By Biswajit Banerjee
Requiring no complex wisdom of wave propagation, An advent to Metamaterials and Waves in Composites makes a speciality of theoretical facets of metamaterials, periodic composites, and layered composites. The booklet provides newcomers a platform from which they could commence exploring the topic in additional detail.
After introducing innovations relating to elasticity, acoustics, and electrodynamics in media, the textual content provides aircraft wave strategies to the equations that describe elastic, acoustic, and electromagnetic waves. It examines the aircraft wave growth 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 functions 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, provides 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 e-book offers a release pad for learn into elastic and acoustic metamaterials. some of the rules offered have not begun to be discovered experimentally―the ebook encourages readers to discover those principles and convey them to technological maturity.
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Additional resources for An Introduction to Metamaterials and Waves in Composites
The media are at rest. Define Z t D(x,t) := D(x,t) + −∞ J f (x, τ) dτ . 81) and using Maxwell’s equation connecting the magnetic field to the total current, we have ∂D ∂D = +Jf = ∇ ×H . 67) and Maxwell’s equation relating the electric displacement to the free charge density, we get ∇ ·D = ∇ ·D+ Z t −∞ ∇ · J f dτ = ∇ · D − Z t ∂ρ f −∞ ∂t dτ = ∇ · D − ρ f = 0 . 83) Therefore, we can write Maxwell’s equations in terms of D as ∇ ×E = − ∂B ; ∇ ·B = 0 ; ∂t ∇ ×H = ∂D ; ∇ ·D = 0 . 84) This reduction reflects the fact that it is difficult to distinguish the free current density J f from currents arising from the electric displacement field through ∂D/∂t.
We can express the acoustic equation for that situation in terms of components with respect to a rectangular Cartesian basis (e1 , e2 , e3 ) as (after dropping the hats for convenience) ∂ ∂xi 1 ∂p ρ0 ∂xi + ω2 p=0. κ 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 .
The above relation for the permittivity tensor can be obtained as follows. Recall that Z t D(x,t) = D(x,t) + J f (x, τ) dτ . −∞ Differentiation with respect to time gives ∂D ∂D = + J f (x,t) . ∂t ∂t 37 Elastodynamics, Acoustics, and Electrodynamics Assuming harmonic solutions of the form D(x,t) = D(x) exp(−iωt) ; D(x,t) = D(x) exp(−iωt) J f (x,t) = J f (x) exp(−iωt) ; E(x,t) = E(x) exp(−iωt) and plugging into the differential equation, we get (−iω)D(x) = (−iω)D(x) + J f (x) . If we assume that the medium obeys Ohm’s law and has a “reference” dielectric constant ε0 , the free current density J f and the electric displacement D are related to the electric field E by J f = σ · E ; D = ε0 · E .
An Introduction to Metamaterials and Waves in Composites by Biswajit Banerjee