By Biswajit Banerjee
Requiring no complicated wisdom of wave propagation, An advent to Metamaterials and Waves in Composites makes a speciality of theoretical elements of metamaterials, periodic composites, and layered composites. The publication provides newbies a platform from which they could commence exploring the topic in additional detail.
After introducing recommendations with regards to elasticity, acoustics, and electrodynamics in media, the textual content provides airplane wave recommendations to the equations that describe elastic, acoustic, and electromagnetic waves. It examines the airplane wave enlargement of resources in addition to scattering from curved interfaces, particularly spheres and cylinders. the writer then covers electrodynamic, acoustic, and elastodynamic metamaterials. He additionally describes examples of alterations, elements of acoustic cloaking, and purposes of pentamode fabrics to acoustic cloaking. With a spotlight on periodic composites, the textual content makes use of the Bloch-Floquet theorem to discover the potent habit of composites within the quasistatic restrict, offers the quasistatic equations of elastodynamic and electromagnetic waves, and investigates Brillouin zones and band gaps in periodic constructions. the ultimate bankruptcy discusses wave propagation in easily various layered media, anisotropic density of a periodic layered medium, and quasistatic homogenization of laminates.
This ebook offers a release pad for study into elastic and acoustic metamaterials. a few of the rules offered have not begun to be discovered experimentally―the publication encourages readers to discover those rules and produce them to technological maturity.
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Extra resources for An Introduction to Metamaterials and Waves in Composites
85) K¯ E (x,t − t ) · E(x,t ) dt where causality requires that K¯ B (x, τ) = K¯ E (x, τ) = 0 if τ = t − t < 0. 5 Maxwell’s equations at fixed frequency To obtain Maxwell’s equations at fixed frequency (ω) we Fourier transform the equations. This is equivalent to assuming that all fields depend harmonically on time. 84), we have Z ∞ −∞ iωt ∇ ×E e Z ∞ ∂B iωt dt = − e dt −∞ ∂t or ∇× Z ∞ −∞ Eeiωt dt = − Beiωt ∞ + iω −∞ Z ∞ −∞ Beiωt dt . If we assume that the magnetic induction B is zero at ±∞ we get, using the usual hat notation for the Fourier transformed quantities, ∇ × E(x, ω) = iωB(x, ω) .
In the two-dimensional case p, ρ0 and κ depend only on x2 and x3 and the acoustic wave equation can be written as ∂ ∂x2 1 ∂p ρ0 ∂x2 + ∂ ∂x3 1 ∂p ρ0 ∂x3 + ω2 p=0. κ That means that the three-dimensional gradient operator ∇ can be replaced with the two-dimensional gradient operator ∇ , and we get ∇· 1 ω2 1 ∇p + p=0; u= 2 ∇p . 53) This equation is similar to the elastodynamic wave equation for antiplane shear with the substitutions 1 1 p ↔ u1 ; ↔µ; ↔ρ. ρ κ Therefore, the solutions of the two problems will lead to similar conclusions if we make the appropriate interpretations of the quantities involved.
These equations consist of simplified forms of the conservation of mass and the Navier-Stokes equations for the conservation of linear momentum in fluids. In this section we consider only the linear equations (also called the “small signal” equations) for acoustic waves. The equations of elastodynamics implicitly assume that mass is conserved within the body Ω and there is no mass flow in and out of the body. Such a description of the dynamics of a continuum is called Lagrangian. However, a Lagrangian description usually proves inconvenient for acoustics and a spatial or Eulerian description is preferred.