# Internal wave crystals

Sasan J. Ghaemsaidi  Massachusetts Institute of Technology, Department of
Mechanical Engineering, Cambridge, Massachusetts 02139, U.S.A.  Michel
Fruchart  Department of Physics and James Franck Institute, University of
Chicago, Chicago, Illinois 60637, U.S.A.  Severine Atis  Massachusetts
Institute of Technology, Department of Mechanical Engineering, Cambridge,
Massachusetts 02139, U.S.A.  Department of Physics and James Franck Institute,
University of Chicago, Chicago, Illinois 60637, U.S.A.

###### Abstract

Geophysical fluids such as the ocean and atmosphere can be stratified: their
density depends on the depth. As a consequence, they can host internal gravity
waves that propagate in the bulk of the fluid, far from the surface [1]. These
waves can transport energy and momentum over large distances, thereby
affecting large-scale circulation patterns [2], as well as the transport of
heat, sediments, nutrients and pollutants in the ocean [3]. When the density
stratification is not uniform, internal waves can exhibit wave phenomena such
as resonances, tunneling, and frequency-dependent transmissions [4, 5, 6].
Spatially periodic density profiles formed by thermohaline staircases are
commonly found in stratified fluids ranging from the Arctic Ocean [7] to giant
planet interiors [8, 9], and can produce extended regions with periodically
stratified fluid. Here, we report on the experimental observation of band gaps
for internal gravity waves, ranges of frequencies over which the wave
propagation is prohibited in the presence of a periodic stratification. We
show the existence of surface states controlled by boundary conditions and
discuss their topological origin. Our results suggest that energy transport
can be profoundly affected by the presence of periodic stratifications in
geophysical fluids ranging from Earth’s oceans to gas giants.

Internal gravity waves are carried in fluids that present density
stratification, such as the atmosphere and the oceans [1]. They can be
generated by the wind, or near the ocean’s floor by tidal flow over topography
[2]. Because they can travel long distances before being dissipated, they play
an important role in redistributing heat and momentum over large scales, and
play a consequent role in the oceanic general circulation and the evolution of
climate [2, 10, 11]. The propagation of internal waves can be strongly
affected by a non-uniform stratification.

In the ocean, the interplay between heat diffusion and salt diffusion can lead
to double-diffusive instability and produce spatially periodic density
profiles called thermohaline staircases over spatially extended regions [12,
7]. These periodic structures have also been suggested to exist in
astrophysical bodies, such as in giant planet interiors [8, 9], again as the
result of the coexistence of a double-diffusive phenomena [12]. When the size
of the steps is small compared with the internal wave wavelength, internal
wave propagation can be described by ray theory [1, 5]. However, ray theory
does not account for wave-like behaviors such as diffraction and tunneling,
that can occur when the stratification varies over length scales comparable
with the wavelength [4, 13, 14]. In the Canada Basin, for instance, the period
of the thermohaline staircases can be comparable to the wavelength of internal
waves [15, 7, 16]. In this regime, it is expected that a periodic variation of
the stratification will strongly affect the propagation of internal waves [17,
18, 6, 19].

Figure 1: Experimental observation of a band gap and surface states. (a)
Schematic of the experimental setup with periodic stratification. The
submerged wave generator located near the top of the tank creates incident
internal waves that propagate downward. (b) Experimentally measured density
profile. (c) Stratification obtained from panel b. The raw signal is show in
gray and a sliding average over a $3\text{\,}\mathrm{mm}$ window is shown in
blue. (d) Transmitted internal wave amplitude versus $\omega$ for $k_{x}=27.6$
m-1 obtained from the experiment (see Fig. 2 for details). (e-i) Selected
snapshots of the flow velocity field determined from PIV measurements for
different values of $\omega$. The wave is transmitted to bottom of the tank
when the excitation frequency $\omega$ is in the bulk bands (panels e and g),
but it quickly decays as a function of $z$ when $\omega$ is in a band gap. In
panels h and i, we show the wave profiles corresponding to surface states. The
wave amplitude decays exponentially (see also Fig. 2), but the amplitude is
appreciably high. This is particularly visible in panel h: artifacts in the
PIV due to turbulence are visible near the top.

Motivated by these considerations, we perform laboratory experiments to
investigate the dynamics of internal waves in controlled periodic
stratifications. A five meter long wave tank is filled with a mixture of salt
water (Fig. 1a). The density $\rho_{0}(z)$ of the fluid as a function of depth
is varied with the salt concentration. The stratification of the fluid is then
characterized by the Brunt-Väisälä frequency, defined as

$N(z)=\sqrt{-\frac{g}{\rho_{0}}\,\frac{\text{d}\rho_{0}}{\text{d}z}}$ (1)

where $g$ is the acceleration due to gravity, $\rho_{0}(z)$ is the background
density, and the vertical axis $z$ points up. The stratification is uniform
when $N(z)$ is a constant. In a nonuniform stratification, the fluid density
becomes a non-linear function of $z$ and the buoyancy frequency $N(z)$ can
vary with the depth in the fluid.

In the experiment, we prepared a spatially periodic stratification composed of
alternating layers with buoyancy frequencies
$N_{1}\simeq$1.1\text{\,}{\mathrm{s}}^{-1}$$ and $N_{2}\simeq 0$, as
represented on Fig. 1a-c (see Methods for experimental details). A total of
$4$ layers with $N=N_{2}$ separate $3$ layers with $N=N_{1}$. In addition, a
upper and lower layer with $N=N_{1}$ are present on the top and on the bottom
of the tank. The measured density profile and the resulting buoyancy profile
(determined from Eq. (1)) are shown in Fig. 1b-c, and exhibit a spatial period
$a=$10\pm 1.\text{\,}\mathrm{cm}$$. We use a submerged linear wave generator
[20] to generate internal waves with an amplitude $A=$3\text{\,}\mathrm{mm}$$
and a dominant horizontal wavenumber $k_{x}=$27.6\text{\,}{\mathrm{m}}^{-1}$$
at a given frequency $\omega$. The resulting waves propagate in the $x-z$
plane, and the corresponding velocity field $\bm{v}=(v_{x},0,v_{z})$ is
measured using particle image velocimetry (PIV). Selected snapshots of the
recorded wave field $v_{z}(x,z)$ are displayed in Fig. 1e-g.

To estimate how much energy is transmitted through the periodic
stratification, we show in Fig. 1d the average $V_{z}(\omega)$ of the wave
amplitude $|v_{z}|$ over the region
$-z=$0.3$\text{--}$0.5\text{\,}\mathrm{m}$$ indicated by the dashed white
lines in Fig. 1e. The internal wave transmission exhibits a large drop in an
extended range of frequencies
$\omega\approx$0.45$\text{--}$0.58\text{\,}{\mathrm{s}}^{-1}$$. For a wave
frequency in this range, the transmission is impeded by the stratification,
and a rapid decrease of the wave amplitude with depth is seen in Fig. 1f. On
the contrary, the wave is fully transmitted through the structure at
frequencies outside of this specific range as shown in Figs. 1e and g.

Figure 2: Band structure and experimental spectrum of the internal wave
crystal. (a) Internal waves band structure in a periodic stratification
determined from the eigenvalues of the operator $\mathcal{L}$ defined by Eq.
(3) (see Methods). (b) Experimental wave amplitude
$\tilde{v}_{z}=v_{z}/(A\omega)$, in which $A$ and $\omega$ are the amplitude
and frequency of motion of the internal wave generator, as a function of depth
$z$ and the frequency $\omega$. (c) To estimate the transmission through the
periodic stratification, we compute the average $V_{z}(\omega)$ of the
(normalized) experimental wave amplitude $|\tilde{v}_{z}|$ over the region
between $z\simeq$-0.3\text{\,}\mathrm{m}$$ and
$z\simeq$-0.5\text{\,}\mathrm{m}$$, at the bottom of the tank. (d) Numerical
simulation of the wave amplitude as a function of $z$ and $\omega$. (e) In
this panel, we plot the same quantity as in panel c, this time obtained from
numerical simulations. (f) Exponential decay (accompanied by oscillations) of
the surface state measured experimentally, obtained from slicing panel b at
$\omega\approx$0.54\text{\,}{\mathrm{s}}^{-1}$$. (g) Equivalent of panel f in
numerical simulations.

This phenomenon can be traced to the existence of a band gap at these
frequencies in an infinite version of the crystal realized in our experiment
[21]. To describe the propagation of small-amplitude internal waves, it is
convenient to introduce the scalar streamfunction $\psi(x,z,t)$, defined such
that $v_{x}=-\partial_{z}\psi$ and $v_{z}=\partial_{x}\psi$. The propagation
of small-amplitude internal waves is then described by the equation (see Ref.
[1])

$\partial_{t}^{2}\nabla^{2}\psi+N(z)^{2}\partial_{x}^{2}\psi=0.$ (2)

For harmonic horizontally periodic solutions of the form
$\psi(x,z,t)\propto\phi(z)e^{i(k_{x}x-\omega t)}$, Eq. (2) reduces to the
ordinary differential equation

$\phi^{\prime\prime}(z)+k_{x}^{2}\left[\frac{N^{2}(z)}{\omega^{2}}-1\right]\phi(z)=0.$
(3)

In a region where $N(z)=N_{0}$ is constant, the vertical wavenumber $k_{z}$ of
a wave with frequency $\omega$ is given by the dispersion relation
$k_{z}^{2}(\omega)=k_{x}^{2}\left[N_{0}^{2}/\omega^{2}-1\right]$. In the
presence of a periodic stratification $N(z+a)=N(z)$, the dispersion relation
organises into distinct bands $\omega_{i}(k_{z})$ (with $i=1,2,\dots$)
separated by band gaps where internal waves cannot propagate. We use Eq. (3)
to compute in Fig. 2a the band structure of an infinite crystal that
approximates the experimental stratification profile (see Methods), in which
band gaps are indicated in yellow. Figure 2b, displays the experimental wave
amplitude as a function of the depth $z$ and the excitation frequency
$\omega$. Comparing Figs. 2a and b-c, we find that the low-transmission
regions in the experimental spectrum of Figs. 2b qualitatively match the band
gaps predicted analytically. However, the experimental band gap appears to be
smaller than the one predicted analytically. This observation persists in
direct numerical simulations of internal wave propagation shown in Figs. 2d-e
(see Methods). Taking a closer look at Fig. 2b and d, we note that a large
wave amplitude is present inside the predicted bulk band gap around
$\omega\sim$0.6\text{\,}{\mathrm{s}}^{-1}$$ suggesting the existence of a
resonance at this frequency. The exponential decay with depth of the wave
amplitude in Fig. 2f-g shows that this resonance is localized near the upper
boundary of the system.

Figure 3: Topological invariant in the internal wave crystal. To compute the
Zak phase of the one-dimensional internal wave crystal, we measured
experimentally the Bloch functions $v_{z}(x,z;k_{z})$ of the two highest bands
at momentum $k_{z}=0$ and $k_{z}=\pi/a$ (normalized by their maximal values
$v_{z}^{\text{max}}$) in panels a-d. The boundaries of the unit cells are
marked by white dashed lines. Panels a-b (c-d) describe the highest (second
highest) band. In both cases, the Bloch function $v_{z}(x,z;k_{z})$ at
$k_{z}=0$ is antisymmetric with respect to the center of the unit cell, while
the Bloch function at $k_{z}=\pi/a$ is symmetric. Hence, the Zak phases of
both bands are $\pi$. This means that the Wannier centers of the bands are
located at the boundary of the unit cell (called the Wyckoff position 1b), as
represented in panel e. In panel f, we represent the other possible situation
(not occurring in the experiment) in which the Wannier centers are located in
the center of the unit cell (called the Wyckoff position 1a).

The surface states shown in Fig. 2 are reminiscent of surface states found in
topological insulators [22, 23], in contexts ranging from electrons in solids
[22] to optics [24, 25] and geophysical fluids [26, 27]. To understand this
connection, notice first that the periodic stratification $N(z)$ is
approximately invariant under spatial inversion: $N(z_{0}+z)=N(z_{0}-z)$,
where $z_{0}$ is at the center of a unit cell (see Fig. 1c). In this case, one
can define a topological invariant called the Zak phase $\varphi_{\text{Zak}}$
(there is one Zak phase per band). It is constrained to take values $0$ or
$\pi$, which label the two topologically distinct states. Qualitatively, the
Zak phase gives the average position of the wave in the unit cell (called the
Wannier center of the band, see Refs. [28, 23, 29, 30] for details). In a
system with inversion symmetry, the Wannier center can only be located at the
center ($\varphi_{\text{Zak}}=0$) or at the edge ($\varphi_{\text{Zak}}=\pi$)
of the unit cell. Consider a finite system terminated at the edge of a unit
cell (black dashed lines in Fig. 3e-f). A surface state occurs when a Wannier
center is exposed at the boundary, like in Fig. 3e. In contrast, no surface
state occurs when the Wannier center is in the middle of the unit cell, like
in Fig. 3f.

The Zak phase can be directly measured experimentally from the wave fields
using its expression [28]

$\frac{\varphi_{\text{Zak}}}{\pi}=\frac{\xi_{0}-\xi_{\pi}}{2}\,\text{mod
{\hbox{2}}}.$ (4)

Here, the quantities $\xi_{k_{z}}$ take the values $\pm 1$ depending on
whether the Bloch wave $\phi(k_{z})$ at $k_{z}=0,\pi/a$ are symmetric or
antisymmetric by reflection about the center of the unit cells. Figures 3a-d
are showing snapshots of the experimental Bloch waves in which the boundaries
of the unit cells are indicated by white dashed lines. Starting with the
highest frequency band $B_{1}$, we find from Fig. 3a-b that $\xi_{0}=-1$ and
$\xi_{\pi}=1$, leading to $\varphi_{\text{Zak}}=\pi$. Similarly, we find
$\varphi_{\text{Zak}}=\pi$ for the second highest frequency band $B_{2}$ (Fig.
3c-d). In this configuration, we would expect to observe a surface state if
the experimental system was terminated at the edge of a unit cell (i.e., at
one of the white dashed lines in Fig. 3c-d).

Figure 4: Effect on the upper layer depth on surface states. (a) Scattering
matrix model of the finite stratified structure. From the point of view of the
waves, the finite system composed of the internal wave crystal, plus walls (at
the bottom of the tank, and at the wave generator). It can be modelled by a
graph of scattering matrices: $S_{\text{sys}}$ represents the propagation in
the periodic stratification (plus the upper and lower layers), while the
reflection coefficients ($1\times 1$ scattering matrices) $r_{\text{wall}}$
represent the effect of the hard walls. When the system is gapped, it acts as
a frequency-dependent mirror from the point of view of the outside. Hence, the
top and bottom of the systems are decoupled and $S_{\text{sys}}$ can be
replaced by frequency-dependent reflection coefficients $r(\omega)$ and
$r^{\prime}(\omega)$ on each end. (b) We show a plot of the transmission
through the system obtained from numerical simulations (in the same way as
Fig. 2e) as a function of the depth of the last layer $\ell/a$. The bulk bands
are marked by dashed patterns. The frequency of the surface state changes from
$\omega\simeq$0.45\text{\,}{\mathrm{s}}^{-1}$$ (bottom of the gap) to
$\omega\simeq$0.68\text{\,}{\mathrm{s}}^{-1}$$ (top of the gap) when $\ell/a$
increases, as the distance of Wannier center to the surface changes. We note
that there are also surface states in the second gap (below
$0.4\text{\,}{\mathrm{s}}^{-1}$).

However, in the experiment, the wave generator is not located at the edge of a
unit cell: there is an additional layer of size $\ell$ with stratification
$N_{1}$ (see Fig. 1). Hence, the Wannier centers are nor at the center,
neither at the boundary of the unit cell. Although it is tempting to expect
that a surface state will still exist in these intermediate situations, this
cannot be determined from considerations of topology alone. Instead, the
existence of surface states can be phrased as a general condition of
constructive interference [31, 32, 33]. In our experiments, the upper layer
$\ell$ can be viewed as a cavity bounded by two mirrors with reflection
coefficients $r(\omega)$ and $r_{\text{wall}}$, as represented in Fig. 4a (see
also Methods). The reflection coefficient
$r(\omega)=\text{e}^{\text{i}\varphi(\omega)}$ represents the periodic
stratification, that acts as a frequency-dependent mirror in the gap. The
boundary condition $\psi=0$ at a wall (such as the wave generator) fully
reflects the wave with a reflection coefficient $r_{\text{wall}}=-1$. Resonant
modes then occur when a wave interferes constructively with itself over a
round-trip, i.e. when the total phase
$\varphi_{\text{tot}}=\varphi(\omega)+2k_{1}(\omega)\ell+\pi$ is an integer
multiple of $2\pi$.

To monitor the evolution of surface states with the depth $\ell$ of the upper
layer, we artificially vary $\ell$ in numerical simulations. The wave
amplitude near the surface is shown in Fig. 4b as a function of $\omega$ and
$\ell$. We observe that the frequency of the surface states fully crosses the
band gap as $\ell/a$ is increased. In the Methods, we analyse a simplified
model of the experiment, in which these features can be reproduced.
Remarkably, the edge states always flow through the band gap in the same
direction as we vary the parameter $\ell/a$, a bit like in a Thouless pump
[34]. Although we focus here on the surfaces states occurring near the top
surface, because they can easily be accessed experimentally, we emphasize that
edge modes can in general also exist at the bottom of the periodic
stratification, as well as at the interface between two different periodic
stratifications, in the absence of any hard boundary.

Our work demonstrates the existence of internal waves featuring band gaps
described by band theory as well as surface states described by topological
band theory in periodic stratifications.

A band gap could completely prevent the propagation of internal waves at
certain frequencies in regions with periodic stratification. The presence of
an interface state would allow horizontal propagation, but only in a
restricted vertical region near the interface and in a narrow range of
frequencies, in a way similar to internal wave ducting [1]. These effects, if
they indeed occur outside of the lab, could have noticeable consequences in
regions such as the Arctic ocean, where staircase stratifications with the
appropriate length scale exist [15, 7]. Similar phenomena are expected when
the periodicity of the stratification is imperfect. Irregular thermohaline
staircases can in principle exhibit Anderson localization, another phenomenon
that hinders the transmission of waves [35].

## Methods

###  .1 Experimental setup

We first investigate the effect of periodically stratified fluid on internal
wave propagation with a laboratory experiment in a $H\times L\times
W=0.54\times 5.46\times$0.55\text{\,}{\mathrm{m}}^{3}$$ wave tank filled with
salt-stratified water. To avoid reflections from the sides of the tank to
interact with the waves generated in the working section, parabolic reflection
barriers are positioned at both ends of the tank and deviate the waves to the
rear section of a partition running along the length of the tank [5], as shown
in the schematic of Fig. 1(a). We use a traditional double-bucket method to
fill the tank and create a non-uniform stratification composed of 5 layers of
linearly varying density giving $N_{1}\simeq 1.1$ s-1, separated by 4 mixed
layers of constant density where $N_{2}\simeq 0$, see Fig. 1(b). The density
profile $\rho_{0}(z)$ shown in Fig. 1(c) is measured using a calibrated
Precision Measurements Engineering probe attached to a vertically held Parker
linear traverse. The resulting buoyancy profile determined from Eq.(1)
displayed in Fig. 1(c) exhibits a three layer structure with a periodicity
$a=10\pm$1\text{\,}\mathrm{cm}$$ surrounded by upper and lower layers with
identical $N$ on top and bottom of the tank respectively. Note that the last
$5.6\text{\,}\mathrm{cm}$ of the density profile is not shown since our probe
doesn’t reach the bottom of the tank. We generate an internal wave field with
a dominant characteristic horizontal wavenumber
$k_{x}=$27.6\text{\,}{\mathrm{m}}^{-1}$$ using a submerged linear wave
generator with a sinusoidal profile with an amplitude
$A=$3\text{\,}\mathrm{mm}$$ placed at the top of the tank. The waves are
horizontally forced from left to right and propagate downward through the
periodic stratification in a range of frequency
$0.32<\omega<$0.84\text{\,}{\mathrm{s}}^{-1}$$ set by the generator. The flow
velocity field across a $50\times$50\text{\,}{\mathrm{cm}}^{2}$$ domain is
then measured with particle image velocimetry (PIV), allowing a direct
visualization of the internal wave spatial structure inside the periodic
stratification as a function of the wave frequency $\omega$. Selected
snapshots of recorded wave fields are displayed in Fig. 1(c) for 4 different
frequencies.

###  .2 Numerical simulations

We impose a downward propagating internal wave in the upper boundary and use
the experimentally measured stratification profile to solve for the
transmitted wave amplitude at the lower boundary. The wave at the top boundary
is imposed to be equal to the horizontal wave produced by the generator
$\Psi(x,t)=\Psi_{0}\exp(ik_{x}x)$ and evaluate Eq. 2 as a boundary problem.
The subsequent set of equations are then numerically solved using MATLAB bvp4c
function. The resulting transmitted wave amplitude is shown in 1(d) and is in
good agreement with the experimental band structure. Figure 2(c) displays the
resulting wave field amplitude as a function of $z$ and $\omega$, and exhibits
two distinct maxima localized near the upper boundary located near similar
frequencies, $\omega\approx 0.33$ and $0.6\text{\,}{\mathrm{s}}^{-1}$, with
the experimental surface states. We now vary the thickness $\ell$ of the top
layer in the numerical model, and re-iterate the same frequency sweep shown in
Fig. 2(c); see Supplemental material for the corresponding wave fields.
Changing $\ell$, we expect the resonance conditions to shift accordingly and
Fig. 4(c) shows the resulting amplitude of the internal wave in the upper
layer as we change $\ell$. While the band structure appears unchanged, the two
edge modes frequency location seems to shift as $\ell$ is increased.
Remarkably, both of the band gap crossings are happening in the same
direction, while additional edge modes continue to appear as we vary the
parameter $\ell/a$.

###  .3 Computation of the band structures

In this section, we describe the computation of the band structure of a model
of the experiment based on the experimentally measured density profile. As the
stratification is not exactly piecewise constant, we found more convenient to
directly compute the band structure by diagonalizing a linear operator, rather
than using the method based on transfer matrices used to analyse the
simplified model of section .4.

We start from the one-dimensional differential equation (3), and assume that
the stratification $N(z)=N(z+a)$ is spatially periodic with period $a$. The
propagation of internal waves in an infinite system with such a periodic
stratification is described by Bloch theory [21, 17]. The main result of this
theory is that the streamfunction takes the form of a plane wave modulated by
a function with the same periodicity as the unit cell (called a Bloch wave)

$\phi(z)=\text{e}^{\text{i}k_{z}z}u(z)$ (5)

in which $u(z)=u(z+a)$. Here, the quasi-momentum (or wavevector) $k_{z}$ lives
in the Brillouin zone $[-\pi/a,\pi/a]$. The propagation of waves in finite
crystals is usually well-approximated by the infinite crystal (except from
interface states that are discussed in Methods Sec. .4). Equation (3) then
takes the form of an eigenvalue problem

$\mathcal{L}(k_{x},k_{z})\ket{u}=\lambda\ket{u}$ (6)

in which $\mathcal{L}(k_{x},k_{z})$ is a differential operator acting on
periodic functions $[0,a]\to\mathbb{R}$ defined by

$\mathcal{L}(k_{x},k_{z})\ket{u}=\frac{1}{N^{2}(z)}\left[k_{x}^{2}-(\partial_{z}+\text{i}k_{z})^{2}\right]\ket{u}$
(7)

and where the eigenvalue is $\lambda=k_{x}^{2}/\omega^{2}$. Diagonalizing
$\mathcal{L}(k_{x},k_{z})$ gives the dispersion relations
$\omega_{i}(k_{x},k_{z})$ where the integer $i$ labels the band. To do so, we
discretize Eq. (7) using the open-source pseudospectral solver Dedalus [36].
The one-dimensional domain $[0,a]$ is discretized with $n_{\text{C}}=2^{8}$
Chebyshev modes under periodic boundary conditions. The resulting matrix is
diagonalized. Because of the nature of internal waves, the band structure is
bounded from above (there is a maximum frequency). Low-frequency bands
correspond to modes that oscillate fast at the scale of a unit cell. Hence,
the finite number $n$ of Chebyshev modes allows us to resolve only so many
bands, starting from the highest-frequency ones (this is in contrast with the
spectral theory of the Laplacian and derived problems, in which the lowest-
frequency bands correspond to slow oscillations in the unit cell and would be
resolved first). As seen in Fig. 2a, the low-frequency bands accumulate near
$\omega=0$ to form a quasi-continuum.

We computed the band structure of the infinite periodic stratification that
would arise from an infinite repetition of one unit cell of the stratification
measured in the experimental system (Fig. 2a). The stratification $N(z)$ is
obtained by fitting the experimental density profile and computing the
derivative of the fit analytically, while the other parameters $a$ and $k_{x}$
are measured directly.

###  .4 Simplified model of interface states

In this section, we provide a simplified model for stratifications with
interface states as well as general arguments based on scattering matrices for
their existence adapted from Refs. [31, 37, 38, 32, 33, 39, 40].

####  .4.1 Scattering matrices

Figure 5:  Scattering at an interface. The interface is drawn in black.
Incoming amplitudes are drawn in blue while outgoing amplitudes are drawn in
red. The amplitudes of right-moving (left-moving) waves are labelled with a
$+$ ($-$).

We want to keep track of the phases over propagation but also decompose the
function into left/right-moving fields, so to get the amplitude of the field
at $z_{0}$, we decompose the field as

$\phi(z)=\phi_{+}\text{e}^{\text{i}k(\omega)(z-z_{0})}+\phi_{-}\text{e}^{-\text{i}k(\omega)(z-z_{0})}$
(8)

Consider an interface between two media with dispersions
$k_{\text{L/R}}(\omega)$, where L/R correspond to the left/right of the
interface. (To match with usual notations for scattering matrices, we consider
that the propagation is horizontal to define the notations. To get back to
physical directions, one can substitute left/right with bottom/top,
respectively.)

A plane wave with frequency $\omega$ on the left/right of the interface takes
the form

$\phi^{\text{L/R}}(z)=\phi_{+}^{\text{L/R}}\text{e}^{\text{i}k_{\text{L/R}}(\omega)z}+\phi_{-}^{\text{L/R}}\text{e}^{-\text{i}k_{\text{L/R}}(\omega)z}$
(9)

(the full time-dependent wave also has a phase $\text{e}^{-\text{i}\omega t}$,
already factored out of $\phi$).

The scattering matrix $S$ relates incoming waves (incident on the interface)
and outgoing waves (scattered by the interface) by

$\begin{pmatrix}\phi_{+}^{\text{R}}\\\
\phi_{-}^{\text{L}}\end{pmatrix}=S\begin{pmatrix}\phi_{-}^{\text{R}}\\\
\phi_{+}^{\text{L}}\end{pmatrix}.$ (10)

The notations are summarized in Fig. 5. We refer the reader to Refs. [41, 42]
for more details.

The scattering matrix can be expressed as

$S=\begin{pmatrix}r&t^{\prime}\\\ t&r^{\prime}\end{pmatrix}$ (11)

in which $r$ and $t$ ($r^{\prime}$ and $t^{\prime}$) describe the reflection
and transmission of waves arriving from the left (from the right). In general,
$r$, $r^{\prime}$, $t$, $t^{\prime}$ can be matrices (here we only consider
one channel, so they are just numbers).

Equivalently, the transfer matrix

$T=\begin{pmatrix}(t^{\dagger})^{-1}&r^{\prime}(t^{\prime})^{-1}\\\
(r^{\prime})^{\dagger}(t^{\dagger})^{-1}&(t^{\prime})^{-1}\end{pmatrix}$ (12)

(in which $\dagger$ is the conjugate-transpose) relates the fields on the left
to the fields on the right by

$\begin{pmatrix}\phi_{+}^{\text{R}}\\\
\phi_{-}^{\text{R}}\end{pmatrix}=T\begin{pmatrix}\phi_{+}^{\text{L}}\\\
\phi_{-}^{\text{L}}\end{pmatrix}.$ (13)

The transfer matrix of two adjacent regions $(1)$ and $(2)$ with transfer
matrices $T_{1}$ and $T_{2}$ is $T_{2}T_{1}$ (matrix multiplication). The
composition law of scattering matrices can be deduced from that of scattering
matrices [42, Eq. (10.35)].

####  .4.2 Model and transfer matrix of the unit cell

Each unit cell of the crystal is composed of two layers of depth $a/2$ with
uniform stratifications $N_{1}$ and $N_{2}$ (Fig. 8). Within each layer, the
dispersion relation of internal waves is

$k_{z,a}(\omega)=\sqrt{k_{x}^{2}\left(\frac{N_{a}^{2}}{\omega^{2}}-1\right)}$
(14)

in which $a=1,2$. To describe the system, we use the formalism of transfer
matrices: the transfer matrix $T_{\text{uc}}$ of a single unit cell maps the
fields $\psi_{n-1}^{(1)}=(\phi_{+,n-1}^{(2)},\phi_{-}^{(1)})$ in the first
layer of the $(n-1)$th unit cell to the fields $\psi_{n}^{(1)}$ in the first
layer of the $n$th unit cell by

$\psi_{n}^{(1)}=T_{\text{uc}}\psi_{n-1}^{(1)}$ (15)

It is decomposed into (see Fig. 6)

$T_{\text{uc}}=T_{1}^{\text{prop}}(\ell_{1})\,T_{12}^{\text{int}}\,T_{2}^{\text{prop}}(\ell_{2})\,T_{21}^{\text{int}}$
(16)

in which

$T_{a}^{\text{prop}}(\ell)=\begin{pmatrix}\text{e}^{\text{i}k_{a}(\omega)\ell}&0\\\
0&\text{e}^{-\text{i}k_{a}(\omega)\ell}\\\ \end{pmatrix}$ (17)

describes the propagation within a layer of size $\ell$ with dispersion
$k_{a}(\omega)$ and $T_{ab}^{\text{int}}$ the interface between the layers $a$
and $b$. We assume that the density $\rho$ is continuous at each interface, so
that the streamfunction $\psi$ and its derivative $\psi^{\prime}$ are
continuous [1]. Hence, we find that at an interface between layers with
dispersions $k_{L}(\omega)$ on the left of the interface and $k_{R}(\omega)$
on the right,

$\displaystyle\phi_{+}^{\text{L}}+\phi_{-}^{\text{L}}$
$\displaystyle=\phi_{+}^{\text{R}}+\phi_{-}^{\text{R}}$ (18)
$\displaystyle\frac{1}{k_{L}}\left[\phi_{+}^{\text{L}}-\phi_{-}^{\text{L}}\right]$
$\displaystyle=\frac{1}{k_{R}}\left[\phi_{+}^{\text{R}}-\phi_{-}^{\text{R}}\right].$
(19)

These relations can be written in the form

$\begin{pmatrix}\phi_{+}^{\text{R}}\\\
\phi_{-}^{\text{R}}\end{pmatrix}=T_{RL}^{\text{int}}\begin{pmatrix}\phi_{+}^{\text{L}}\\\
\phi_{-}^{\text{L}}\end{pmatrix}$ (20)

in which we have defined the transfer matrix

$T_{RL}^{\text{int}}=\frac{1}{2k_{\text{L}}}\,\begin{pmatrix}k_{\text{L}}+k_{\text{R}}&k_{\text{L}}-k_{\text{R}}\\\
k_{\text{L}}-k_{\text{R}}&k_{\text{L}}+k_{\text{R}}\end{pmatrix}$ (21)

####  .4.3 Bulk band structure

The properties of the infinite crystal (and of the bulk of a large enough
finite crystal) are described by Bloch theory. Bloch waves satisfy

$\psi_{n}^{(1)}=\text{e}^{\text{i}k_{z}\,a}\psi_{n-1}^{(1)}$ (22)

in which $a=\ell_{1}+\ell_{2}$ is the size of the unit cell, and $k_{z}$ the
quasi-momentum in the direction of propagation $z$. Comparing with Eq. (15),
we identify the Bloch phase factors $\text{e}^{\text{i}k_{z}\,a}$ with the
eigenvalues $\lambda$ of the transfer matrix $T_{\text{uc}}$ that satisfy
$|\lambda|=1$. The corresponding eigenvectors $\ket{\Psi}$ satisfying

$T_{\text{uc}}\ket{\Psi}=\lambda\ket{\Psi}$ (23)

are the Bloch eigenvectors. We can therefore obtain the dispersion relation
$k_{z}(\omega)$ [and by inverting this relation, $\omega(k_{z})$] by
diagonalizing $T_{\text{uc}}$ and selecting the relevant eigenvalues. The band
structure obtained in this way is plotted in Fig. 7. As illustrated in Fig.7,
the band structure features band gaps, a internal waves with frequencies in
the band gap cannot propagate through the crystal.

Figure 6:  Unit cell of the simplified model. Figure 7:  Band structure of the
simplified model. The band structure is obtained by diagonalizing
$T_{\text{uc}}(\omega)$. The eigenvalues satisfying $|\lambda|=1$ correspond
to Bloch modes with a wavevector $k_{z}$ such that
$\lambda=\text{e}^{\text{i}k_{z}\,a}$. Frequencies $\omega$ for which no
eigenvalue satisfies $|\lambda|=1$ correspond to band gaps. We have set
$N_{2}/N_{0}=$1.2$$, $N_{1}/N_{0}=$0.8$$, $k_{x}/a=$3.0$$.

####  .4.4 Surface modes

Now, we consider a finite structure composed of $n$ unit cells, but in which
the last (upper) layer has a depth $a/2+\ell$ instead of $a/2$ (see Fig. 8a).
At the top and bottom of the structure, a wall imposes the boundary condition
$\psi=0$. We decompose this structure into (i) the lower wall, (ii) the
crystal, (iii) the additional layer of depth $\ell$ and (iv) the upper wall;
and focus on what happens at the top.

Figure 8:  Scattering matrix model of the finite stratified structure. (a)
Schematic representation of the model. A unit cell of length $a$ composed of
two layers with stratifications $N_{1}$ and $N_{2}$ is repeated. In the
infinite crystal, the unit cell is repeated an infinite number of times. In a
finite system, it is only repeated a certain number $n$ of times. Here, we
assume that the system is terminated by hard walls (like in the experiment),
and allow the possibility that the last layer (at the top) has a thickness
$\ell_{2}+\ell$ instead of $\ell_{2}$ (again, like in the experiment). (b)
From the point of view of internal waves, this finite system can be modelled
by a graph of scattering matrices: $S_{\text{sys}}$ represents the propagation
in the periodic stratification (plus the last layer), while the reflection
coefficients ($1\times 1$ scattering matrices) $r_{\text{wall}}$ represent the
effect of the hard walls. When the system is gapped, the top and bottom of the
systems are decoupled and $S_{\text{sys}}$ can be replaced by a reflection
coefficient on each end ($r$ and $r^{\prime}$).

When $\omega$ is in the bulk band gap of the stratified structure, there is no
transmission of internal waves through the crystal. Hence, the scattering
matrix of the crystal is of the form

$S_{\text{crystal}}=\begin{pmatrix}r_{\text{crystal}}&0\\\
0&r^{\prime}_{\text{crystal}}\end{pmatrix}$ (24)

(as $t=t^{\prime}=0$; when the crystal is finite, the transmission
coefficients do not strictly vanish and taking them to zero is an
approximation). If there is no absorption, the reflection coefficients are
simply phases $r=\text{e}^{\text{i}\varphi_{0}}$ and
$r^{\prime}=\text{e}^{\text{i}\varphi_{0}^{\prime}}$. In the additional layer
of thickness $\ell$, internal waves freely propagate: this is described by the
transfer matrix
$T_{\text{layer}}=\operatorname{diag}(\text{e}^{\text{i}\varphi},\text{e}^{-\text{i}\varphi})$
or equivalently by the scattering matrix

$S_{\text{layer}}=\begin{pmatrix}0&\text{e}^{\text{i}\varphi}\\\
\text{e}^{\text{i}\varphi}&0\end{pmatrix}$ (25)

in which $\varphi=k_{2}(\omega)\ell$. Hence, the scattering matrix for the
full structure is

$S_{\text{sys}}\equiv\begin{pmatrix}r&0\\\
0&r^{\prime}\end{pmatrix}=\begin{pmatrix}r_{\text{crystal}}&0\\\
0&\text{e}^{2\text{i}\varphi}r^{\prime}_{\text{crystal}}\end{pmatrix}$ (26)

so that the reflection phase of waves arriving from the right (physically,
from the top) is $\varphi_{0}^{\prime}+2\varphi$.

As there is no transmission of internal waves through the stratification
described by $S_{\text{sys}}(\omega)$ at frequencies $\omega$ in the bulk band
gap of the crystal, we can analyse the top and the bottom independently (Fig.
8b). For concreteness, let us focus on the top (a similar analysis applies to
the bottom). The reflection phase due to the upper wall is $\pi$ 111At the
wall, located e.g. at $z=0$, we impose $\psi=0$. Hence, $\phi_{+}+\phi_{-}=0$.
The scattering matrix of the wall is simply the reflection coefficient
$S_{\text{wall}}=r_{\text{wall}}$ such that
$\phi_{-}=r_{\text{wall}}\phi_{+}$, so $r_{\text{wall}}=-1$.. Hence, the
resonance (constructive interference) condition in the cavity formed by the
upper wall and the gapped stratification is

$r^{\prime}(\omega)\times r_{\text{wall}}=1$ (27)

i.e. $\varphi_{0}^{\prime}(\omega)+2k_{1}(\omega)\ell+\pi=2\pi n$ with
$n\in\mathbb{Z}$. This condition is usually satisfied only for discrete values
of $\omega$ that are the frequencies of the surface modes. Note that in
general, the condition (27) can hold even when $\ell=0$ (see Fig. 9 and
discussion below for an example).

We emphasize that the walls are not a necessary ingredient: the same mechanism
can take place at an interface between two gapped stratifications. It is also
not necessary that the gapped stratifications are spatially periodic: they can
also be quasiperiodic (see Ref. [33] and references therein for examples in
photonic crystals) or disordered.

Surface states can also be obtained by propagating an initial condition that
encodes the boundary condition into the crystal with the transfer matrix of
the unit cell [39, 40] (see also Refs. [44, 45, 46, 47], in particular for the
relation with topological band theory). States that decay from the boundary
correspond to surface states, while those with constant amplitude are bulk
modes. This perspective is convenient for the numerical (or analytical)
computation of the frequencies of surface states. To do so, it is more
practical to start from the bulk and then match the boundary condition. First,
consider the eigenvectors $\Psi$ of $T_{\text{uc}}(\omega)$ with eigenvalues
$\lambda$ satisfying $|\lambda|>1$, so that they decay in the crystal from an
edge located on the right side of the system (provided that the appropriate
boundary condition is satisfied). Second, express the boundary condition
$\phi^{\text{wall}}=0$ as
$\mathcal{W}(\Psi^{\text{wall}})\equiv\phi_{+}^{\text{wall}}+\phi_{-}^{\text{wall}}=0$.
Hence, surface states correspond to the eigenvectors $\Psi$ of $T_{\text{uc}}$
with eigenvalues $|\lambda|>1$ that satisfy $\mathcal{W}(\Psi)=0$ when the
crystal is terminated at the end of a unit cell. Solving for the frequencies
$\omega_{\text{ss}}$ such that both conditions are simultaneously satisfied
gives the frequencies of the surface modes. When the last unit cell is
incomplete of modified, we have to first express the fields
$\Psi^{\text{wall}}$ at the wall in terms of the fields $\Psi^{\text{end}}$ at
the end of the crystal (i.e., of the last whole unit cell) by an appropriate
transfer matrix $T_{\text{surf}}$ such that
$\Psi^{\text{wall}}=T_{\text{surf}}\Psi^{\text{end}}$, and to ask that the
eigenvector satisfies $\mathcal{W}(T_{\text{surf}}\Psi)=0$ instead.

In Fig. 9, we compute in this way the bulk band structure (bulk bands are
drawn in grey) and the frequencies of the top surface states (in red) in a
generalization of our simplified model, in which the two layers have different
thicknesses $\ell_{1}$ and $\ell_{2}$ (with $a=\ell_{1}+\ell_{2}$) and the
additional layer is not present ($\ell=0$; the effect of a non-zero $\ell$ is
represented on Fig. 10). When $\omega$ is in a bulk gap, the scattering matrix
corresponding to the transfer matrix for $n$ unit cells
$[T_{\text{uc}}(\omega)]^{n}$ is approximately diagonal (or block-diagonal if
there are more than one channel) as the transmission coefficients on the off-
diagonal decay exponentially with $n$. The reflection coefficients on the
diagonal converge to the value of the infinite crystal as $n\to\infty$. In
practice, a few unit cells are enough to obtain good approximate convergence.
In the inset of Fig. 9, we plot the phase $\arg r^{\prime}$ of the reflection
coefficient at the surface for a fixed $\omega$ corresponding to the dashed
blue line in Fig. 9 and $n=$20$$. As can be seen on the figure, a surface
state indeed occurs when $\arg r^{\prime}=\pi$, which corresponds to the
constructive interference condition $r^{\prime}(\omega)\times
r_{\text{wall}}=1$ (with $r_{\text{wall}}=-1$) discussed above. We also
observe on Fig. 9 that continuously changing the parameter
$\ell_{2}/[\ell_{1}+\ell_{2}]$ moves a surface state from one band to another:
each surface state (in red) connects the band above its gap to the band below
as the parameter is changed 222Expanding $k_{z}(\omega)$ and
$\varphi_{0}^{\prime}(\omega)$ in series around $\omega^{0}$ by writing
$\omega=\omega^{0}+\Delta\omega$, the condition of constructive interference
(27) becomes $\alpha+\beta\Delta\omega=2\pi n$ (in which $\alpha$ and $\beta$
are coefficients obtained from the expansion). Starting from a solution
$\Delta\omega_{0}$ of this equation, we change the reflection phase of the
crystal by $\Delta\varphi$ (independently of frequency for simplicity). This
amounts to the change $\alpha\to\alpha+\Delta\varphi$. Hence, the frequency of
the surface state becomes $\Delta\omega_{0}-\Delta\phi/\beta$.. Formally, this
is similar to a Thouless pump [34]. Physically, it means that a two-
dimensional system in which the reflection phase is continuously changed in a
dimension orthogonal to the direction of periodicity (for instance by shifting
the internal wave crystal linearly) will generically exhibit non-zero first
Chern numbers; this has been demonstrated in optics in Ref. [49] (see also
Ref. [50], in which the additional dimension is a control parameter).

Figure 9:  Surface states of the simplified model. Surface states of the
simplified model with fixed $\ell=$0.2$$ as a function of the relative
thickness $\ell_{2}/[\ell_{1}+\ell_{2}]$ of the two layers in a unit cell,
computed using the transfer matrix method [39, 40]. The bands are marked as
grey regions, and the surface states as red curves. We have only drawn the
surface states (in red) at a single wall, where the boundary condition
$\psi=0$ is imposed. Inset: reflection phase $\arg r^{\prime}$ of a finite
crystal with $n=$20$$ unit cells for $\omega/N_{0}=$0.41$$ (corresponding to
the blue dashed line in the main figure). Surface states occur when the
reflection phase crosses $\pi$. The corresponding value of
$\ell_{2}/[\ell_{1}+\ell_{2}]$ is marked by a gray dotted line both in the
inset and in the main figure. We have set $N_{1}/N_{0}=$1.2$$,
$N_{2}/N_{0}=$0.8$$, $k_{x}/a=$3.0$$.  Figure 10:  Surface states of the
simplified model. Surface states of the simplified model as a function of the
excess thickness of the last layer $\ell$ (see schematic on Fig. 8), computed
using the transfer matrix method [39, 40]. Here, $\ell_{1}/a=\ell_{2}/a=1/2$.
The bands are marked as grey regions, and the surface states as red curves. We
have only drawn the surface states at a single wall, where the boundary
condition $\psi=0$ is imposed. Unless otherwise specified, the parameters are
the same as in Fig. 9.

We now comment on the relation between surface states and topology. In their
simplest form, topological insulators are spatially periodic systems
(crystals) in which (i) the band structure describing wave propagation
exhibits gaps and (ii) certain topological properties (winding numbers and
their generalizations) can be ascribed to the bulk Bloch bands. These
topological states occur in various contexts including electrons in solids
[22], cold atoms [51], optics [24, 25], acoustic and mechanics [52, 53], but
also in biochemical networks [54], active matter [55], and geophysical fluids
[26, 27]. In many cases (but not all), a relation called the bulk-boundary
correspondence connects the topological properties of the bulk bands to the
existence of surface states at an interface. In continuum models, however, the
bulk-boundary correspondence is often modified to account for subtleties not
present in lattice models, and is therefore less easily applicable [56, 57,
58, 59, 60, 61, 62].

Our simplified model with alternating stratifications $N_{1}$ and $N_{2}$ is
similar to continuum models of photonic crystals made of alternating
dielectric slabs with permittivities $\epsilon_{1}$ and $\epsilon_{2}$. The
surface states are similar to Tamm-Shockley states [63, 64] that exist in
electronic, optical, and other wave systems. The topological nature of these
surface states in optics and acoustics has been discussed in Refs. [32, 33,
65] (see also references therein). The continuum photonic crystal analysed in
Refs. [32, 33, 65] can indeed be mapped [65] to the Su-Schrieffer-Heeger (SSH)
model, a standard example of symmetry-protected topological insulator [22, 66]
(which involves some subtleties [67, 68, 69]). The same applies to our models
of internal waves in periodic stratifications. The relevant topological
invariant is called the Zak phase $\phi_{\text{Zak}}$ (there is one Zak phase
per band). In general, the Zak phase can be defined as the integral of the
Berry connection across the one-dimensional Brillouin zone [28], and it can
take any value. In a system with spatial inversion symmetry (here, $z\to-z$),
the Zak phase is constrained to take values $0$ or $\pi$, and can be expressed
as [28, 70]

$\frac{\phi_{\text{Zak}}}{\pi}=\frac{\xi_{0}-\xi_{\pi}}{2}\,\text{mod
{\hbox{2}}}$ (28)

in which $\xi_{q}$ is the inversion eigenvalue (or parity eigenvalue) of the
band at wave-vector $q=0,\pi/a$, which is $\pm 1$ depending on whether the
eigenvector $\psi(k)$ describing the band is symmetric or antisymmetric (i.e.,
even or odd under inversion). The two possible values
$\phi_{\text{Zak}}=0,\pi$ can be considered as topologically inequivalent. To
see why, note that the Zak phase gives, qualitatively, the average position of
the wave in the unit cell. Formally, this quantity is called the Wannier
center of the band (see Refs. [28, 23, 29, 30] for details). It turns out that
when the system is invariant under spatial symmetries, the Wannier center must
be located at special points in the unit cell called (maximal) Wyckoff
positions. In a system with inversion symmetry, the Wannier center can only be
located at the center or at the edge of the unit cell (the Wyckoff positions
1a and 1b, see Fig. 12), that correspond respectively to $\phi_{\text{Zak}}=0$
and $\phi_{\text{Zak}}=\pi$. In Fig. 11, we show the band structure of the
simplified model labelled with their Zak phases calculated from Eq. (28). A
topological transition occurs between the cases (a) and (b), when the bulk
band gap around $\omega/N_{0}\sim$0.25$$ closes at
$\ell_{2}/(\ell_{1}+\ell_{2})\sim$0.6$$ (see Fig. 11). In the framework of
topological quantum chemistry [23, 29, 30], the bands with
$\phi_{\text{Zak}}=\pi$ (equivalently, with Wannier centers at Wyckoff
position 1b) are in an obstructed atomic limit. In contrast, those with
$\phi_{\text{Zak}}=0$ (Wyckoff position 1a) are in a trivial atomic limit.
(For more details, see Fig. 1 of Ref. [30], Sec. III of Ref. [71], and SI Sec.
IV of Ref. [23].)

We can also ask to what extent does it matter in practice that a surface state
is topological, besides the fundamental interest of the underlying
mathematical structure. In certain contexts (such as the quantum Hall effect),
the topological origin of edge states confers them a robustness against
defects, disorder, and changes in the system’s parameter. Here, however, the
existence and properties of surface states depends strongly on how the system
is terminated (this can already be seen in Fig. 9).

Figure 11:  Zak phases in the simplified model. The band structures of the
simplified model are computed with the spectral method described in Sec. .3
for two values of $\ell_{2}/[\ell_{1}+\ell_{2}]$, respectively in panels (a)
and (b). We also compute the Zak phases of the four first (highest-frequency)
bands using Eq. (28). In the bottom panels, we show the (real part of) the
Bloch wavefunction of the second highest-frequency band (highlighted in pink
in the top panels), at $q=0$ (dashed blue lines) and $q=\pi/a$ (continuous red
lines) in both cases. In (a), they have opposite symmetries with respect to
inversion $z\to-z$, so the Zak phase is $\pi$. In (b), they have the same
symmetry, so the Zak phase is $0$. We have set $N_{1}/N_{0}=$1.2$$,
$N_{2}/N_{0}=$0.8$$, $k_{x}/a=$3.0$$.  Figure 12:  Topological phases and
maximal Wyckoff positions. We show in (a) the two maximal Wyckoff positions 1a
and 1b in a one-dimensional unit cell with spatial inversion symmetry. Panels
(b) and (c) show a sketch of the two possible atomic limits, corresponding to
Wannier centers (in red) pinned either at the Wyckoff position 1a (at the
center of the unit cell) or at the Wyckoff position 1b (at the edge of the
unit cell). (Both edges of the unit cell are equivalent; the spacing between
the unit cells has been added for clarity and is not really present.)

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