Selecting the Direction of Sound
Steven A. Cummer
A device containing a circulating fluid breaks
the symmetry of acoustic waves and allows
one-way transmission of sound.
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Acknowledgments: Supported in part by the UK Natural
Environment Research Council MAMM and Tropical Methane projects, the European Union’s Ingos project, and Royal Holloway.
Structures that admit flow in only one direction are commonplace—con- sider one-way streets, insect traps,
and the staple of the police procedural story,
the one-way mirror. However, creating a
device that allows waves to pass in only one
direction, termed an isolator, is challenging
because of the inherently symmetric physics of wave phenomena. On page 516 of this
issue, Fleury et al. (1), taking inspiration
from a natural electromagnetic phenomenon, designed and demonstrated an engineered structure that allows one-way transmission of sound waves.
Creating a one-way, or nonreciprocal,
structure for general wave flow is more challenging than one might think. The one-way
mirror, for example, is not truly a nonreciprocal optical wave device, as the effect is created primarily by a trick of unequal lighting.
True nonreciprocity in linear materials is
tied to breaking of time-reversal symmetry
(2). A system exhibits time-reversal symmetry if one solution of the entire system, but
run backward in time, is a second solution.
This condition is equivalent to interchanging
the source and receiver sides of the problem,
as illustrated in panel A of the figure. Wave
phenomena by their nature generally exhibit
time-reversal symmetry (consider how circular ripples on a pond surface can be either outwardly expanding or inwardly converging),
so creating a nonreciprocal device or medium
takes something special.
Engineering wave propagation nonreci-
procity into materials is an area of substan-
tial recent research. There are several differ-
ent ways to do this, which are summarized in
clear and thorough reviews in the context of
acoustic (3) and electromagnetic (4) waves.
Interestingly, many efforts that have demonstrated asymmetric power transmission for
specific input and output field distributions
are not truly nonreciprocal and cannot be used
to create general nonreciprocal devices (3, 4).
Carefully designed nonlinear structures can
exhibit nonreciprocity without time-reversal
asymmetry (3, 4), but this approach results
in constraints like amplitude dependence that
limit its value in wave isolator applications.
It turns out that linear systems that contain a directional bias that is defined by some
form of internal motion (a so-called odd vector under time reversal) can be made nonreciprocal (2). In such a system, strict time
reversal reverses the direction of that internal
motion and reverses the bias direction as well.
Such systems would be reciprocal if the inter-
nal motion, and thus the directional bias, is
also reversed when the input and outputs are
swapped, in accord with time reversal. Inter-
changing the input and output ports without
reversing the direction of the bias creates a
nonreciprocal device by breaking time-rever-
sal symmetry, as illustrated in panel B of the
Some materials naturally contain this kind
of directional bias and are inherently non-
reciprocal—for example, the ionized gas of
Earth’s upper atmosphere permeated by the
directional bias of the steady geomagnetic
field (5). An external magnetic field can also
be applied to magnetically active materials,
such as ferrites to create nonreciprocity. This
approach is used in many practical nonreciprocal optical or radio-frequency isolators (6).
In contrast, linear acoustic nonreciprocity had not, until now, been demonstrated
except in weak or large-scale forms not suitable for compact applications. Fleury et al.
have now done precisely that by borrowing
the basic physics of the Zeeman
effect, in which a biasing magnetic field creates a strongly birefringent medium in which different polarization states interact with different medium resonances. Because this is an effect
created by a fixed bias field, such a
medium is nonreciprocal. Fleury
et al. (1) create analogous acoustic resonance splitting in a compact circular cell that contains a
rotational mean air flow. Counterpropagating acoustic waves
in this cell experience different
resonant frequencies, an effect
that can be derived both from the
basics of acoustic wave propagation in a steady mean flow and
from a quantum-mechanical
operator approach [see the supplementary materials of (1)].
Department of Electrical and Computer Engineering, Duke
University, Durham, NC 27708, USA. E-mail: cummer@
Original Time reversal A
Original Time reversal B
Obeying and breaking time-reversal symmetry. (A) In a material
or device without a preferred direction, the fundamentally symmetric
behavior of waves means that a solution runs backward in time and is
still a solution. This condition is equivalent to interchanging the input
and output waves in the system. (B) However, if the system contains
a flow-based directional bias, and that bias is not flipped when the
input and output waves are interchanged, then time-reversal symmetry is broken. Wave transmission through such a system can be
dramatically different when the input and outputs are interchanged,
and thus nonreciprocal.