# Sound diffusion with spatiotemporally modulated acoustic metasurfaces

Janghoon Kang<EMAIL_ADDRESS>Walker Department of Mechanical
Engineering, The University of Texas at Austin, Austin, TX, USA, 78712-1591
Michael R. Haberman<EMAIL_ADDRESS>Walker Department of Mechanical
Engineering, The University of Texas at Austin, Austin, TX, USA, 78712-1591

###### Abstract

Traditional sound diffusers are quasi-random phase gratings attached to
reflecting surfaces whose purpose is to augment the spatiotemporal incoherence
of the acoustic field scattered from reflective surfaces. This configuration
allows one to cover a large reflecting surface by periodically tiling the
diffuser unit cells to cover a large area while reducing undesirable specular
reflection for incident plane waves. However, the periodic arrangement of the
unit cells leads to coherent constructive and destructive interference in the
scattered field in some directions which is undesirable for achieving acoustic
diffusivity. The spatial uniformity of acoustic energy scattered from
conventional diffusers constructed in this way is a fundamental limitation of
the traditional approach which is not easily overcome when one wishes to cover
large reflecting surfaces. In this work, we investigate spatiotemporal
modulation of the surface acoustic admittance of an acoustic diffuser as a new
approach to improve the sound diffusion. We develop a semi-analytical model
that employs Fourier series expansion to determine the scattered sound field
from a surface admittance consisting of a quadratic residue diffuser (QRD)
design whose individual well admittances are modulated with a traveling wave
with modulation frequency, $\omega_{\mathrm{m}}$, amplitude, $Y_{\mathrm{m}}$,
and a wavenumber that matches the unit cell length, $\Lambda$. We observe
significant improvement in diffusion performance due to the fact that the
spatiotemporal modulation scatters sound into additional frequency–wavenumber
pairs associated with harmonics of the modulation frequency and their
diffraction orders. The semi-analytical model results are verified using time-
domain finite element model.

††preprint: AIP/123-QED

Conventional sound diffuser designs employ surface irregularity, usually in
the form of acoustic cavitiesSchroeder (1975, 1979); D’Antonio and Konnert
(1984); Cox and Lam (1994), to maximize the randomness in the phase of the
acoustic field scattered from each point on the diffuser. Randomization in the
scattered field results in a minimization in the angular dependence of the
scattered field intensity when subjected to plane wave incidence, thereby
increasing the diffusivityBallestero  _et al._ (2019). Traditional diffuser
unit cell designs are then tiled over larger areas using a periodically
repeating arrangement in space. As a result, grating lobes are generated in
the field scattered from the collection of units, which significantly degrades
acoustic diffusion performanceAngus (2000); Cox and D’Antonio (2003); Cox and
d’Antonio (2016). Conventional diffuser performance can therefore be improved
by modifying the design to disrupt periodicity in unit cell tiling, for
example, by adding different types of sequencesAngus (2000); Cox and d’Antonio
(2016). However, these approaches require increased diffuser size to introduce
additional elements or additional depth to create a quasi-random geometric
sequences with sufficiently different phase distribution. This work presents
an unique approach to enhance acoustic diffusion using spatiotemporal
modulation of the input impedance of traditional diffuser design geometries.
We show that this approach greatly improves diffuser performance while
maintaining the geometry of traditional diffuser designs such as the quadratic
residue diffuser (QRD), first introduced by SchroederSchroeder (1979).

Spatiotemporal modulation of material properties has recently gained interest
as a mechanism to generate nonreciprocal wave propagating in the
electromagnetic, acoustic, and elastic wave propagation thereby enabling new
avenues to control wave fields Nassar  _et al._ (2020). Early work considered
the effects of spatiotemporal modulation in the form of a pump wave imposed by
an external source in an unbounded medium on electromagnetic wave
propagationCassedy and Oliner (1963), while more recent research studied
spatiotemporal modulation of stiffness or density in the form of a traveling
wave to induce nonreciprocal propagation in acoustic and elastic
domainsTrainiti and Ruzzene (2016); Nassar  _et al._ (2017a, b); Vila  _et
al._ (2017); Riva  _et al._ (2019); Trainiti  _et al._ (2019); Goldsberry,
Wallen, and Haberman (2020); Sugino, Ruzzene, and Erturk (2020); Shen  _et
al._ (2019); Li  _et al._ (2019); Zhu  _et al._ (2020). Modulation has also
been employed as a means to mimic symmetry-breaking momentum bias present in
fluid-flowFleury  _et al._ (2014) in order to create nonreciprocal acoustic
and electromagnetic circulatorsWang and Fan (2005); Fleury, Sounas, and Alù
(2015); Goldsberry, Wallen, and Haberman (2022). Modulation has also been used
in metasurfaces to induce nonreciprocal reflection and transmission of elastic
and acoustic wavesZhu  _et al._ (2020); Adlakha  _et al._ (2020) as well phase
shifting Wang  _et al._ (2020), total power combiningWang  _et al._ (2021),
nonreciprocal scatteringGuo  _et al._ (2019), and the Doppler-like frequency
shift of reflected optical waves Shaltout, Kildishev, and Shalaev (2015);
Shaltout, Shalaev, and Brongersma (2019).

Spatiotemporal modulation of bounded and unbounded media leads to the
generation of harmonics in propagating and forced standing wave fields,
respectively, at multiples of the modulation frequency and wavenumberVila  _et
al._ (2017); Wallen and Haberman (2019); Nassar  _et al._ (2017a); Goldsberry,
Wallen, and Haberman (2019). While the presence of these harmonics can
complicate the design of nonreciprocal devicesGoldsberry, Wallen, and Haberman
(2020), the generation of additional frequencies and wavenumbers is highly
useful if one wishes to efficiently diffuse acoustic energy in time and space.
The idea of scattering energy into multiple frequencies and wavelengths via
nonlinear interactions is a well-known approach to efficiently absorb
acousticKonarski, Haberman, and Hamilton (2020) and vibrational energyMotato
_et al._ (2017), but this concept has not been considered as a means to
improve acoustic diffusion. Further, the use of spatiotemporal modulation to
scatter energy into different frequencies does not require nonlinear behavior
or high amplitude signals. In this case, the surface input impedance becomes a
linear, time varying parameter for every point no the surface, which requires
a generalization of the conventional grating equation to model the diffuser
performance.

In this work, we adapt the approach that Goldsberry et al.Goldsberry, Wallen,
and Haberman (2020) employed to investigate nonreciprocal vibrations in Euler
beams to the present case of surface acoustic admittance modulation of a
traditional Schroeder diffuser scheme to improve sound diffusion. We derive a
theoretical model that considers a spatiotemporally modulated surface
admittance in the form of a traveling wave with modulation frequency,
$\omega_{\mathrm{m}}$, amplitude, $Y_{\mathrm{m}}$, and a wavenumber that
matches the unit cell length, $\Lambda$. With the properly selected modulation
parameters, the far-field scattering is determined using a semi-analytical
model shows that the directions of the diffraction modes are rotated from the
original ones due to the addition of wavenumber and frequency harmonics which
do not have symmetric amplitudes with respect to the incident frequency and
wavenumber. Further, we show that the diversity of scattered wavenumbers
generated by the modulation leads to additional diffracted modes, thereby
reducing the prominence in grating lobes of the classical diffuser design and
significantly improving diffuser performance. These results demonstrate that
the spatiotemporal modulation of the phase grating on a sound diffuser is a
new approach to tailor the scattered sounds and to improve the diffusion
performance in order to disperse scattered sound power.

Consider the classical, Schroeder-type one-dimensional (1D) sound diffuser
shown in Fig. 1a, where the local input admittance $Y(x)$ is a function of the
depth of the cavity at the position. The position-dependent admittance can be
determined using various approaches to optimize sound diffusionCox and Lam
(1994). We have used a QRD with seven cavities for this demonstration, but the
approach presented here can accommodate any admittance profile. We consider a
generalized case of a diffuser shown in Fig. 1b where the local input
admittance is decomposed into a static and modulated components as
$Y(x)=Y_{0}(x)+Y_{\mathrm{m}}(x,t)$, where $Y_{0}(x)$ is the surface
admittance of the unmodulated sound diffuser and $Y_{\mathrm{m}}(x,t)$ is the
modulation. The scattering problem is solved by decomposing the sound field
into incident and scattered fields which are represented by the summation of
the diffraction modesBerkhout, van Wulfften Palthe, and De Vries (1979);
Mechel (1995); Wu, Cox, and Lam (2000) as shown in Eq. 1. To account for the
modulation in the analytical model, the acoustic field and the acoustic
surface admittance of the diffuser are expressed as the function of time and
spaceGoldsberry, Wallen, and Haberman (2019). The total acoustic field is
therefore described as

$\displaystyle p\left(x,z,t\right)=$
$\displaystyle\,p_{\mathrm{I}}\left(x,z,t\right)+p_{\mathrm{s}}\left(x,z,t\right)$
$\displaystyle=$
$\displaystyle\,a_{\mathrm{I}}e^{j\left(\omega_{0}t-k_{x}x+k_{z}z\right)}+\sum_{n=-\infty}^{\infty}a_{n}(t)e^{-j\left(\beta_{n}x+\gamma_{n}z\right)},$
(1)

where $p_{\mathrm{I}}\left(x,z,t\right)$ and
$p_{\mathrm{s}}\left(x,z,t\right)$ represent the incident and the scattered
sound respectively, $\omega_{0}$ is the frequency of the incident plane wave,
$k_{x}=k\sin\theta_{\mathrm{I}}$ and $k_{z}=k\cos\theta_{\mathrm{I}}$ are the
wave numbers of the incident sound in the $x$ and $z$-direction,
$\theta_{\mathrm{I}}$ is the angle of incidence, $\beta_{n}$ and $\gamma_{n}$
are the wave numbers of the $n^{\mathrm{th}}$ diffraction mode in the $x$\-
and $z$-direction, respectively, and $\theta_{n}$ is the angle of the
propagation of the $n^{\mathrm{th}}$ diffracted mode. The scattering
amplitudes, $a_{n}(t)$, are represented as the function of time to consider
the effect of the temporal modulation. By using the classical grating
equation, $\beta_{n}$ can be related to the geometry of the sound diffuser as

$\displaystyle\beta_{n}=k_{x}+\frac{2\pi}{\Lambda}n,$ (2)

where $\Lambda$ is the length of one period of the acoustic diffuser.

Figure 1:  Diagram of the incident and scattered sound in front of a sound
diffuser. The scattered field is composed of plane-waves reflected from (a) an
unmodulated surface, or (b) a modulated surface.

The momentum equation at the diffuser interface, Eq. 3, must consider the
interaction of the incident field with the spatiotemporally modulated surface
impedance through the normal component of the particle velocity,
$u_{z}(x,z=0,t)$ and the acoustic pressure, $p(x,z=0,t)$. Using the definition
of the surface impedance, the normal velocity can be related to the sound
pressure at $z=0$ and the surface impedance, $Z(x,t)$.

$\displaystyle\rho_{0}\frac{\partial u_{z}}{\partial
t}\Big{|}_{z=0}=-\frac{\partial p}{\partial z}\Big{|}_{z=0},\quad
u_{z}(x,t)|_{z=0}=-\frac{p(x,t)}{Z(x,t)}\Big{|}_{z=0}.$ (3)

The sound pressure, normal velocity, and surface impedance must all be
expressed as the function of time and space to account for spatiotemporal
modulation. The normalize surface admittance, $Y(x,t)=\rho_{0}c_{0}/Z(x,t)$,
where $\rho_{0}c_{0}$ is the characteristic impedance of the background
medium, is represented by the expansion of the harmonic terms in the form of
the admittance

$\displaystyle
Y(x,t)=Y_{0}(x)+Y_{\mathrm{m}}\sum_{\begin{subarray}{c}p=-\infty\\\ p\neq
0\end{subarray}}^{\infty}\hat{\chi}^{p}(x)e^{jp\omega_{\mathrm{m}}t},$ (4)

where $Y_{\mathrm{m}}$ and $\omega_{\mathrm{m}}$ are the modulation magnitude
and the modulation frequency and $\hat{\chi}^{p}(x)$ represents the spatial
variation of the modulation amplitude due to the $p^{\mathrm{th}}$ harmonic
component of the modulation function. In order to conserve momentum in the
normal direction in the presence of spatiotemporal modulation, the surface
admittance as expressed by Eq. 4 requires the introduction of modulation
harmonics in the scattered wave field. The complex amplitude of the scattered
field must therefore be expressed with a Fourier expansion of the harmonics of
the modulation frequency. Specifically, the the scattering amplitudes
$a_{n}(t)$ in Eq. (1) are represented as

$\displaystyle
a_{n}(t)=\sum_{p=-\infty}^{\infty}\hat{a}_{n}^{p}e^{j\left(\omega_{0}+p\omega_{\mathrm{m}}\right)t}=\sum_{p=-\infty}^{\infty}\hat{a}_{n}^{p}e^{j\omega_{p}t},$
(5)

where $\hat{a}_{n}^{p}$ represents the amplitudes of $p^{\mathrm{th}}$
modulation harmonic and $\omega_{p}=\omega_{0}+p\omega_{\mathrm{m}}$ which is
the scattered frequency shifted by the integer harmonic of the modulation
frequency. The wavenumber of the scattered sound in the $z$-direction can also
be expressed using the shifted frequency, $\omega_{p}$, as

$\gamma_{n}^{p}=\sqrt{\left(\frac{\omega_{p}}{c_{0}}\right)^{2}-\beta_{n}^{2}}.$
(6)

Furthermore, the field of each diffraction order is scattered at the angle

$\theta_{n}^{p}=\sin^{-1}{\left(\frac{\beta_{n}}{\omega_{p}/c_{0}}\right)}=\sin^{-1}{\left(\frac{\sin{\theta_{\mathrm{I}}}+\frac{\lambda_{0}}{\Lambda}n}{1+p\frac{\omega_{m}}{\omega_{0}}}\right)},$
(7)

where $n$ is the diffraction order associated with the diffuser periodicity
and $p$ is the modulation harmonic index. When spatiotemporal modulation is
absent, $p=0$ and $\omega_{\mathrm{m}}=0$, and Eq. (6) and Eq. (7) simplify to
the standard grating formula. The shifted frequency of the scattered sound,
$\omega_{p}$, caused by the modulation clearly affects the directions of each
diffraction mode by shifting them from what one would observe with with an
unmodulated diffuser surface. We clearly observe that the primary effect of
the spatiotemporal modulation on the scattered sound is to scatter energy into
modulation harmonics and their associated diffraction angles which leads to
significant improvement of the diffusion performance as shown below. Finally,
we note that diffraction mode having the purely imaginary $\gamma_{n}^{p}$ are
evanescent and cannot propagate into the far-field. Propagating diffraction
modes are therefore bounded by the relations

$-\frac{\Lambda}{\lambda_{0}}\left(1+\sin{\theta_{\mathrm{I}}}+p\nu\right)\leq
n\leq\frac{\Lambda}{\lambda_{0}}\left(1-\sin{\theta_{\mathrm{I}}}+p\nu\right),$
(8)

where $\nu:=\frac{\omega_{m}}{\omega_{0}}$. Equations (1)–(8) allow the
calculation of the scattering amplitudes $\hat{a}_{n}^{p}$, for the entire
range of the index $n$ and $p$ which represent the diffraction modes and the
harmonics of the modulation frequency, respectively, given the incident
amplitude and frequencysup . Once the $\hat{a}_{n}^{p}$ are obtained, the
scattered sound pressure on the surface of the sound diffuser can be
calculated using Eqs. (1) and (5). We then can then employ angular spectrum
methods approximation to estimate the far-field directivity using the non-
evanescent components of the scattering amplitudes at $z=0$Williams (1999).

To demonstrate the ability spatiotemporal modulation to improve sound diffuser
performance, consider two periods of a one dimensional QRD with seven cavities
and design frequency of 5 kHz as a baseline model. The cavity depths (left to
right) of the design shown in Fig. 2a are
$4.9\mathrm{mm},19.6\mathrm{mm},9.8\mathrm{mm},0\mathrm{mm},9.8\mathrm{mm},19.6\mathrm{mm},4.9\mathrm{mm}$.
The surface admittance in Fig. 2a is approximated as
$Y_{0}(x)=j\tan{\left(k_{0}\,d(x)\right)}$ because each cavity of the QRD can
be considered as the closed-end tube. Figure 2b shows the normalized surface
admittance $Y_{0}(x)$ when assuming negligible wall thickness and that the
surface admittance is assumed to be constant across the opening of each
cavity. The modulation function is similarly discretized in space and
modulated in time and space using Eq.9 in Fig. 2c. Figure 2d shows the QRD
admittance and the total spatiotemoral admittance for discrete times $t=0$ to
$t=T_{\mathrm{m}}/2$ where $T_{\mathrm{m}}$ is the period of the modulation.
The incident wave is assumed to be normal to the diffuser surface and
$f_{0}=5$ kHz. Considering the normal incidence and the geometrical symmetry
of the sound diffuser about $x=0$, the nonreciprocity and the effects of the
spatiotemporal modulation can be observed by assessing the change in the
angular dependence of the scattered sound when modulating in the $+x$ or $-x$
directions, such that $+\beta_{n}$ for “forward modulation” and $-\beta_{n}$
for “backward direction.”

Figure 2: (a) Diagram of the original QRD with seven cavities in one period.
(b) Imaginary part of $Y_{0}(x)$. Exact equation (solid blue line) and Fourier
expansion with $N=14$ (solid orange line). (c) Spatial modulation function at
$t=0$. (d) Surface acoustic admittance as a function of space for different
instants in time.

We now consider a single frequency, single wavenumber spatiotemporal surface
admittance given by

$\displaystyle
Y(x,t)=Y_{0}(x)+jY_{\mathrm{m}}\cos(\omega_{\mathrm{m}}t-k_{\mathrm{m}}x),$
(9)

where $k_{\mathrm{m}}=2\pi/\Lambda$. The term for the modulation is assumed to
be purely imaginary assuming no losses in the QRD. A purely reactive
admittance modulation may be achieved by continuously varying cavity depth of
an QRD element or altering the termination stiffness of the cavity. Based on
Eq. (4), the modulation function in Eq. (9) can be approximated as using a
Fourier series to match the expansion of the scattered and momentum equations
as

$Y_{\mathrm{m}}\cos(\omega_{\mathrm{m}}t-k_{\mathrm{m}}x)=\frac{Y_{\mathrm{m}}}{2}\sum_{\begin{subarray}{c}p=-1\\\
p\neq 0\end{subarray}}^{1}\hat{\chi}^{p}(x)e^{jp\omega_{\mathrm{m}}t},$ (10)

where

$\displaystyle\hat{\chi}^{-1}(x)$ $\displaystyle=$ $\displaystyle
e^{j\frac{2\pi}{\Lambda}x},\quad\quad p=-1,$ $\displaystyle\hat{\chi}^{1}(x)$
$\displaystyle=$ $\displaystyle e^{-j\frac{2\pi}{\Lambda}x},\quad\quad p=1,$
$\displaystyle\hat{\chi}^{p}(x)$ $\displaystyle=$ $\displaystyle
0,\quad\mathrm{elsewhere}.$

The surface admittance of the unmodulated QRD elements, $Y_{0}(x)$, can also
be represented as a Fourier seriesBerkhout, van Wulfften Palthe, and De Vries
(1979); Mechel (1995) expansion on the spatial $2\pi n/\Lambda$ as shown in
Fig. 2b using the following expression

$Y_{0}(x)=\sum_{n=-\infty}^{\infty}y_{0,n}\,e^{-j\frac{2\pi n}{\Lambda}x}.$
(11)

Assembling Eqs. (10) and (11), we can write the full expression for the
modulated surface admittance required to establish conservation of momentum at
$z=0$. We substitute the pressure from the impedance relation in Eq. (3) for
the normal velocity in Eq. (3). The resulting momentum equation is then
rewritten as

$-\frac{1}{c_{0}}\frac{\partial}{\partial
t}\left(Y(x,t)p(x,z=0,t)\right)=-\frac{\partial}{\partial
z}\left(p(x,z=0,t)\right),$ (12)

where the surface admittance and pressure are Fourier expansions. In order to
calculate the scattered sound field, we must truncated the number of terms of
the diffraction modes, $n$, and the modulation harmonics, $p$ to lie within
$n\in[-N,\,\,N]$ and $p\in[-P,\,\,P]$. Then, using the truncated series and
the expression for the scattering amplitude from the modulation provided in
Eq. (5), we find a matrix equation for the $(2N+1)(2P+1)$ scattering
amplitudes which is provided in Eq. (S14) of the supplementary materialsup .
In this derivation, $N$ was chosen as twice of the number of elements in one
period for the convergence of the expansion of the original normalized
admittance provided in Eq. (11)Mechel (1995); Cox and d’Antonio (2016). In a
similar way, the number of the modulation harmonics was truncated at $P=10$
which is selected to include all scattering amplitudes above $-50$ dB relative
to the maximum amplitude. The modulation frequency and amplitude were selected
to be $f_{\mathrm{m}}=530$ Hz and $Y_{\mathrm{m}}=1.3$, respectively, to show
the improvement of the diffusion performance.

The magnitude of the calculated scattering amplitudes using the input
parameters described here are plotted according to the indices of the
diffraction modes and the modulation harmonics for two opposite spatial
modulation directions in Fig. 3.

Figure 3:  Square of the scattering amplitudes normalized by the maximum
scattering amplitude for all $n$ and $p$, $|{a}_{\rm n}^{\rm
p}|^{2}/\\{\mathrm{max}\left[{a}_{\rm n}\right]\\}^{2}$, for (a) unmodulated
QRD, (b) backward modulation, and (c) forward modulation. Propagating modes
are enclosed by the solid red lines. Color scale in decibels.

For the unmodulated QRD, the frequency of the scattered sound is the same as
the incident frequency and there are no harmonics and the only three
diffracted orders propagate to the far-field, which can be observed in the
three lobes in the scattering pattern for the unmodulated diffuser in Fig. 4.
When spatiotemporal modulation is present, we observe non-zero scattering
amplitudes at harmonics of the modulation frequency and multiple diffraction
orders for each frequency. The wavenumber in the $z$-direction,
$\gamma_{n}^{p}$ in Eq (6) is dependent on both the diffraction orders and
modulation harmonics, so that each pair of modulation harmonics and diffracted
modes represents a different direction for each component of the scattered
acoustic field. As a result, each scattering amplitude contributes to the
total scattered sound field, thereby distributing scattered sound power more
efficiently in space when compared to the reference QRD diffuser. We can
determine the scattered pressure at the surface of the diffuser using Eq. (1)
and Eq. (5) to yield

$p_{\mathrm{s}}(x,z=0,t)=\sum_{n=-N}^{N}\left(\sum_{p=-P}^{P}\hat{a}_{n}^{p}e^{j\omega_{p}t}\right)e^{-j\beta_{n}x}=\sum_{p=-P}^{P}\hat{a}^{p}(x)e^{j\omega_{p}t},\\\
$ (13)

where we have defined the spatial distribution of the scattered pressure
amplitudes of the $p^{\mathrm{th}}$ modulation harmonics on the diffuser
surface as

$\hat{a}^{p}(x)=\sum_{n=-N}^{N}\hat{a}_{n}^{p}e^{-j\beta_{n}x}.$ (14)

Using the Fraunhofer far-field approximation, by taking the spatial Fourier
transform of $\hat{a}^{p}(x)$, the far-field directivity for each modulation
harmonic for $p\in[-P,P]$ are obtainedsup  and plotted in Fig. 4. It is
observed that the locations of the grating lobes and nulls shift relative to
the grating lobe directions of the unmodulated case and that the propagation
directions of the diffraction modes are significantly diversified when
compared to the unmodulated QRD. The generation of additional diffraction
orders associated with the modulation harmonics is seen to significantly
reduce minima scattered field as a function of the angle $\theta$. We note
that each of the scattering patterns in Fig. 4 represent the angular
dependence of the propagating scattering amplitudes, each with their own
shifted frequency $\omega_{p}$, due to a normally-incident single-frequency
plane wave. To compare the response from the modulated sound diffuser with
that of the unmodulated case, all modulation harmonics in Fig. 4 were combined
to give a single scattering amplitude at each angle since all propagating
modulation harmonics contribute the scattered sound energy. Figure 5a compares
the polar responses of the modulated and unmodulated QRD, showing that the
spatiotemporal modulation significantly improves diffusion performance. The
diffusion coefficient is used as an objective measure for assessing the
diffusion performance. It is defined as in Eq. (15), which is an auto
correlation for spatial similarityCox and d’Antonio (2016). Therefore, the
higher values means the better diffusion performance.

$d_{\psi}=\frac{\left(\sum^{n}_{i=1}10^{L_{i}/10}\right)^{2}-\sum^{n}_{i=1}\left(10^{L_{i}/10}\right)^{2}}{(n-1)\sum^{n}_{i=1}\left(10^{L_{i}/10}\right)^{2}}$
(15)

From the polar responses in Fig. 5a, the diffusion coefficient of the forward
and backward modulation for an incident frequency of 5kHz are calculated as
0.75, while the original QRD’s coefficient is 0.53. These numbers clearly show
the improvement of the diffusion performance by spatiotemporal modulation.

(a) Backward modulation

(b) Forward modulation

Figure 4: Scattered fields for all modulation harmonics, $p\in[-P,P]$, for
forward and backward modulation. The thick red dotted lines for unmodulated
QRD.

(a) Analytical model (b) Numerical simulation (c) Scattered field from
numerical simulation: Modulated (d) Scattered field from numerical simulation:
Unmodulated

Figure 5:  Combining all responses of the modulation harmonics in (a) and of
frequency bands in (b). Each curve is normalized by its maximum value. Non-
modulation (red dotted line), backward (blue solid line) and forward
modulation (red solid line). Transient numerical simulation for (c) the
modulated surface and (d) the unmodulated surface.

For the verification of the developed semi-analytical model, the numerical
finite element model was established in Comsol including the modulated surface
admittance of Eq. (9). The geometry of the numerical model is the same as the
image in Fig. 2a and also has two periods of elements. By using the transient
acoustic module, the far-field polar responses for all frequency bands
including the incident frequency were obtained for the modulation and non-
modulation casessup . By combining all those responses, the far-field polar
responses from the numerical simulation in Fig. 5b show that the diffusion
from the modulated QRD has been improved compared to the original one. The
diffusion coefficient for the modulation case is 0.83, and 0.57 for the
original QRD in the numerical simulation. The improved diffusion performance
due to the sptiotemporal modulation can also be observed in the FEM
simulation. From the transient numerical simulation in a realistic
conditionsup , the scattered sound fields at $t=T_{\rm m}/2$ for both cases
were plotted in Fig. 5c and Fig. 5d, which also shows more dispersed scattered
field from the modulated sound diffuser compared to the three distinct
propagation directions of diffraction modes from the original QRD. The images
of time series were also recorded as an animationsup , in which the true
temporal scattered field can be observed.

The semi-analytical model for the spatiotemporal modulation has been developed
by representing the surface admittance and the scattered amplitudes with the
expansion of the modulation harmonics. For the demonstration, the scattering
amplitudes for multiple diffraction orders and modulation harmonics were
calculated for a representative QRD geometry along with a specified modulation
frequency and amplitude. The presence of modulation harmonics lead to a shift
in the frequency of the scattered acoustic field which affects the propagation
direction for each of the diffraction orders which significantly improves
diffusion performance. The resulting diversity in scattering directions in the
presence of spatiotemporal modulation leads to a significantly more uniform
far-field scattering pattern when compared to the unmodulated case. This
approach can therefore mitigate the minima and maxima in the scattering
directivity present in conventional diffuser designs. The validity of the
semi-analytical model was demonstrated by comparison with the far-field
scattering response obtained using finite element methods, which showed very
similar directivity and improved diffusion behavior when spatiotemporally
modulating the input admittance of a conventional diffuser design.

###### Acknowledgements.

JK acknowledges support from Samsung Electronics. MRH acknowledges partial
support from NSF EFRI Award No. 1641078.

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