Problem decomposition for symmetric and antisymmetric waves 

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Love waves in a vertically periodic coated substrate


Acoustic wave spectra in periodic laminates with structural defects has been studied from different perspectives; see [2]–[10]. One of the topical problems studies the effect of coat-ing on SH waves in a periodically layered substrate. It is well known that if a semi-infinite substrate is homogeneous then its coating by ’slow’ layer is a means to achieve the SH wave localisation (decay into the depth). These are the classical Love waves whose phase-velocity branches v (kx) = ω/kx must be subsonic with respect to the substrate, i.e. be confined between the shear speeds of the coating layer and the substrate cl < cs (here kx is the wavenumber along the surface, and cl means min cl (y) if the coating is transversely inhomogeneous); see [46]. The situation is different if the substrate is periodic. For any kx (possibly kx = 0), taking ω in the Floquet stopbands ensures by itself decaying waves without setting upper limit on their velocity v = ω/kx [2, 3, 47, 48]. At the same time, it is also possible that SH surface waves in a free (uncoated) periodic substrate are restricted to a fairly narrow velocity range, as detailed in [49]. For instance, this may be often the case in substrates with a bilayered or monotonic profile on a period; moreover, there is no SH surface waves at all if the profile is even. In such cases, coating the substrate creates or drastically enriches the spectrum of SH localised waves and thus plays a similar role as in the classical setting with a homogeneous substrate, though in a more complicated context (e.g. the coating does not have to be ’slow’). In view of different terminology used in the literature, let us agree to refer to the SH localised waves in the uncoated and coated periodic substrate as the surface waves and the Love waves, respectively. The former are related to a strictly periodic halfspace, the latter imply perturbed periodicity. Similar in that they both lie in the Floquet stopbands of a substrate and decay into the depth, these are actually two different types of wave solutions. One essential difference is that any periodic substrate admits no more than a single SH surface wave per a stopband, while there is no such limitation for Love waves. This feature of possible wave mutliplexing is another interesting functional capacity of coating.
The present Chapter studies the dispersion spectrum of Love waves in a coated sub-strate with arbitrary periodic piecewise constant or functionally graded (see [49, 50]) variation of material properties in the depth direction. The motivation is to gain better understanding into the spectral effect of coating and to obtain some general analytical re-sults. The structure and main results of the Chapter are as follows. The background of the problem is outlined in §2.2. Here we state the SH wave equation, introduce the notion of propagator for 1D periodic structure (also referred to as monodromy matrix) and explain how it defines spectrum of the propagating waves. Unlike commonly used formulation via the transfer matrix, the dispersion equation is set as equality of scalar SH impedances of the coating layer and the substrate. The advantage of using impedances lies in their helpful analytical properties such as monotonicity in ω and kx which is demonstrated in §2.3 and exploited in §2.4.1. It enables insightful graphical visualisation of roots of the dispersion equation that clarifies the nature and connectivity of the broken branches of Love waves (§, and it also underpins the relation of the number of Love waves in a stopband to the number of resonances of the coating layer (§ The monotonicity of Love dispersion branches ω (kx) is proved in § and their lower bound is discussed in § The criterion for existence of the fundamental Love branch with an origin at zero ω and kx is derived in §2.4.2. General findings are illustrated in §2.5 by examples of Love waves which arise due to coating of periodically layered and functionally graded substrates. Concluding remarks are presented in §2.6.


Propagator (monodromy matrix)

Consider the SH wave uz (x, y, t) = u (y) ei(kxx−ωt) in an isotropic medium with piecewise continuous density ρ (y) and shear modulus µ (y) . The wave equation  ′ 2 2 u (2.1) (µu ) − µkxu = −ρω may be recast as the system Qη = η′ with Q (y) = i 0 −µ−1 , η (y) = u , (2.2) µkx2 − ρω2 0 if where f = µu′ is the traction and ′ ≡ d/dy. For any initial condition η (y0) , the solution of
the first-order differential system (2.2) is M1 M2 y η (y) = M (y, y0) η (y0) with M (y, y0) ≡ M3 M4 = y0 (I + Qdy) , (2.3)
where M is the matricant (propagator or transfer matrix) and is the multiplicative integral which expands in the Peano series and reduces to a product 1 j=n exp Qj (Δy)j if Q (y) takes piecewise constant values Qj on [y0, y]; see [51]. In view of the explicit form of Q in (2.2), it follows that det M = 1 and that ImM1,4 = 0, ReM2,3 = 0.
In the case of periodic properties ρ (y) = ρ (y + T ) and µ (y) = µ (y + T ), the central role is played by the matricant over the period, or the monodromy matrix, M (y + T, y) [= M (y, 0) M (T, 0) M−1 (y, 0)] with eigenvectors W1,2 = W1,2 (y) and with eigenvalues q1,2 (q1 = q2−1) independent of y.
The monodromy matrix gives the projection of the spectrum of the propagating waves on the plane {ω, kx} in the following way. The so-called Lyapunov function Δ (ω2, kx2) = 12 traceM (T, 0) partitions the {ω, kx}-plane into alternating passband and stopband areas where, respectively, |Δ (ω2, kx2)| < 1 ⇒ complex q1 = q2 with |q1,2| = 1, W1+TW2 = 0;
|Δ (ω2, kx2)| > 1 ⇒ real q1,2 ≷ 1, W1+TW1 = W2+TW2 = 0
(+ means Hermitian conjugation and T is a matrix with zero diagonal and unit off-diagonal components). Equation |Δ (ω2, kx2)| = 1 defines the set of curves {ωe (kx)} of band edges where a degenerate eigenvalue qd = ±1 corresponds to a single eigenvector Wd so that W1, W2 → Wd at |Δ| → 1 (barring the exceptional case of a zero-width stopband and also the point ω = 0, kx = 0 where (2.1) has a trivial solution u (y) = const). Note that |Δ (0, kx2)| > 1 at kx = 0. Indeed the first stopband area [0, ωe(1) (kx)) for the case of periodic media is what is the subsonic interval for homogeneous media.
In the following, we refer to eqs. (2.1)-(2.3) with ρl, µl for a coating layer y ∈ [−d, 0] and
with ρs, µs for a T -periodic substrate y 0. Denote
2 2 1 0 • dy,
cl (y) = µl/ρl, c¯l = µl / ρl with • ≡
d −d (2.5)
2 2 1 T
cs (y) = µs/ρs, c¯s = µs / ρs with • ≡ • dy,
where c (y) implies ’local’ speed of short waves and c¯ implies effective speed of long waves in the direction x. Note that we will use the same symbol • for an average across the layer and across the substrate period (the label l or s of the enclosed quantities suffices to clarify which is the case).

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Definition of conditional and modal SH impedances

Assume that a layer y ∈ [−d, 0] with ρl (y) and µl (y) is free of traction at the upper face and, under this condition, introduce via (2.3) the impedance zl at the lower face: f (−d) = 0, f (0) = −zlu (0) ⇒ zl = iM3 (0, −d) M1−1 (0, −d) . (2.6)
The ’conditional’ impedance zl = zl (ω2, kx2) is a real function, whose zeros and poles define the spectra ωF/F (kx) and ωF/C (kx) of SH waves in the given layer with free/free or free/clamped faces (f (−d) = 0 and f (0) = 0 or u (0) = 0), respectively. The least value ωF(1)/F (kx) among ωF/F (kx) , i.e. the first zero of zl, satisfies kx miny∈[−d,0] cl (y) ωF(1)/F (kx) kxc¯l , (2.7) while the first pole of zl is ωF(1)/C (kx) ∼ O (1) at any kx. The definition (2.6)2 obviously reduces to zl = µlky tan ky d with ky = ω2/cl2 − kx2 (2.8) if the layer is homogeneous (has constant ρl, µl). Next, consider a periodic half-space y 0. With reference to (2.3), define the Floquet modes η1,2 (y) via the eigenvectors W1,2 of, specifically, M (T, 0) :
ηα (y) = M (y, 0) Wα, where M (T, 0) Wα = qαWα; α = 1, 2. (2.9)
By this definition, ηα (nT ) = qαnWα for n = 0, 1, … Using the notations ηα = (uα ifα)T and
Wα = (aα bα)T , Wd = (ad bd)T (T means transposition), introduce the modal impedances
Zsα at the substrate surface as follows
fα (0) = −Zsαuα (0) ⇔ Zsα = ibαa−α1, α = 1, 2;
Zs1 = Zs2 = ibda−d1 at |Δ| = 1 if ad = 0 ,
ibα iM3 (T, 0) M1 (T, 0) − qα √
= = , q = Δ Δ2 − 1. (2.11)
qα − M4 (T, 0) iM2 (T, 0)
aα α ±
Our case of interest is when ω, kx lie in the stopbands. The impedances Zsα (ω2, kx2) at |Δ| 1 (hence Wα+TWα = 0, see (2.4)2) are real functions, with zeros and poles on the sets of curves {ωN (kx)} and {ωD (kx)} which yield bα = 0 and aα = 0 and thus fulfil the Neumann f (0) = 0 (⇔ u′ (0) = 0) and Dirichlet u (0) = 0 conditions, respectively. It is known that each closed stopband contains one Neumann branch ωN (kx) and one Dirichlet branch ωD (kx) (missing in the first stopband only), and that these two branches coincide with the opposite edges of each stopband if ρ (y) and µ (y) are even about the midpoint of a period [0, T ]. Similarly to (2.7), the first Neumann branch ωN(1) (kx) lying in the first stopband ω ∈ [0, ωe(1) (kx)) satisfies kx miny∈[0,T ] cs (y) ωN(1) (kx) ωe(1) (kx) kxc¯s, (2.12)
while the second stopband and hence the first Dirichlet branch occur at finite frequency whatever small kx is. For more details, see [52].

Statement of the Love wave problem

Consider Love waves in a periodic substrate y 0 coated by a layer y ∈ [−d, 0]. The Love-wave spectrum {ωL (kx)} follows from the SH wave equation under the traction-free condition f (−d) = 0 at the upper surface, continuity of displacement u (y) and traction f (y) at the layer/substrate interface, and the requirement that the solution must vanish at y → ∞. The latter implies that the branches ωL (kx) are confined to the stopbands and that the Love waves must incorporate the decreasing Floquet mode in the substrate (|qα| < 1). In the following, we choose to label the two Floquet modes so that α = 1 and α = 2
always correspond to, respectively, decreasing and increasing ones, i.e. |q1| < 1 < |q2| in any stopband. In this regard, we will refer to the impedances Zs1 and Zs2 in the stopbands as physical and non-physical one.

Table of contents :

1 Introduction 
2 Love waves in 1D PC 
2.1 Introduction
2.2 Background
2.2.1 Propagator (monodromy matrix)
2.2.2 Definition of conditional and modal SH impedances
2.2.3 Statement of the Love wave problem
2.3 Properties of the impedances
2.3.1 Layer impedance
2.3.2 Substrate impedances
2.4 Love wave spectrum
2.4.1 Overview of the spectrum
2.4.2 Number of Love waves per a stopband
2.4.3 Monotonicity of the Love wave branches
2.4.4 Lower bound of the Love wave spectrum
2.4.5 Fundamental branch
2.5 Numerical examples
2.6 Conclusion
3 Love waves in 2D PC 
3.1 Introduction
3.2 Problem statement
3.3 The resolvent
3.4 The spectral projector
3.4.1 The eigenspaces
3.4.2 The spectral projector definition via propagator eigenvalues
3.4.3 The alternative definition of the spectral projector via path integral
3.5 The dispersion equation
3.5.1 Love wave spectrum
3.5.2 Projection of the spectrum of the propagating waves
3.6 Numerical examples
3.7 Conclusion
4 Plate waves in 2D PC
4.1 Introduction
4.2 Problem statement
4.3 The dispersion equation in terms of the propagator
4.4 The dispersion equation in terms of the resolvent
4.5 The dispersion spectrum for mirror symmetric profiles
4.5.1 Problem decomposition for symmetric and antisymmetric waves
4.5.2 Dispersion equations in stable terms
4.6 Numerical examples
4.6.1 One array of inclusions
4.6.2 Two arrays of inclusions
4.7 The displacement and traction field
4.7.1 The wave field equation
4.7.2 Numerical examples
4.8 Conclusion
5 Guided waves in 2D PC 
5.1 Introduction
5.2 Problem statement
5.3 Dispersion equation with resolvent and projector
5.3.1 The localized waves spectrum
5.3.2 Projection of the spectrum of the propagating waves
5.4 The dispersion equation for the mirror-symmetric case
5.4.1 Problem decomposition for symmetric and antisymmetric waves
5.4.2 Dispersion equations in stable terms
5.5 Numerical examples
5.5.1 Square inclusions and 2D substrates
5.5.2 Circular inclusions and layered substrates
5.6 The displacement and traction field
5.6.1 The propagators for decreasing modes
5.6.2 The wave field equation
5.6.3 Numerical examples
5.7 Conclusion
6 Conclusions


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