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arXiv:1710.11273v1 [physics.app-ph] 30 Oct 2017 otpol eldrcl uigerhuks Surface . 1 during as slowly directly travel what feel have is this waves people horizontal and L most motion causes transverse earthquakes. that and during fact longitudinal both Earth this is the It of guided shifting layer. waves shear elastic polarized an are Earth- waves by L mainly waves. are R surface motion[1–3]. to ground due the transverse at observed have motions quake and boundary a body of that the than distance of 3]. the waves[2, but with the depth slowly The the is more with decays epicenter exponentially generation. the decrease and the waves surface earth of ar- the waves point of body main surface the the when waves. at generated surface rive is waves, are waves body surface Love(L) The two are and are waves Rayleigh(R) there waves[1].and Secondary(S) and surface and wave and waves seis- acoustic body A Primary(P) of waves: Earth. seismic kind the of a of types mantle is at the wave located earth- of mic are parts the hypocenters upper and very Most where released the point first begins. the is rupture is energy quake focus strain or stored seis- hypocenter the produces The that crust waves. 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FIG. 1: Concept of the negative density and its model of the negative ball. The white arrows are the direction of the acceleration to the from the left side. FIG. 2: Model of meta-box which has negative modulus and its electrical analogy[6]. The empty spaces among the cylin- modulus E could be any type: Young’s modulus at 1D, ders act as a large capacitor. Shear modulus at 2D, and at 3D. If one of the acoustical constituent parameter is neg- ative effectively at some specific frequency ranges, then buildings. The negative belt is to build the damper re- the velocity in Eq. (1) becomes imaginary in the fre- gion underground between epicenters and residential ar- quency range and it makes the wave decay exponentially. eas. There are enough data of epicenters for one hundred Therefore, negative density or negative modulus creates years, and new earthquakes take place around previous the stop-band of the wave and the key of the seismic neg- epicenter region again like volcanoes. The underground ative belt that dissipates seismic energy. Note that the negative belt composed of effective negative struc- acoustical impedance Z = ρv = √ρE becomes imaginary tures dissipates seismic energy and weakens the earth- because it is an energy absorption. If the two constituent quakes. parameters are negative simultaneously, it creates a back- An effective negative modulus has been realized by ward wave without the dissipation. mechanical [13, 14]. The of accu- The effective negative density has been realized mulated waves in the reacts against the ap- already[9–12]. The concept of the negative density or plied at some specific frequency ranges. Then, negative inertia is pretty simple. From the Newton’s law, the negative modulus is realized by passing the acoustic the mass m and moment of inertia I are related with force wave through an array of Helmholtz . In this and torque as way the acoustic intensity decays just above the resonant F τ frequencies[6, 7]. Multi resonance frequencies creates m = and I = , (2) multi stop-band of the wave. The Helmholtz resonator − a − α − − is a realization of an electrical resonance circuit by me- where a and α are the linear and angular acceleration. If chanical correspondence. A pipe or tube with open both the mass responds against to the applied force or torque, ends corresponds to an inductor. On the other hand, one it acts as a negative mass or negative moment of inertia open and one closed end corresponds to a capacitor. The effectively. The negative density or inertia creates an resonance frequency of a Helmholtz resonator is[6, 7] anti-phase motion and decays of the wave. An effective negative density or negative mass is a mul- S ω = c . (3) tiple mass or mass-in-mass system connected by springs o l′V as in Fig. 1. If a force or torque is applied to the outer r bone structure, then the inertia of the ball inside reacts S is the area of the entrance, V is the volume, and c is against the force or torque. They oscillate against exter- the background velocity. l′ is the effective length which nal force creating an anti-phase motion, and then cancel is given by l′ l +0.85d, where l is the length of the hole out the seismic vibration. A simple harmonic motion of or thickness of≃ the resonator and d is the diameter of the 2 a spring-mass system which acts as a = ωox is the sim- hole[15, 16]. plest case of the negative density. A membrane− structure We designed a meta-box based on the Helmholtz res- behaves an effective negative density, too. The mass of onator Fig. 2 to match the seismic surface frequencies[6]. the central ball in Fig. 1 is 102 103 kg, and the spring The size can be estimated by analogy between electric constants are 105 107 N/m. The∼ oscillation frequency circuits and mechanical pipes. The diameter of the hole of the structure matches∼ with the seismic surface waves: and the thickness of the cylinder is order of foot, and the 3 ωo = keff /m, where keff is the effective spring con- volume inside is order of 10 100m . Because the wave- stant. The system should have different oscillation fre- length of the is∼ much longer than the size of quenciesp because the surface waves are inhomogeneous. the holes, the incoming wave will be diffracted strongly The application of the negative density is found ev- into the meta-boxes. Note that the more the number of erywhere. It has been used as a damper for very high holes, the higher the resonant frequencies. The shape of 3

dex is 2i for the negative belt. The impedance of the belt is imaginary or absorption of energy. It roles as an “attenuator” to the seismic waves. The left border of the belt in Fig. 3 is breaking by the impedance difference, but could be fixed after the earthquakes. Besides the large scale of the protection, there is more advantage of the negative belt. It is the construction cost at large magnitude of earthquakes. The common form of the seismic magnitude, M, in Richter-scale defined by comparing the two amplitudes as A M = log , (4) Ao where A is the maximum amplitude of the seismic wave FIG. 3: Pressure distribution by a negative belt. Acoustic which is rescaled at 100km from the epicenter. Ao is the wave comes from the left side. Freq.= 10Hz. The units are background amplitude to be set Ao = 1µm. In princi- m. ple the conventional earthquake engineering is to reduce the amplitude of buildings be seismic wave. Then, the cost increase exponentially as the magnitude M is in- the meta-box is neither necessary to be circular nor to creasing. Magnitude 3 or lower earthquakes are almost have 6 holes. It could be any form of a concrete box with imperceptible and rarely cause damage. On the other a few side holes such as cubic or hexagonal shapes. hand, magnitude 6 or higher ones are very destructive, Resonances are common in everyday life. Laser has the and it is expensive to build. resonance of atoms, microwave oven has the resonance of Let the initial seismic wave, that is, before entering molecules and the wind instrument has the resonance of the waveguide, have amplitude Ai and magnitude Mi, artificial atoms. The negative modulus method is some- and final seismic wave, that is after leaving the waveg- thing like to build hundred thousands of huge concrete- uide, have amplitude Af and magnitude Mf following made saxophones underground and play them with seis- the Eq. (4). If we assume that the plain seismic wave of mic energy. The energy turns into sound and heat. The wavelength λ propagates in x direction, the amplitude negative modulus method has been used for long time in of the wave reduces exponentially− as civil engineering. There has occurred a large earthquake −2π|n|x/λ in Mexico in 1985, and it was found that some build- Af = Aie , (5) ings of specific size destroyed completely compared with other buildings were relatively safe. It was the seismic where and λ is the wavelength of the seismic surface energy dissipation by resonances. It is common sense for wave, and n is the relative of the neg- civil engineers to avoid resonances from external ative belt. The amplitude of the seismic wave reduces or torques including seismic waves. The negative belt of exponentially as passing the waveguide of metamateri- the effective negative modulus is the reverse concept of als. We can rewrite Eq. (5) with the definition of the it. magnitude in Eq. (4) as We build a negative belt or an attenuator of the seis- Mf Mi −2π|n|x/λ mic surface wave by filling up many resonators or “meta- Ao10 = Ao10 e , (6) boxes” underground. The amplitude of the seismic wave where W is the width of the belt or x. Taking logarithms that passes through the negative belt is reduced as in both sides of Eq. (6), we obtain the width as Fig. 3. Epicenters are located in the left side of the belt and the other side is the seismic shadow zone. The en- 0.366λ W = ∆M, (7) ergy of the seismic waves is dissipated inside of the meta- n boxes, and the absorbed energy is converted into sound | | and heat. The use of a variety of resonators will cover a where ∆M = Mi Mf . For example, if n = 1.5 and wide range of seismic wave frequencies. λ = 100m, then W−= 25m for ∆M = 1. Therefore, the A numerical simulation of the negative belt is shown cost of the negative belt is linearly increasing as the mag- in Fig. 3. We applied some typical material parame- nitude is increasing. ∆M = 4 can be engineered by at ters. Density and modulus are ρ = 1.0 103kg/m3, most W = 100m in Eq. 7. If seismic wave of magnitude 8 E = 1.0GP a for the background. It makes× the in- comes in the belt, then it leaves the belt with magnitude coming seismic wave velocity of 1km/sec. The back- 4. The magnitude 4 is not serious and can be defend- ground could be rocks, water, or mixed. We applied able by conventional methods of earthquake engineering. ρ = 2.0 103kg/m3 and E = 0.50GP a for the neg- The depth of the belt should be at least the same as the ative− belt,× and E = 0.50GP a with− the same density for foundation of the building, but it is not necessary to be the normal barrier. Note that the relative refractive in- more than the wavelength of the seismic surface waves. 4

ing technology produces an artificial seismic shadow zone that reduces the seismic wave using effective negative density or modulus. Some engineering models of negative density and negative modulus were shown. The negative belt is constricted underground anyplace between possi- ble epicenters and the areas to be protected. It changes the seismic wave-vector to an imaginary one and reduces the amplitude of the seismic wave exponentially. A numerical simulation of the seismic shadow zone was conducted and a possible application to protect San Fran- cisco in U. S. A. was suggested. The seismic range of the structure can be upgraded or downgraded by adjusting the width of the belt. The biggest advantage is that all structures behind the barrier be protected, which de- FIG. 4: Possible position of the negative belt to protect San creases the cost of seismic protection. We hope to test Francisco, U. S. A. The epicenters are located in the Ring of this method with engineers using small scale test models Fire. soon. Tsunamis are commonly accompanied by earthquakes Note that a high refractive index (soft but heavy) mate- and causes damage far greater than earthquakes. There- rial is desirable for the narrow seismic barrier. Composite fore, we can not separate tsunami from earthquakes. Re- materials could be good candidate for this belt. We sug- cently, we have published a totally different tsunami wall gested a negative belt to protect residential area in San method by meta-lens in the sea[17]. This is a removable Francisco from earthquakes in Fig. 4. The epicenters are tsunami wall that is based on an optical Eaton lens. The a part of the Ring of Fire and located in the left side of principle is to transform the incoming wave into a rotat- the belt. ing wave and the impedance matching is used to reduce A negative belt method of acoustic metamaterials for the pressure of the tsunami without reflecting the wave. the attenuation of seismic waves are introduced. This It is just a beginning and will be a next hot subject in method which is different from the conventional cloak- acoustic metamaterials beyond earthquake.

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