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Errata: Metamaterials and Waves in Composites
1. Section 5.6.1: Ensemble averaging: The definition of ensemble averaged displacements should read
Z
p j (x) hu(x, t)i j =
χ j (x; α) u(x, t; α) d[p(α)] .
A
2. Section 5.6.2: The Willis equations: The convolution operator was not clearly defined in the text. Here is an
example of how the operator should be interpreted when both space and time are involved
Z
G ? (∇ · S − ṁ) =
dx0 [G(x0 − x, t) ? (∇ · S − ṁ)(x0 , t)]
Ω
#
"Z t
Z
dτ G(x0 − x, t − τ) · (∇ · S − ṁ)(x0 , τ) .
dx0
=
Ω
−∞
3. Section 5.7: Forces The expression for the elastic force in equation (5.111) has the wrong sign. The correct
equation is
felastic
= hk [(1 − A3 )1 + (A3 − A1 )D41 + (1 − A2 )D42 ] · u4 .
4
4. Section 5.7: A simpler model: The dispersion relation given in the text is wrong. The correct equation is
"
!#
hk2
9c2 k2 − 3k cos(hk2 ) 2c2 k[1 + 2 cos(hk1 )] + (1 + c2 )2 δhω2 sin2
2
!
!#
"
4 hk2
2 hk2
2 2 2 2 4
2
2 2
2
2 2
2
+ (1 + c ) δ h ω sin
+ cos (hk2 ) c k [1 + 2 cos(hk1 )] + (1 + c ) δhkω [1 + 2 cos(hk1 )] sin
2
2
!
hk2
+ (1 + c2 )2 (δh − 2m)2 ω4 sin4
sin2 (hk2 ) = 0 .
2
When δ = 0, we have
9c k − 6c k cos(hk2 )[1 + 2 cos(hk1 )] + c k cos (hk2 )[1 + 2 cos(hk1 )] + 4(1 + c ) m w sin
2 2
2 2
2 2
2
2
2 2
2
4
4
!
hk2
sin2 (hk2 ) = 0 .
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In the limit h → 0, the above equation is degenerate.
5. Section 5.7: A simpler model: The expression for the elastic force at node 4 is wrong. The correct expression is
h
i
felastic
=hk {1 − e−ihk1 cos(hk2 )}u4x + ie−ihk1 sin(hk2 )u4y e1
4
n
o
h
i
+ hk ie−ihk1 sin(hk2 )u4x + 2 − eihk2 − e−ihk1 cos(hk2 ) u4y e2 .