# Seismic Waveguide of Metamaterials

**ABSTRACT** 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.

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**ABSTRACT:**In this work, we develop a general mathematical framework on regularized approximate cloaking of elastic waves governed by the Lam\'e system via the approach of transformation elastodynamics. Our study is rather comprehensive. We first provide a rigorous justification of the transformation elastodynamics. Based on the blow-up-a-point construction, elastic material tensors for a perfect cloak are derived and shown to possess singularities. In order to avoid the singular structure, we propose to regularize the blow-up-a-point construction to be the blow-up-a-small-region construction. However, it is shown that without incorporating a suitable lossy layer, the regularized construction would fail due to resonant inclusions. In order to defeat the failure of the lossless construction, a properly designed lossy layer is introduced into the regularized cloaking construction . We derive sharp asymptotic estimates in assessing the cloaking performance. The proposed cloaking scheme is capable of nearly cloaking an arbitrary content with a high accuracy.10/2014;

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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.

I.INTRODUCTION

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: shkim@mmu.ac.kr

†Electronic address: mpd105@rsphysse.anu.au

verse motion and this is what most people feel directly

during earthquakes.

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

region.

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

Ao, (1)

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.

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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

as

∇2

sp +ω2

v2p = 0,(4)

where the velocity of the seismic wave is

v =

?

G

ρ.

(5)

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,

?ρ/G,

Then, the nega-

-8

-6

-4

-2

0

2

4

6

0 0.5 1 1.5

ω/ωο

2 2.5 3

Geff/G

F=1

γ=0.1

real part

imag. part

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 [9].

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 [15]

The

ǫ = ǫo

?

1 −

ω2

p

ω(ω + iΓ)

?

, (6)

where ωpis the plasma frequency and Γ is a loss by damp-

ing.

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]

1

Geff

=1

G

?

1 −

Fω2

o

ω2− ω2

o+ iΓω

?

, (7)

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

Page 3

3

loss, is small compared with real part, the effective shear

modulus has negative value at 1 < ω/ωo<√1 + F.

III.SEISMIC ATTENUATOR

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

∆x =ln10

2π|n|

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].

λ∆M

=0.366λ

|n|

∆M,(11)

L =ρl′

S,

C =

V

ρv2, (12)

FIG. 2:

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

ωo≃

1

√LC

=

?

S

l′Vv.

(13)

In the meta-cylinder l′is the effective length which is

given by l′≃ l + 0.85d [17], where l is the length of the

hole or thickness of the cylinder, and d is the diameter of

the hole.

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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4

FIG. 3:

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

many meta-cylinders.

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.

IV. SUMMARY

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

rail-roads, etc.

Acknowledgments

This research was supported by Basic Science Research

Program through the National Research Foundation of

Korea(NRF) funded by the Ministry of Education, Sci-

ence and Technology(2011-0009119).

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