\rho}$, with $\rho=mN/S$ the 2D mass
density, where $N$ is the number of particles inside the region of interest.
In the solid phase, linear elasticity predicts
$C_{\textrm{L}}=\sqrt{(K+G)/\rho}$ and $C_{\textrm{T}}=\sqrt{G/\rho}$ for the
transverse waves. Therefore, consistently with both theories, we take
$K=\rho\,(C_{\textrm{L}}^{2}-C_{\textrm{T}}^{2})$ and
$G=\rho\,C_{\textrm{T}}^{2}$ throughout all four regimes of our granular
system. We obtain the elastic coefficients from the QLCA theory,
$K_{\textrm{Q}}$ and $G_{\textrm{Q}}$, by substituting $C_{\textrm{L}}$ and
$C_{\textrm{T}}$ with ${C_{\textrm{Q,L}}}$ and ${C_{\textrm{Q,T}}}$,
respectively; and we deduce experimentally adjusted elastic coefficients,
$K_{\textrm{A}}$ and $G_{\textrm{A}}$, similarly from the fitted sound
velocities ${C_{\textrm{A,L}}}$ and ${C_{\textrm{A,T}}}$.

All three of our bulk modulus estimates increase with $\varepsilon$,
demonstrating the increasing rigidity of the granular assembly as magnetic
interactions become stronger (figure 4a). Specifically, $K_{\textrm{Q}}$ and
$K_{\infty}$ are nearly equal in the regimes of strong coupling
($\varepsilon>10.7$), which reflects the consistency of the method as the
velocities deduced from the QLCA theory rely on physical principles equivalent
to those we use to determine the instantaneous elastic coefficients.
$K_{\textrm{A}}$, which is the relevant experimental bulk modulus estimate in
the regimes of strong coupling, shows a similar behaviour, although with
smaller values for $\varepsilon>10.7$.

We also find that our shear modulus estimates grow with $\varepsilon$,
indicating an increasing transverse stiffness in the system (figure 4b). All
three bulk moduli take similar values in the solid phase. We note that,
similarly to $K_{\infty}$ and $K_{\textrm{Q}}$, $G_{\infty}$ and
$G_{\textrm{Q}}$ are nearly equal when the coupling is strong. For
$\varepsilon<10.7$, $G_{\infty}$ measures the visco-elastic response of the
fluid phase. In contrast, $G_{\textrm{A}}$ vanishes in the absence of
transverse waves ($\varepsilon<7$), thus showing the typical response of a
fluid, in which shear excitations are damped due to viscosity and do not
propagate. In the dipolar fluid regime, $G_{\textrm{A}}$ is significantly
smaller than $G_{\infty}$ and $G_{\textrm{Q}}$.

Finally, let us point out that the high-frequency moduli $K_{\infty}$ and
$G_{\infty}$ convey the mechanical response of a static ensemble of particles
to stress. In particular, $G_{\infty}$ can be expected to be nonzero in the
granular gas regime. Indeed, in a fluid phase, $G_{\infty}$ measures the
instantaneous viscoelastic response, whereas the shear modulus reaches zero at
a time given by the viscous relaxation of shear stress.

To summarize, the analysis of the mechanical waves that we experimentally
study here enables to non-intrusively estimate the elastic coefficients of the
granular system. We find elastic coefficient values within $[10^{-5};10^{-1}]$
Pa m, which is consistent with the theoretical estimates from the
thermodynamics of interacting dipoles. This confirms the relevance and
accuracy of the wave spectroscopy method presented here.

Figure 4:  2D elastic coefficients. (a) Bulk modulus and (b) shear modulus as
functions of $\varepsilon$. The solid black curves show the “instantaneous”
bulk and shear moduli, $K_{\infty}$ and $G_{\infty}$, which are computed from
the pair correlation function and the magnetic interaction potential (see Eqs.
19 and 20). The red circles show ${K_{\textrm{A}}}$ and ${G_{\textrm{A}}}$,
the moduli estimated within the hydrodynamics description of granular gases,
from ${C_{\textrm{A,L}}}$ and ${C_{\textrm{A,T}}}$. The solid grey curves show
${K_{\textrm{Q}}}$ and ${G_{\textrm{Q}}}$, the moduli obtained with
${C_{\textrm{Q,L}}}$ and ${C_{\textrm{Q,T}}}$, which are derived from the QLCA
theory.

##  6 Conclusions

In this article, we have shown experimentally that particle velocity
fluctuations in quasi-two-dimensional, driven granular media correspond to the
propagation of mechanical waves in the presence of significant energy
dissipation. The random motions of the particles are self-organized into
collective modes, which are the macroscopic counterpart of the phonons of
condensed matter. When dipolar particle interactions are weak or turned off,
we only observe longitudinal, non-dispersive, compression waves—as expected
for a fluid phase. This study constitutes the first step in investigating at
more depth the physics of granular gases through their collective dynamics, in
relation to fluctuating hydrodynamics with dissipative collisions Vollmayr-Lee
et al. (2011); Gradenigo et al. (2011); Brito et al. (2013). In contrast, when
the coupling due to the dipolar interactions is dominant, propagating waves
are both longitudinal and transverse, as well as dispersive. The experimental
dispersion relations are satisfyingly modelled using the spatial distribution
of particles and their interaction potential. Notably, a specificity of our
experimental system is the presence of a cut-off frequencies at $k=0$. We
interpret these cut-offs as optical dispersion relations arising from the
coupling between horizontal motions and small vertical oscillations, the
latter of which we do not have the ability to measure. For all coupling
strengths, we compute characteristic sound velocities, which we use to non-
intrusively estimate the elastic parameters of the granular medium.

In our system, in-plane mechanical forcing—the horizontal bead velocities
acquired from rebounds on the rough bottom surface of the cell—is random, thus
is non-coherent and has no characteristic frequency. We demonstrated that, in
response to this forcing, strongly damped mechanical waves propagate through
the system. This result agrees with the linearised hydrodynamic theory of
particulate systems, which predicts wave propagation as the first-order
mechanical response to random forcing, for both driven granular media without
remote interactions Vollmayr-Lee et al. (2011); Brito et al. (2013) and
interacting molecules Hansen & McDonald (2013). As an extension of our system
which we would suggest to implement next, a piston could be added at a cell
boundary to generate either large-scale sinusoidal or pulsed mechanical waves
into the granular medium Buttinoni et al. (2018), to bridge the gap between
the large-scale sound waves and the incoherent random waves that we evidenced
in this driven, interacting granular material. Another interesting direction
would be to compare the large scale elastic coefficients with those obtained
here, at the particle level, by subjecting the granular quasi-monolayer to an
external mechanical perturbation, such as shear or compression.

##  7 Methods

### Experimental setup, protocol, and image analysis

The granular particles are chromed alloy steel (AISI 52100) spheres of
diameter $a=1$ mm and of mass $m=4.07$ mg. They are ferromagnetic with high
magnetic permeability and a low remnant magnetic field, so that their coercive
field is small compared with the values of the magnetic field $B$ used in our
experiments. A total of $N_{0}=2000$ particles are placed in a square
aluminium cell with sandpaper covering its bottom surface, and sealed by a
smooth, transparent acrylic lid. The area between the aluminium walls is
$S_{0}=9$ cm $\times$ 9 cm and the gap between the rough bottom surface and
the lid is $1.42\,a$. To avoid any electrostatic effects, the top transparent
lid is made of polycarbonate with an antistatic coating. Linked to an electric
ground, this lid releases the electrostatic charges that may be temporarily
carried by particles. The cell is mounted horizontally on and vibrated
vertically by an electromagnetic shaker. The mechanical forcing is sinusoidal,
of frequency 300 Hz, and with a root mean square acceleration of
$\Gamma=1.6\,g$, with $g$ the gravitational acceleration. A pair of coils
generate a nearly homogeneous vertical magnetic field across the cell area
(the magnetic field is roughly 3% smaller at the edges of the cell than at its
center). Immersed in this magnetic field, the particles are magnetized into
induced, vertical dipoles. The average area fraction of particles is
$\phi_{0}=\frac{N_{0}\,\pi a^{2}}{4S_{0}}\approx 0.2$.

An annular light-emitting diode (LED) array is centred above the cell and
produces a high-contrast circular signal by reflection on the chromed
particles. We record pictures of the particle assembly from above using a
high-speed video camera (Phantom V10) at high frequency (780 Hz) and
resolution ($1152\times 1152$ pixels). The window of observation around the
cell center is of area $S=50.36$ $\times$ 50.36 mm2. One particle diameter
corresponds to roughly $20$ pixels.

We determine the positions of particle centres using a convolution-based
least-square fitting routine Reis et al. (2007) completed by an intensity-
weighted center detection algorithm (accuracy estimated to better than $0.3$
pixel). The trajectories and velocities of each particle projected in the
horizontal plane are reconstructed using a tracking algorithm Crocker & Grier
(1996) in the window of observation ${S}$.

In each experimental run, first, we raise the amplitude of the mechanical
shaker to $1.6g$ and the magnetic field to the desired value, second, we wait
an equilibration time of $100$ s, and finally, we record 3000 frames using the
high-speed camera during $3.85\,$s. For each value of $\varepsilon$, we run
five independent and identical experiments; the data that we present in this
article is obtained from their average.

The area fraction of particles within $S$ is computed for each run as
$\phi=\frac{N\,\pi a^{2}}{4S}$. Note that due to inelastic clustering at small
$\varepsilon$ and non-repelling boundaries, $\phi$ decreases from $0.34$ to
$0.17$ across the full range of $\varepsilon$ Merminod et al. (2014); Castillo
et al. (2020).

### Magnetic potential energy, kinetic energy, and hexagonal order parameter

From the positions and velocities of the $N$ particles that we track, we
compute the mean magnetic energy per particle as

${E_{\textrm{m}}}=\left\langle\,\frac{1}{N}\sum_{i=1}^{N}\sum_{j=i+1}^{N}U_{i,j}{(r_{i,j})}\right\rangle\,,$
(4)

and the mean kinetic energy per particle as

${E_{\textrm{K}}}=\left\langle\frac{m}{2N}\sum_{i=1}^{N}v_{i}^{2}\right\rangle$
(5)

with $r_{i,j}$ the distance between particles $i$ and $j$ in the horizontal
plane, $U_{i,j}$ their interaction potential (Eq. 1), $v_{i}$ the velocity of
particle $i$, and the brackets denoting time averaging.

We monitor the degree of hexagonal order through the sixfold bond-
orientational order parameter $\Psi_{6}^{j}$, also called the local hexagonal
order parameter,

$\Psi^{j}_{6}=\frac{1}{n_{j}}\sum_{k=1}^{n_{j}}\mathrm{e}^{6i\theta_{jk}},$
(6)

where $n_{j}$ is the number of nearest neighbours of particle $j$, and
$\theta_{jk}$ is the angle between the neighbour $k$ of particle $j$ and a
reference axis. By definition, $\Psi_{6}^{j}=1$ when the nearest neighbours
form a regular hexagon. The corresponding global average,

$|\Psi_{6}|=\left|\left\langle\frac{1}{N}\sum_{j=1}^{N}\Psi^{j}_{6}\right\rangle\right|,$
(7)

where the vertical bars denote a modulus, measures the degree of hexagonal
order of the particle assembly. We refer to $|\Psi_{6}|$ as the global
hexagonal order parameter of the system.

### Current correlation functions

The velocities of individual particles are analysed in the spatial Fourier
space. With the hypothesis of homogeneous and isotropic system, according to
the linearised hydrodynamics theory of molecular fluids Hansen & McDonald
(2013), the coarse-grained velocity field can be inferred from the computation
of longitudinal current correlation function and of the transverse current
correlation function. This approach has been extended to the case granular
systems, for which energy injection and dissipation are also included van
Noije et al. (1999); Vollmayr-Lee et al. (2011); Gradenigo et al. (2011);
Brito et al. (2013). Here, we define in 2D, the longitudinal current
correlation function $J_{\mathrm{L}}(\mathbf{k},t)$ and the transverse current
correlation function $J_{\mathrm{T}}(\mathbf{k},t)$, as Hansen & McDonald
(2013); Castillo (2013); Puglisi et al. (2012)

$J_{\textrm{L}}(\mathbf{k},t)=\left\langle\frac{1}{N}\sum_{i,j=1}^{N}(\mathbf{\hat{k}}\cdot\mathbf{v}_{i}(t))\,(\mathbf{\hat{k}}\cdot\mathbf{v}_{j}(0))\,\mathrm{e}^{\imath\,\mathbf{k}\,(\mathbf{r_{i}(t)-r_{j}(0))}}\right\rangle$
(8)

and

$J_{\textrm{T}}(\mathbf{k},t)=\left\langle\frac{1}{N}\sum_{i,j=1}^{N}(\mathbf{\hat{k}}\times\mathbf{v}_{i}(t))\,(\mathbf{\hat{k}}\times\mathbf{v}_{j}(0))\,\mathrm{e}^{\imath\,\mathbf{k}\,(\mathbf{r_{i}(t)-r_{j}(0))}}\right\rangle$
(9)

where $\mathbf{\hat{k}}$ is the unitary vector directed along the wave vector
$\mathbf{k}$. These functions are computed on a $150\times 150$ with node
values $0.0624\,$mm${}^{-1}<k_{x,y}<9.36\,$mm-1. Each pair $(k_{x},k_{y})$ is
discretized according to $(n_{x}\pi/L_{x},n_{y}\pi/L_{y})$, where
$n_{x},n_{y}\in\mathbb{N}$ and $L_{x}=L_{y}=50.36$ mm. The brackets
$\left\langle\right\rangle$ denote time averaging.

Then assuming system isotropy, we define the angle-averaged current
correlation functions,

$J_{\mathrm{\alpha}}(k,t)=(2\pi)^{-1}\,\int_{0}^{2\pi}J_{\mathrm{\alpha}}(\mathbf{k},t)\,\mathrm{d}\theta\,,$
(10)

where the subscripts $\alpha$ should be substituted by “L” for longitudinal
waves, and “T” for transverse waves. These functions are also called dynamical
structure factors of longitudinal and transverse velocity modes Gradenigo et
al. (2011); Puglisi et al. (2012). Finally, $J_{\mathrm{\alpha}}(k,t)$ is
averaged over five independent runs with identical experimental parameters.

### Space-time spectra

According to the Wiener-Khinchin theorem, the space-time spectra of
excitations $S_{\mathrm{\alpha}}(k,\omega)$ are directly deduced from the
temporal current correlation functions by a time Fourier transform as

$S_{\mathrm{\alpha}}(k,\omega)=\mathrm{Re}\,\,\int_{0}^{\infty}J_{\mathrm{\alpha}}(k,t)\,\mathrm{e}^{\imath\,\omega\,t}\,\mathrm{d}t.$
(11)

Here, the time Fourier transform is performed using the fast Fourier Transform
algorithm and a Hann window as tapering function. The frequencies are
discretised into 2999 values between 0 Hz and 390 Hz.

To deduce experimental dispersion relations from the space-time spectra, we
assume that the current correlations of propagating waves can be modelled as
damped oscillations, that is,

$J_{\mathrm{\alpha}}(k,t)\propto\mathrm{e}^{-\Lambda_{\mathrm{\alpha}}(k)\,t}\,\cos(\omega_{\mathrm{R,\alpha}}(k)\,t)\,,$
(12)

which, using Eq. 11, leads to the space-time spectra being modeled as
proportional to double Lorentzian functions Khrapak et al. (2018); Mistryukova
et al. (2019),

$\displaystyle
S_{\mathrm{\alpha}}(k,\omega)\\!\propto\\!\dfrac{1}{(\omega-\omega_{\mathrm{R,\alpha}}(k))^{2}+\Lambda_{\mathrm{\alpha}}^{2}(k)}\\!+\\!\dfrac{1}{(\omega+\omega_{\mathrm{R,\alpha}}(k))^{2}+\Lambda_{\mathrm{\alpha}}^{2}(k)}$
(13)

where $\omega_{\mathrm{R,\alpha}}(k)$ is the fitted experimental dispersion
relation, and $\Lambda_{\mathrm{\alpha}}(k)$ is the fitted width of the
dispersion relation, which quantifies the wave damping. In practice, for each
value of wavenumber $k$, we fit our experimental time-space spectra with Eq.
13 to obtain $\omega_{\mathrm{R,\alpha}}(k)$ and
$\Lambda_{\mathrm{\alpha}}(k)$.

### QLCA theory for a 2D array of interacting dipoles

To describe the collective excitations in a 2D system of interacting dipoles
in strong coupling regime for fluid and solid phases, Golden et al. Golden et
al. (2008, 2010) proposed the quasilocalized charge approximation (QLCA)
method. It extends the usual calculation of the phonon dispersion relation in
a lattice within the harmonic approximation, by taking into account the actual
average position of particles through the input of the pair correlation
function $g(r)$ in the continuous limit and assuming isotropy. Khrapak et al.
Khrapak et al. (2018) provided explicit expressions of the dispersion
relations, using the Wigner-Seitz radius $a_{\textrm{WS}}$ as a characteristic
distance:

$\displaystyle\omega_{\textrm{Q,L}}^{2}$ $\displaystyle=$
$\displaystyle\dfrac{3{\omega_{\textrm{D}}}^{2}}{2}\,\int_{0}^{\infty}\,\dfrac{g(x)\,\mathrm{d}x}{x^{4}}\,[3-J_{0}(q\,x)+5\,J_{2}(q\,x)]$
(14) $\displaystyle\omega_{\textrm{Q,T}}^{2}$ $\displaystyle=$
$\displaystyle\dfrac{3{\omega_{\textrm{D}}}^{2}}{2}\,\int_{0}^{\infty}\,\dfrac{g(x)\,\mathrm{d}x}{x^{4}}\,[3-J_{0}(q\,x)-5\,J_{2}(q\,x)]$
(15)

with $J_{0}(x)$ and $J_{2}(x)$ the Bessel functions of the first kind,
$x=r/a_{\textrm{WS}}$ the reduced distance, $q=k\,a_{\textrm{WS}}$ the reduced
wavenumber, and ${\omega_{\textrm{D}}}$ the characteristic dipole oscillation
angular frequency. The Wigner-Seitz radius $a_{\textrm{WS}}$ is directly
related to the particle surface density $\rho_{\mathrm{N}}=N/S$, by
$a_{\textrm{WS}}=(\pi\,\rho_{\mathrm{N}})^{-1/2}$. From the QCLA theory,
${\omega_{\textrm{D}}}$ reads:

${\omega_{\textrm{D}}}=\left(\dfrac{2\pi\,\rho_{\mathrm{N}}}{m\,(a_{\mathrm{WS}})^{3}}\,\dfrac{4\,\pi\,B^{2}}{\mu_{0}}\,\left(\dfrac{a}{2}\right)^{6}\right)^{1/2}\,.$
(16)

Knowing that in our system $a_{\mathrm{WS}}=a/(2\sqrt{\phi})$ and
$\rho_{\mathrm{N}}=4\phi/(\pi\,a^{2})$, the characteristic frequency can be
expressed as a function of the experimental parameters:

${\omega_{\textrm{D}}}=B\,\phi^{5/4}\,\sqrt{\dfrac{4\pi\,a}{\mu_{0}\,m}}\,.$

In the limit $k\rightarrow 0$, the QLCA theory predicts an acoustic behaviour,
i.e., a non dispersive wave propagation, $\omega\propto k$, with the
longitudinal and transverse sound velocities respectively:

$\displaystyle
C_{\textrm{Q,L}}=\sqrt{\dfrac{33\,M}{16}}\,{\omega_{\textrm{D}}}\,a_{\mathrm{WS}}\,\int_{0}^{\infty}\dfrac{g(r)}{r^{2}}\mathrm{d}r$
(17) $\displaystyle
C_{\textrm{Q,T}}=\sqrt{\dfrac{3\,M}{16}}\,{\omega_{\textrm{D}}}\,a_{\mathrm{WS}}\,\int_{0}^{\infty}\dfrac{g(r)}{r^{2}}\mathrm{d}r\,,$
(18)

with $M=(1/2)\,\sum_{i}(r_{i}/a_{\mathrm{WS}})^{-3}\approx 0.798512$ the
Madelung constant for a perfect hexagonal lattice. For the set of experiments
presented here, the experimental distribution of the particles in the
different regimes gives: $\int_{0}^{\infty}{g(r)}\,r^{-2}\,\mathrm{d}r\approx
0.85$.

Finally, the bulk and shear moduli obtained from the QLCA theory, $K_{\infty}$
and $G_{\infty}$, are given by:

$\displaystyle
K_{\infty}\\!=\\!2\,\rho_{\mathrm{N}}\,{E_{\textrm{K}}}-\dfrac{\pi\,\rho_{\mathrm{N}}^{2}}{4}\int_{0}^{\infty}\,r^{2}\,\left[\dfrac{\mathrm{d}U(r)}{\mathrm{d}r}-r\,\dfrac{\mathrm{d}^{2}U(r)}{\mathrm{d}r^{2}}\right]\,g(r)\,\mathrm{d}r$
(19) $\displaystyle
G_{\infty}\\!=\\!\,\rho_{\mathrm{N}}\,{E_{\textrm{K}}}+\dfrac{\pi\,\rho_{\mathrm{N}}^{2}}{8}\int_{0}^{\infty}\,r^{2}\,\left[\dfrac{3\,\mathrm{d}U(r)}{\mathrm{d}r}+r\,\dfrac{\mathrm{d}^{2}U(r)}{\mathrm{d}r^{2}}\right]\,g(r)\,\mathrm{d}r$
(20)

These moduli are qualified of high frequency, or instantaneous, because they
characterize the mechanical response for a static arrangement of particles in
space.

The data corresponding to the article ’s figures are available at:
http://www.msc.univ-paris-
diderot.fr/~berhanu/DataRepositoryBerhanuRSPA2022.html

M.B.: conceptualization, formal analysis, funding acquisition, software,
resources, writing original draft; S.M.: conceptualization, investigation,
formal analysis, software, writing review and editing; G.C.:
conceptualization, funding acquisition, formal analysis, software, writing
review and editing; E.F.: conceptualization, funding acquisition, resources,
writing review and editing.

This research was supported by Université Paris Cité and Fondecyt Grant No.
11200464 (G.C.).

###### Acknowledgements.

We thank at MSC, Université Paris Cité, Thierry Hocquet and Martin Devaud for
granting us access to their computing facilities; and Marc Durand, Michel
Saint-Jean, Christophe Coste and François Gallet for discussions. We
acknowledge Nicolás Mujica and Rodrigo Soto from University of Chile for
discussions.

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