By Biswajit Banerjee
Requiring no complex wisdom of wave propagation, An creation to Metamaterials and Waves in Composites makes a speciality of theoretical elements of metamaterials, periodic composites, and layered composites. The publication offers newbies a platform from which they could begin exploring the topic in additional detail.
After introducing ideas regarding elasticity, acoustics, and electrodynamics in media, the textual content provides aircraft wave ideas to the equations that describe elastic, acoustic, and electromagnetic waves. It examines the aircraft 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 modifications, features 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 powerful 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 buildings. 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 presents a release pad for study into elastic and acoustic metamaterials. a number of the rules provided haven't begun to be learned experimentally―the booklet encourages readers to discover those principles and convey them to technological maturity.
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Extra resources for An Introduction to Metamaterials and Waves in Composites
10), the balance of momentum for a homogeneous medium becomes ∇ p + (λ + 2µ) ∇ (∇ ∇ · v) − µ ∇ × (∇ ∇ × v) + ρ b(x) = ρ −∇ ∂v ∇v)T . , ∇ × v = 0 and b(x) = 0. Therefore the momentum equation reduces to ∇ p + (λ + 2µ) ∇ (∇ ∇ · v) = ρ −∇ ∂v ∇v)T . , the bulk and shear viscosities are zero, we get ∇p = ρ −∇ ∂v ∇v)T + v · (∇ ∂t . 41) 19 Elastodynamics, Acoustics, and Electrodynamics If we introduce the assumption of small disturbances that we had used in the derivation of the mass balance equation and neglect products of fluctuation terms, we can write ∂v˜ ˜ [ v ·∇ v ] ∇ [ p + p] + [ ρ + ρ] −∇ ˜ = ρ ∂t ∇v) ∇ v )T .
The Biot-Savart law relates the elemental magnetic flux density (dB) produced at a point x due to the current in the element located at point y and can be expressed as dB = k J dl × (x − y) x−y 3 where k is a constant. In SI units k = µ0 /(4π) where µ0 is the magnetic permeability of free space and has units of henry/m (H/m). Linear superposition also applies for many magnetic phenomena. If we assume that the magnetic induction produced by elementary current elements can be superposed, the total magnetic induction produced at a point x by a current density J located in a region Ω is given by µ0 Z x−y µ0 Z B(x) = J(y) × J(y) × ∇x dy .
70) are Hi = µ−1 ij Bj ; The expression for the curl of the curl of a vector in Cartesian coordinates is ∇ × ∇ × v = empq eqi j v j,ip em . 75) in terms of the Hertz superpotentials are erpq eqi j Πej,ip + εµ ∂2 Πer = f,r ; ∂t 2 erpq eqi j Πmj,ip + εµ ∂2 Πm r = f,r . ∂t 2 32 An Introduction to Metamaterials and Waves in Composites To express Maxwell’s equations in terms of spatial curvilinear coordinates we start with the definition of the curl of a vector (v), ∇ × v) · a = ∇ · (v × a) (∇ where a is an arbitrary constant vector.