arXiv:1202.1586v1 [physics.class-ph] 8 Feb 2012
Seismic Waveguide of Metamaterials
Sang-Hoon Kima∗and Mukunda P. Dasb†
aDivision of Marine Engineering, Mokpo National Maritime University, Mokpo 530-729, R. O. Korea
bDepartment of Theoretical Physics, RSPhysSE, Institute of Advanced Studies,
The Australian National University, Canberra, ACT 0200, Australia
(Dated: February 9, 2012)
We have developed a new method of an earthquake-resistant design to support conventional
aseismic designs using acoustic metamaterials. We suggest a simple and practical method to reduce
the amplitude of a seismic wave exponentially. Our device is an attenuator of a seismic wave.
Constructing a cylindrical shell-type waveguide that creates a stop-band for the seismic wave, we
convert the wave into an evanescent wave for some frequency range without touching the building
we want to protect.
PACS numbers: 81.05.Xj,91.30.Dk,43.20.Mv
Keywords: metamaterial; seismic wave; acoustic properties.
Earthquakes are the result of sudden release of huge
amount of energy in the Earth’s crust that produces
seismic waves.A sudden forthcoming of the seismic
waves with large amplitudes and low frequencies have
been great hazards to life and property. It is the col-
lapse of bridges, dams, power plants, and other struc-
tures that causes extensive damage and loss of life during
earthquakes. Aseismic capabilities are highly relevant to
public safety and a large amount of research has gone
into establishing practical analysis and design methods
for them. Numerous earthquakeproof engineering meth-
ods have been tried to resist earthquakes, but still we are
not so safe.
Seismic waves are a kind of inhomogeneous acous-
tic wave with various wavelengths.
types of seismic waves: body waves and surface waves.
P(Primary) and S(Secondary) waves are the body waves
and R(Rayleigh) and L(Love) waves are the surface
waves. Surface waves travel slower than body waves and
the amplitudes decrease exponentially with the depth. It
travels about 1 ∼ 3km/sec with lots of variety within the
depth of a wavelength [1, 2]. The wavelengths are in the
order of 100m and the frequencies are about 10 ∼ 30Hz,
that is, low end and just below the audible frequency.
However, they decay slower than body waves and are
most destructive because of their low frequency, long du-
ration, and large amplitude.
Rayleigh waves can exist only in an homogeneous
medium with a boundary and have transverse motion.[1,
2] Earthquake motions observed at the ground surface
are mainly due to R waves. On the other hand, L waves
are polarized shear waves guided by an elastic layer. It
is this that causes horizontal shifting of the Earth during
earthquakes. L waves have both longitudinal and trans-
There are two
∗Electronic address: firstname.lastname@example.org
†Electronic address: email@example.com
verse motion and this is what most people feel directly
Recent development of metamaterial science opens a
new direction to control the seismic waves. Farhat et al.
proposed a design of a cloak to control bending waves
propagated in isotropic heterogeneous thin plates [3–5].
Their cloaking of shear elastic waves pass smoothly into
the material rather than reflecting or scattering at the
material’s surface. However, the cloaked seismic waves
are still destructive to the buildings behind the cloaked
In this paper we introduce a method to control seismic
waves by using a new class of materials called metamate-
rials. Metamaterials are artificially engineered materials
which has a special property of negative refractive index.
We present here a solution that a metamaterial acts as
an attenuator by converting the destructive seismic wave
into an evanescent wave by making use of the imaginary
velocity of stop-band of the wave.
There are many representations to express the scale
of earthquakes. Among them the magnitude that comes
from the amplitude of the seismic waves is most impor-
tant. The common form of the magnitude, M, is the
Richter-scale defined by comparing the two amplitudes
in a logarithmic scale as
M = logA
where A is the maximum amplitude of the seismic wave
and Ao is the maximum amplitude of the background
vibration and order of µm. The equipment measures a
transformed magnitude of the intensity. Another factor
of the strength is PGA(Peak Ground Acceleration) but it
is not expressed in a closed form. We focus here on how
to reduce the amplitude of the seismic wave by using the
properties of the metamaterials.
II. NEGATIVE MODULUS
Acoustic waves are created by compressibility or elas-
ticity of the medium.Young’s modulus, Y , is a one-
dimensional compressibility defined by ∆P = Y ∆l/l,
where ∆P is the pressure or stress and l is the length.
Shear modulus, G, is a two-dimensional one for a surface
wave defined by ∆P = G∆x/h, where ∆x is the horizon-
tal shift and h is the height of the object. Bulk modulus,
B, is a three-dimensional one for a body wave defined by
∆P = −B∆V/V .
Seismic medium of Earth crust can be considered as
an accumulation of infinite number of elastic plates. Al-
though the seismic surface wave is not pure two dimen-
sional, the velocity is mainly dependent upon the density,
ρ, and shear modulus, G, of the seismic medium. Seis-
mic wave is a kind of acoustic wave and every acoustic
wave propagates following two wave equations in princi-
ple. Assuming the plane wave time dependence eiωt, the
pressure, p, and the velocity, ? v of the wave in the two
dimensions are expressed as the Newton’s 2nd law
∇sp = iωρ? v,(2)
and the continuity equation
iωp = G∇s·? v,(3)
where ∇s is the Laplacian operator at the surface, p is
the pressure, ω is the anguar frequency of the wave, and
? v is the velocity.
The Eq. (2) and Eq. (3) generates the wave equation
v2p = 0, (4)
where the velocity of the seismic wave is
If the shear modulus becomes negative, the velocity be-
comes imaginary. Then, so does the refractive index, n,
or the inverse of the velocity as n = vo/v = vo
where vo is the background velocity. Since k = 2πn/λ,
the wavevector becomes imaginary, too. Therefore, the
imaginary wavevector makes the amplitude of the seismic
wave become an evanescent wave. We call it a barrier or
attenuator. Note that the impedance Z = ρv =√ρG
becomes imaginary because it is an absorption.
Negative shear modulus of elastic media has been stud-
ied and realized very recently [6–8]. The key was the
Helmholtz resonators similarly with the negative bulk
modulus. The Resonance of accumulated waves in the
Helmholtz resonator reacts against the applied pressure
at some specific frequency ranges.
tive modulus is realized by passing the acoustic wave
through an array of Helmholtz resonators. Therefore,
Then, the nega-
0 0.5 1 1.5
2 2.5 3
FIG. 1: Real and imaginary parts of the effective shear modu-
lus. The negative peak of the imaginary part means an energy
absorption. γ = Γ/ωo.
the sound intensity decays exponentially at some reso-
nant frequency ranges. In general the elastic material
is described by three independent effective parameters of
G, B, and ρ. Therefore, sometimes the G is replaced by a
linear combination of G, B, and the Lam´ e constant [1, 2],
but it will not change the structure of the theory. Acous-
tic waves from the modulus share fundamental properties
of sound waves.
From the formalism of electromagnetic response in
metamaterials, effective electric permittivity and effec-
tive magnetic permeability show negative values at some
specific frequency ranges around resonances .
Helmholtz resonator is a realization of an electrical reso-
nance circuit by mechanical correspondence. It is known
as that the plasmon frequency in metals or in an array
of metal wires produces the electric permittivity as 
ǫ = ǫo
ω(ω + iΓ)
where ωpis the plasma frequency and Γ is a loss by damp-
The Eq. (2) is the counterpart of the Faraday’s law and
Eq. (3) is of the Ampere’s law by the analogy of electro-
magnetism and mechanics. The inverse of the modulus
in mechanical system corresponds to the electric permit-
tivity in electromagnetic system. Considering the struc-
tural loss, the general form of the effective shear modulus,
Geff, is given similarly with the general form of the bulk
modulus as [10–14]
where ωois the resonance frequency and F is a geometric
factor [15, 16]. The real and imaginary part of the Geff
is plotted in Fig. 1 at specific values of F and Γ. The real
part can be negative at resonance and slightly increased
frequency ranges. The negative range of the real part is
the stop-band of the wave. When the imaginary part, the
loss, is small compared with real part, the effective shear
modulus has negative value at 1 < ω/ωo<√1 + F.
We can built an attenuator or an earthquakeproof bar-
rier of a seismic wave by filling-up many resonators un-
der the ground around the building that we want to pro-
tect. Then, the amplitude of the seismic wave that passed
the waveguide is reduced exponentially by the imaginary
wavevector at the frequency ranges of negative modu-
lus. Mixing up many different kinds of resonators will
cover many different corresponding frequency ranges of
the seismic waves.
If we assume that the plain seismic wave of wavelength
λ propagates in x−direction, the amplitude of the wave
reduces exponentially as
Aeikx= Aei2πnx/λ= Ae−2π|n|x/λ. (8)
Let the initial seismic wave, that is, before entering the
waveguide, have amplitude Ai and magnitude Mi, and
final seismic wave, that is after leaving the waveguide,
have amplitude Af and magnitude Mffollowing the Eq.
(1). Then, Af is written as Aifrom Eq. (8) as
Aie−2π|n|x/λ= Af. (9)
The amplitude of the seismic wave reduces exponen-
tially as passing the waveguide of metamaterials. We
can rewrite Eq. (9) with the definition of the magnitude
in Eq. (1) as
Ao10Mie−2π|n|x/λ= Ao10Mf. (10)
Taking logarithms both sides of Eq. (10), we obtain the
width of the waveguide, x → ∆x, as
where ∆M = Mi− Mf.
For example, if the refractive index is n = 2 and the
wavelength of the surface wave is λ = 100m, we need the
waveguide of the width ∆x ≃ 18m to reduce ∆M = 1.
If the aseismic level of the building is M = 5 and the
width of the waveguide surrounding the building is about
60m, then the effective aseismic level of the building is
increased to M = 8. Therefore, a high refractive index
material is desirable for a narrow waveguide. In civil
engineering earthquakeproofing methods must be practi-
cal, that is, clear to manufacture and easy to construct.
The resonator should be easy to build. We designed an
example of a resonator in Fig. 2.
The size of the cylinder can be estimated from the
analogy between electric circuits and mechanical pipes.
A pipe or tube with open ends corresponds to an induc-
tor, and a closed end corresponds to a capacitor [16, 17].
The size of the cylinder is less than the wavelength of the
surface waves. (b) A combined form of the 4 meta-cylinders.
An electrical analogy is shown.
(a)A sample of a meta-cylinder with 4 side holes.
where ρ is the density inside the volume, l′is the effective
length, S is the area of the cross-section, V is the volume,
and v is the velocity inside. From Eq. (12) the resonant
frequency is given as
In the meta-cylinder l′is the effective length which is
given by l′≃ l + 0.85d , where l is the length of the
hole or thickness of the cylinder, and d is the diameter of
An example of the design of the meta-cylinder for the
seismic frequency range is followings: the diameter of the
hole is order of 0.1m, the thickness of the cylinder is or-
der of 0.1m, and the volume inside is order of m3. Since
the meta-cylinder is considerably smaller than the corre-
sponding wavelengthes, the array meta-cylinders behaves
as a homogenized medium.
The shape of the meta-cylinder is neither necessary to
be circular nor to have 4 holes. It could be any form of
a concrete box with several side holes. Cubic or hexago-
nal boxes would be fine. Various kinds of resonators may
cover various kinds of resonance frequencies of the seis-
mic waves. There happens an energy dissipation of the
seismic waves inside of the waveguide and the absorbed
energy will turn into sound and heat. They makes the
temperature of the waveguide increasing depending on
the magnitude of energy that arrives at the waveguide.
A vertical view of the metamaterial barrier with many
meta-cylinders are in Fig. 3. The width of the barrier,
∆x, is predicted in Eq. (11). The depth of the waveguide
should be at least the foundation work of the building to
protect as in Fig. 3, but it is not necessary to be more
than the wavelength of the surface waves. The completed
form of the waveguide is the aseismic cylindrical shell of
many concentric rings in Fig. 4.
Seismic waves cannot pass through water. Then, we
can imagine a water barrier of a big trench filled with wa-
ter. However, it is not stable to stand a series of attacks
of ‘foreshock → main shock → aftershock.’ Because, if
the outer part of the water trench is brought down by a
seismic wave, we do not have enough time to rebuild it
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barrier and the building to protect from the seismic wave.
A vertical landscape of the metamaterial(MTM)
FIG. 4: A sky view of a completed seismic waveguide with
to prepare for the main shock and the after shock. Wa-
ter cannot have a high refractive index n in Eq. (11),
too. Maintaining the depth of the water trench up to the
wavelength of the seismic wave require huge amount of
water and, therefore, not be practical.
We introduced a supportive method for aseismic de-
sign. It is not to add another aseismic system to a build-
ing but to construct an earthquakeproof barrier around
the building to be protected. This barrier is a kind of
waveguide that reduces exponentially the amplitude of
the dangerous seismic waves.
Controlling the width and refractive index of the
waveguide, we can upgrade the aseismic range of the
building as needed in order to defend it, at will, without
touching it. It could be a big advantage of the waveg-
uide method. This method will be effective for isolated
buildings because we need some areas to construct the
aseismic shell. It may be applicable for social overhead
capitals such as power plants, dams, airports, nuclear re-
actors, oil refining complexes, long-span bridges, express
This research was supported by Basic Science Research
Program through the National Research Foundation of
Korea(NRF) funded by the Ministry of Education, Sci-
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