By Biswajit Banerjee
Requiring no complex wisdom of wave propagation, An creation to Metamaterials and Waves in Composites specializes in theoretical features of metamaterials, periodic composites, and layered composites. The ebook provides newcomers a platform from which they could begin exploring the topic in additional detail.
After introducing suggestions concerning elasticity, acoustics, and electrodynamics in media, the textual content offers airplane wave suggestions to the equations that describe elastic, acoustic, and electromagnetic waves. It examines the airplane wave growth of resources in addition to scattering from curved interfaces, in particular spheres and cylinders. the writer then covers electrodynamic, acoustic, and elastodynamic metamaterials. He additionally describes examples of variations, facets 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 powerful 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 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 e-book offers a release pad for examine into elastic and acoustic metamaterials. the various principles provided have not begun to be learned experimentally―the ebook encourages readers to discover those rules and convey them to technological maturity.
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Extra resources for An Introduction to Metamaterials and Waves in Composites
44) This is the acoustic wave equation expressed in terms of a scalar potential. 2 Cartesian and curvilinear coordinates In terms of components with respect to a rectangular Cartesian basis, the equations of linear acoustics are p = −κ ∂ui ∂p ∂vi ∂vi ∂p ; + ρ0 c20 = 0 ; ρ0 + =0. ∂xi ∂t ∂xi ∂t ∂xi 20 An Introduction to Metamaterials and Waves in Composites The acoustic wave equations in terms of the velocity and the pressure are 2 2 ∂2 vi ∂2 p 2 ∂ vj 2 ∂ p − c − c = 0 ; =0. 0 0 ∂t 2 ∂xi ∂x j ∂t 2 ∂x j ∂x j The acoustic wave equation in terms of the scalar potential φ is 1 ∂2 φ ∂2 φ = 2 2 .
We can show that 1 gi × g j = √ eki j gk g where g = det gi j and ei jk = ei jk . Therefore, we get the expression for the vector product of two vectors in curvilinear coordinates, 1 b := v × a = √ eki j vi a j gk = bk gk g ⇔ 1 bk = √ eki j vi a j . g Now, the divergence of the vector (v × a) can be expressed in curvilinear coordinates as 1 ∂ 1 ∂ √ ∇ · (v × a) = ∇ · b = √ bk g = √ eki j vi a j . k g ∂θ g ∂θk Hence we have, using the definition of the curl of a vector, eki j ∂vi ∇ × v) · a = ([∇ ∇ × v] p g p ) · (aq gq ) = √ (∇ aj g ∂θk or, invoking the arbitrariness of a, eki j ∂vi ∇ × v] j = √ [∇ g ∂θk ⇔ eki j ∇ × v] = g j [∇ ∇ × v] j = √ g [∇ g j ∂vi .
55), the average acoustic power density in a cycle of oscillation will be ω Z π/ω ∇ p · v + p∇ ∇ · v) dt (∇ Pden = 2π −π/ω = ω κi ( p2 + p2r ) − ρi (vr · vr + vi · vi ) . 2 κ2i + κ2r i This quadratic form will be non-negative for all choices of p and v if and only if κi = Im(κ) > 0 and ρi = Im(ρ) < 0 for all real ω > 0. Note that the quadratic form does not contain the real part of ρ. Since the work done in a cycle should be zero in the absence of dissipation, this implies that the imaginary parts of the bulk modulus and the density are connected to energy dissipation.