Hybridized Love Waves in a Guiding Layer Supporting an Array of Plates with Decorative Endings

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Hybridized Love Waves in a Guiding Layer Supporting

an Array of Plates with Decorative Endings

Kim Pham, Agnès Maurel, Simon Félix, Sebastien Guenneau

To cite this version:

Kim Pham, Agnès Maurel, Simon Félix, Sebastien Guenneau. Hybridized Love Waves in a Guiding

Layer Supporting an Array of Plates with Decorative Endings. Materials, MDPI, 2020, 13 (7), pp.1632.

�10.3390/ma13071632�. �hal-03080207�

Article

Hybridized Love Waves in a Guiding Layer

Supporting an Array of Plates with

Decorative Endings

Kim Pham

1,

*, Agnès Maurel

2

, Simon Félix

3

and Sébastien Guenneau

4

1

_{IMSIA, ENSTA ParisTech, 828 Bd des Maréchaux, 91732 Palaiseau, France}

2

_{Institut Langevin, CNRS, ESPCI ParisTech, 1 rue Jussieu, 75005 Paris, France; agnes.maurel@espci.fr}

3

_{LAUM, CNRS UMR 6613, Le Mans Université, avenue Olivier Messiaen, 72085 Le Mans, France;}

simon.felix@univ-lemans.fr

4

_{Aix Marseille Univ., CNRS, Centrale Marseille, Institut Fresnel, 13013 Marseille, France;}

sebastien.guenneau@fresnel.fr

*

Correspondence: kim.pham@ensta-paris.fr

Received: 11 November 2019; Accepted: 27 December 2019; Published: 1 April 2020






Abstract:

This study follows from Maurel et al., Phys. Rev. B 98, 134311 (2018), where we reported

on direct numerical observations of out-of-plane shear surface waves propagating along an array

of plates atop a guiding layer, as a model for a forest of trees. We derived closed form dispersion

relations using the homogenization procedure and investigated the effect of heterogeneities at the

top of the plates (the foliage of trees). Here, we extend the study to the derivation of a homogenized

model accounting for heterogeneities at both endings of the plates. The derivation is presented in

the time domain, which allows for an energetic analysis of the effective problem. The effect of these

heterogeneous endings on the properties of the surface waves is inspected for hard heterogeneities.

It is shown that top heterogeneities affect the resonances of the plates, hence modifying the cut-off

frequencies of a wave mathematically similar to the so-called Spoof Plasmon Polariton (SPP) wave,

while the bottom heterogeneities affect the behavior of the layer, hence modifying the dispersion

relation of the Love waves. The complete system simply mixes these two ingredients, resulting in

hybrid surface waves accurately described by our model.

Keywords:

metamaterial; homogenization; elastic metasurface; time domain analysis; elastic energy

1. Introduction

The problem of waves propagating in an elastic half-space supporting an array of beams or plates

is well known in seismology, where the site–city interaction aims at understanding the interaction of

seismic waves with a set of buildings. Starting with the seminal work of Housner [

1

] (see also [

2

]),

the site–city interaction has been intensively studied numerically [

3

–

5

] and analytically [

6

–

11

]. In this

context, seismic shields, or metabarriers, have been considered using resonators buried in soil [

12

–

15

]

or arrays of trees with a gradient in their heights [

16

–

18

]. More generally, this configuration is the

elastic analog of a corrugated interface able to support surface waves, studied in acoustics [

19

] and

in electromagnetism [

20

,

21

], where they are known as Spoof Plasmon Polaritons (SPPs). SPPs play a

fundament role in the extraordinary transmission of long wavelength electromagnetic waves through

metallic gratings [

22

,

23

] and have been studied intensively in the past twenty years for their potential

applications in subwavelength optics, data storage, light generation, microscopy, and bio-photonics;

see, e.g., [

24

]. Such similarities between surface waves in electromagnetism and elastodynamics fuel

research in seismic metamaterials [

25

], as they lead to simplified models that see behind the tree that

hides the forest [

26

].

To describe classical SPPs, the homogenization of a stratified medium is an easy and efficient

tool [

27

,

28

]; the analysis is valid in the low frequency regime, namely owing to the existence of a small

parameter measuring the ratio of the array spacing to the typical wavelength, and it provides, at the

dominant order, the dispersion relation of SPPs. Thanks to the mathematical analogy between the

problem in electromagnetism and in elasticity, this approach was applied in [

18

] accounting for the

presence of a guiding soil layer underlain by an elastic half-space. Simple dispersion relations have

been obtained from the effective model for the resulting spoof Love waves, so-called because of the

characteristics they share with classical Love waves (surface waves supported by the layer on its own)

and SPPs. Next, to account for the presence of heterogeneities (a foliage) at the top of the plates (trees),

a hybrid model was used where the homogenization was performed locally (near the top of the plates)

at the second order.

The present study generalizes and complements this study following two ways: (i) from a physical

point of view, we include the effect of heterogeneities at the bottom of the plates (Figure

1

), and (ii)

from a technical point of view, we derive the full model at second order. This produces a significant

increase in the accuracy of the theoretical prediction: in the reported examples, the model at order two

is accurate up to a 1–2% error margin, while the model previously used in [

18

], at order one, would be

accurate up to 10–30%. The second order model (see Equations (

2

) and (

3

)) provides a one-dimensional

problem along the z-direction with a succession of homogeneous layers: the substrate occupying

a half-space, the guiding layer, and an effective anisotropic layer replacing the region of the plates

(see Figure

2

). The effect of the heterogeneities at the bottom is encapsulated in transmission conditions,

which tell us that the displacement and the normal stress are not continuous; this holds for plates

without ending heterogeneities, a fact that was disregarded in [

18

]. The effect of the heterogeneities

at the top is encapsulated in a boundary condition that differs from the usual stress free condition,

as in [

18

]. We recover that for most of the frequencies, the plates do not interact efficiently with the

layer; in the present case, it results that the surface wave resembles that of the layer only, hence a

wave of the Love type. However, the resonances of the plates produce cut-off frequencies around

which the dispersion relations are deeply affected. For simple plates, this can already produce drastic

modifications in the dispersion relations (hybridization of the Love branches, avoided crossings at the

cut-off frequencies of the plates). When heterogeneities at the endings of the plates are accounted for,

additional changes happen. The heterogeneities at the bottom of the plates modify the behavior of

the layer on its own, resulting in modified Love waves. The heterogeneities at the top of the plates

modify the resonances of the plates, hence the cut-off frequencies. These two simple ingredients

allow us to interpret qualitatively the various dispersion relations obtained in the configuration of

the plates decorated at both ends. Next, the dispersion relations are accurately recovered by our

homogenized model.

x

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h

P

<latexit sha1_base64="qiurQzdWA696ZpDUBeDZr8WC90U=">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</latexit>

H

P

<latexit sha1_base64="4xKWLO2sAvhLcITCbLl+tRw4Fgw=">AAAC1XicjVHLSsNAFD2Nr1pfUZdugkVwVZJa0GXBTZcV7APaUpJ0WofmRTIpltKduPUH3OoviX+gf+GdMQW1iE5Icubce87MvdeJPJ4I03zNaSura+sb+c3C1vbO7p6+f9BMwjR2WcMNvTBuO3bCPB6whuDCY+0oZrbveKzljC9lvDVhccLD4FpMI9bz7VHAh9y1BVF9Xa/1u4LdillX8GBq1Od9vWiWTLWMZWBloIhs1UP9BV0MEMJFCh8MAQRhDzYSejqwYCIirocZcTEhruIMcxRIm1IWowyb2DF9R7TrZGxAe+mZKLVLp3j0xqQ0cEKakPJiwvI0Q8VT5SzZ37xnylPebUp/J/PyiRW4IfYv3SLzvzpZi8AQF6oGTjVFipHVuZlLqroib258qUqQQ0ScxAOKx4RdpVz02VCaRNUue2ur+JvKlKzcu1luind5Sxqw9XOcy6BZLllnpfJVpVitZKPO4wjHOKV5nqOKGupokPcEj3jCs9bS5tqddv+ZquUyzSG+Le3hA0MylhI=</latexit>

Figure 1.

Periodic array of plates decorated at their endings with spacing

` =

1, height h

P

, and thickness

ϕ

P

`

; the substrate occupying a half-space is surmounted by a guiding layer of thickness h

L

able to

support Love waves. The insets show a zoom on the two endings with heterogeneity surfaces

S

t

=

ϕ

t

h

t

z

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transmission conditions

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h

<latexit sha1_base64="YFPIRBNkAQEpM4LN/Xg5PIYYGYs=">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</latexit>

L

H

L

<latexit sha1_base64="doxJWrXqUgtb5fC1uwGQiDjMS3Y=">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</latexit>

h

b

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0

<latexit sha1_base64="5mYUOUymVgUg3BFR6u8Av/Go1nA=">AAACxHicjVHLSsNAFD2Nr1pfVZdugkVwVZJa0GVBEJct2AfUIsl0WofmxWQilKI/4Fa/TfwD/QvvjCmoRXRCkjPn3nNm7r1+EohUOc5rwVpaXlldK66XNja3tnfKu3udNM4k420WB7Hs+V7KAxHxthIq4L1Eci/0A971J+c63r3jMhVxdKWmCR+E3jgSI8E8RVTLuSlXnKpjlr0I3BxUkK9mXH7BNYaIwZAhBEcERTiAh5SePlw4SIgbYEacJCRMnOMeJdJmlMUpwyN2Qt8x7fo5G9Fee6ZGzeiUgF5JShtHpIkpTxLWp9kmnhlnzf7mPTOe+m5T+vu5V0iswi2xf+nmmf/V6VoURjgzNQiqKTGMro7lLpnpir65/aUqRQ4JcRoPKS4JM6Oc99k2mtTUrnvrmfibydSs3rM8N8O7viUN2P05zkXQqVXdk2qtVa806vmoizjAIY5pnqdo4BJNtI33I57wbF1YgZVa2WeqVcg1+/i2rIcPvSWPKw==</latexit>

h

<latexit sha1_base64="qiurQzdWA696ZpDUBeDZr8WC90U=">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</latexit>

P

boundary condition

<latexit sha1_base64="AglfqLJnZSKyy1NSjrRPgDaitdQ=">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</latexit>

x

<latexit sha1_base64="ujWcmwCXK80mrPsr4yEaleaV6WQ=">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</latexit>

Figure 2.

Configuration of the effective problem (

2

) and (

3

): The region of the plates has been replaced

by a homogeneous medium; the effective boundary condition and transmission conditions encapsulate

the effects of the heterogeneities at the decorative endings of the plates.

The paper is organized as follow. Section

2

summarizes the main results of the analysis: the effective

model, Equations (

2

) and (

3

), and the resulting equation of energy conservation, Equation (

10

). The

full derivation of the effective model is detailed in the Appendices

A

and

B

. In Section

3

, we inspect

the characteristics of waves guided by an array of decorated plates. The dispersion relations of these

waves are exhibited numerically and compared to the closed forms provided by the effective model,

Equations (

21

)–(

23

). The heterogeneities have the form of an additional thin hard layer at the bottom of

the plates and a thin hard cap on the top. These simple shapes of heterogeneities allow us to discuss the

Love waves modified by the bottom heterogeneity only and the resonances of a plate modified by the

top heterogeneity.

Throughout the paper, we use the following notations:

-

Material properties: mass density ρ and shear modulus µ, with subscripts “S” for the substrate,

“L” for the guiding Layer, “P” for the Plates, and “b,t” for the heterogeneities at the bottom and at

the top of the plates.

-

Geometrical parameters: the layer has a total height H

L

=

h

L

+

h

b

with h

b

occupied by the

heterogeneities. The array of plates is periodic with spacing

`

, with plate thickness ϕ

P

h

P

and total

height H

P

=

h

P

+

h

t

(h

t

occupied by the heterogeneities). The heterogeneities at the bottom and

top of the plates have surfaces

S

b

=

ϕ

b

h

b

and

S

t

=

ϕ

t

h

t

.

2. Summary of the Main Results

In the actual problem, the Navier equations for shear waves simplify to a wave equation for the

antiplane displacement u

=

u

y

(

x, t

)

and the stress vector σ

(

x, t

)

, of the form [

29

]:

σ

(

x, t

) =

µ

(

x

)

∇

u

(

x, t

)

,

ρ

(

x

)

∂

2

_u

∂t

2

(

x, t

) =

divσ

(

x, t

)

,

(1)

with x

= (

x, z

)

and t the time. The mass density ρ

(

x

)

and the shear modulus µ

(

x

)

are piecewise

constant in the different materials, substrate/layer/plate/heterogeneities; see Figure

1

. At each

boundary between the elastic materials, the continuity of the displacement u and of the normal stress

σ

·

n holds (with n the local normal vector). Eventually, at the boundaries separating elastic media and

air, the stress-free boundary condition σ

·

n

=

0 applies. In this section, we present the effective model

deduced from the asymptotic analysis developed in Appendix

A

.

2.1. Effective Model

In the effective homogenized model, the regions of the substrate z

∈ (−

∞,

−

H

L

)

and of the

z

∈ (

0, h

P

)

is replaced by an equivalent homogeneous region of the same height. In this region,

the medium is highly anisotropic, with propagation being allowed in the vertical direction z only;

this calculation follows from [

18

] and applies almost identically in the acoustic case for arrays of

Helmholtz resonators [

30

]. The boundary condition at the top of the effective medium, z

=

h

P

, is a

condition of the Robin type for the normal stress. The transmission conditions at the bottom of the

effective medium apply across the actual region of the heterogeneity, and they involve four parameters

depending on the geometry of the heterogeneity and of the plates. Specifically, the homogenized

model reads as:



















































for z

∈ (−

∞,

−

H

L

)

,

σ

=

µ

S

∇

u,

ρ

S

∂

2

u

∂t

2

=

divσ,

for z

∈ (−

H

L

,

−

h

b

)

,

σ

=

µ

L

∇

u,

ρ

L

∂

2

u

∂t

2

=

divσ,

for z

∈ (

0, h

P

)

,

σ

=

µ

P

ϕ

P

0

0

0

1

!

∇

u,

ρ

P

ϕ

P

∂

2

u

∂t

2

=

divσ,

(2)

along with the continuity of u and σ

·

n at z

=

−

H

L

and the dynamic effective conditions:























across the region

(

−

h

b

, 0

)

,

JuK

=

`

b

µ

L

σ

z

+

l

b

∂u

∂x

,

Jσ

z

K

=

l

b

∂σ

z

∂x

−

µ

L

L

b

∂

2

u

∂x

2

+

h

b

ˆρ

b

∂

2

u

∂t

2

,

at the top of the plates z

=

h

P

,

σ

z

(

x, h

P

, t

) =

−

L

t

∂σ

z

∂z

(

x, h

P

, t

)

.

(3)

The transmission conditions involve

JuK

=

u

(

x, 0, t

)

−

u

(

x,

−

h

b

, t

)

and u

=

1

₂

(

u

(

x, 0, t

) +

u

(

x,

−

h

b

, t

))

,

being the jump of u across the bottom heterogeneity and its mean value, respectively, and the same

for σ

z

.

Among the five effective parameters

(`

b

, l

b

, L

b

, ˆρ

b

, L

t

)

entering in the effective conditions, two are

known explicitly, while three are defined by elementary problems on

(

V

1

, V

2

)

that satisfy static

problems set in non-dimensional coordinate χ

= (

χ

, ζ

) = (

x/

`

, z/

`)

in the vicinity of z

=

0

(see Figure

A3

in Appendix

A.3

). These problems read as:























div



µ

µ

L

∇

V

1



=

0,

lim

ζ→−∞

∇

V

1

=

e

z

,

ζ→+∞

lim

∇

V

1

=

µ

L

ϕ

P

µ

P

e

z

,

div



µ

µ

L

∇ (

V

2

+

χ

)



=

0,

lim

ζ

→−∞

∇

V

2

=

0,

lim

ζ

→+∞

∇

V

2

=

−

e

x

,

(4)

with V

1

, µ

∇

V

1

·

n continuous at each interface between two elastic media and

∇

V

1

·

n

=

0 at the

boundaries with the air and V

1

and µ

∇

V

1

one periodic with respect to χ for ζ

<

0 (the same for V

2

and

µ

∇ (

V

2

+

χ

)

). Then, we have:

Effective parameters in (3)



















































































`

b

= `

lim

ζ

→+

∞



V

1

−

µ

L

ϕ

P

µ

P

ζ



+

h

b

,

l

b

= `

lim

ζ

→+

∞

(

V

2

+

χ

)

,

L

b

= `

Z

Y

P

µ

P

µ

L

∂

∂χ

(

V

2

+

χ

)

dχ

+ `

Z

Y

b

µ

µ

L

∂

∂χ

V

2

dχ

+

h

b

µ

L

[

ϕ

b

µ

b

+ (

1

−

ϕ

b

)

µ

L

]

,

ˆρ

b

=

ϕ

b

ρ

b

+ (

1

−

ϕ

b

)

ρ

L

,

L

t

=

h

t

ρ

t

ϕ

t

ρ

P

ϕ

P

.

(5)

It is worth noting that the homogenized problem is set in a domain where the regions

(

−

h

b

, 0

)

and

(

h

P

, H

P

)

occupied by the heterogeneities have disappeared. It should be possible to extend the

anisotropic region to

(

0, H

P

)

as done in [

18

]; this would lead to a different, but as accurate effective

model, with slightly different boundary condition at z

=

H

P

(specifically, a different value of L

t

).

However, this is not suitable from an energetic point of view (see Section

2.2

). Similarly, the transmission

conditions involve jumps of the displacement and of the normal stress across a non-zero interface.

It should be possible to express the transmission conditions across a zero thickness interface located

say at z

=

0. Again, this would lead to a different and as accurate effective model, with slightly

different transmission conditions (with different values of

`

b

and L

b

); again, our choice guaranties good

properties of the energy in the effective problem.

2.2. Effective Energy

The solution

(

u, σ

)

of the homogeneous problem is expected to approximate the, say numerical,

solution

(

u

num

, σ

num

)

of the actual problem. Hence, we expect that the actual elastic energy is also

correctly approximated in the effective problem. In the actual problem, the elastic energy simply reads

as [

29

]:

E

num

=

1

2

Z

D

num

1

µ

|

σ

num

|

2

₊

ρ



∂u

num

∂t



2

!

dx.

(6)

We shall now interrogate the equation of energy conservation in the homogenized problem where

the effective boundary and jump conditions in (

3

) make additional energies appear. These terms

appear primarily as fluxes within the bounded region

D

(see Figure

3

), but they can be written as the

time derivative of effective energies supported by the surface γ at the top of the plates and across the

heterogeneities at the bottom of the plates (Γ

±

_).

x

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z

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transmission conditions

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D

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+

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⌃

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⌃

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n

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boundary condition

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Figure 3.

Domain

D

where the energy is conserved in the absence of incoming/outcoming fluxes

through

Σ. The effective boundary condition on γ and jump conditions between Γ

±

in (

3

) result in

additional effective energies

E

t,b

in (

10

).

By simple manipulation of the equations in (

2

), the equation of energy conservation in the

homogenized problem is found to be of the form:

d

dt

(

E

S

+

E

L

+

E

P

) +

Φ

=

0,

(7)

with:

E

S,L

=

1

2

Z

D

S,L

|

σ

|

2

µ

S,L

+

ρ

S,L



∂u

∂t



2

!

dx,

E

P

=

1

2

Z

D

P

|

σ

z

|

2

µ

P

ϕ

P

+

ρ

P

ϕ

P



∂u

∂t



2

!

dx,

(8)

and:

Φ

=

Z

∂D

∂u

∂t

σ

·

n dl

(9)

(here,

Φ is a line integral). The flux Φ has a contribution on Σ and two contributions that do not

cancel even if the region

D

is bounded, that is if

Σ is associated with Neumann or Dirichlet boundary

conditions. Specifically,

Φ

=

Φ

Σ

+

Φ

b

+

Φ

t

with:

Φ

b,t

=

d

dt

E

b,t

and,

E

b

=

1

2

Z

Γ

"

µ

L

L

b



∂u

∂x



2

+

h

b

ˆρ

b



∂u

∂t



2

+

`

b

µ

L

σ

z

2

#

dx,

E

t

=

1

2

Z

γ

ρ

P

ϕ

P

L

t



∂u

∂t



2

dx,

(10)

where n is the normal interior and

D

S,L,P

the parts of

D

occupied by the substrate, the layer, and the

plates, respectively. We have used that σ

z

=

−

L

t

ρ

P

ϕ

P

∂

tt

u on γ from (

2

) and (

3

). We also have that

Φ

b

=

R

Γ

(

∂

t

u

Jσ

z

K

+

∂

t

JuK σ

z

)

dx; hence,

Φ

b

=

R

Γ



[

l

b

∂

x

σ

z

−

µ

L

L

b

∂

xx

u

+

h

b

ˆρ

b

∂

tt

u

]

∂

t

u

+

h

`

b

µ

L

∂

t

σ

z

+

l

b

∂

xt

u

i

σ

z



dx.

The two terms in l

b

cancel after integration by parts of one of them, and we integrate also by parts

the term in L

b

. It is worth noting that the integrations by parts make boundary terms (b.t.) appear.

These terms can be interpreted as concentrated forces at the ending points of

Γ

±

along x; they are

disregarded in the present study. Next,

E

b,t

in (

10

) are energies since they are definite positive quadratic

forms. Indeed, L

t

>

0 from (

5

) and ˆρ

b

>

0 from (

5

), and it is shown in Appendix

B

that

`

b

and L

b

are

positive as well. It is also worth mentioning that choosing a different position for γ would produce a

different and possibly negative value of L

t

. Similarly, expressing the transmission conditions across a

zero thickness interface would produce a possibly negative value of L

b

. Discussions on the effective

energies can be found in [

31

,

32

].

We further stress that the homogenized problem is set on

D

, which differs from

D

num

; the regions

D

b

for z

∈ (−

h

b

, 0

)

and

D

t

for z

∈ (

h

P

, H

P

)

are missing. Intuitively, we expect that the effective energies

E

b,t

represent the elastic energies in

D

b

and

D

t

in the actual problem; specifically, we expect that:

E

b,t

'

1

2

Z

D

b,t

1

µ

|

σ

num

|

2

₊

ρ



∂u

num

∂t



2

!

dx.

(11)

We shall illustrate in Section

3.4

that these intuitive relations are indeed legitimate.

3. Hybrid Love Waves in a Guiding Layer Supporting Decorated Plates

In this section, we inspect the ability of the effective problem (

2

) and (

3

) to reproduce the scattering

properties of an actual array. We consider the geometry of Figure

4

:

` =

1 in arbitrary unit length,

ϕ

P

=

ϕ

t

=

0.5 and ϕ

b

=

1. The total heights H

P

=

h

P

+

h

t

=

12, H

L

=

h

L

+

h

b

=

8 are fixed. When the

heterogeneities are considered, we set h

t

=

1 (hence, h

P

=

11) and/or h

b

=

1 (hence, h

L

=

7). We give

in the tables below the material properties and the values of the effective parameters entering in the

effective conditions (

3

).

We consider the time-harmonic regime with a time dependence e

−iωt

, which is omitted in the

following, and inspect the solution of a scattering problem for a wave coming from z

=

−

∞ with a

wavenumber β along x, resulting in a reflected wave with a complex reflection coefficient R. This scattering

problem allows us to cover the case of an incoming propagating wave, with

|

R

| =

1 for β

≤

ω

/c

S

, and the

case of guided waves, when

|

R

| =

∞ for β

>

ω

/c

S

. The actual problem has to be solved numerically, and

this was done using classical multimodal calculations.

In the rest of this section, we shall use for β the component of the wavenumber along x and make

use of the following quantities:

k

P

=

ω

c

P

,

γ

L

=

s

ω

2

c

2

L

−

β

2

,

γ

S

=

s

ω

2

c

2

S

−

β

2

,

(12)

(c

a

=

p

µ

a

/ρ

a

for

a = P, L, S

).

x

<latexit sha1_base64="ujWcmwCXK80mrPsr4yEaleaV6WQ=">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</latexit>

z

<latexit sha1_base64="SCAG39+GdDGoruf7sFr73NKPOBQ=">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</latexit>

HP

= 12

<latexit sha1_base64="f+FrsqU3B3tdnSpLYh531yiiTlg=">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</latexit>

1

<latexit sha1_base64="nlN2AZUDsk8jKIvvWZK7AKkdsrs=">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</latexit>

R

<latexit sha1_base64="+ILs7ifYHAozGA4JFL3XR1BynLA=">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</latexit>

ht

= 0 or 1

<latexit sha1_base64="hZk23uNn7Dm6OeMMFiFqGmZ/SRk=">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</latexit>

HL

= 8

<latexit sha1_base64="l3tsviC1XDGMOS+NDaOq/3BqKW8=">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</latexit>

` = 1, 'P` = 0.5

<latexit sha1_base64="xHD8Ds/QWkyy5H0h3TM64ScBjbk=">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</latexit>

hb

<latexit sha1_base64="hiOAqwlPAt1aqmAIMJcVdWafH2U=">AAAkUHicrVrNbttGEN6kf6n757S99UJUCdCiqSE7AdpLgaDJwQcdXCtOYkSJIFLUT02KKkk5ogW9RJ+g1/ZxeusbtG/QWzszu6QoUdwZwbFhabk7883Pzg5nd+1Og3GSNpt/3bj51tvvvPverff3Pvjwo48/2b/96dMkmsWef+ZFQRQ/d3uJH4wn/lk6TgP/+TT2e6Eb+M/ci0c4/uzSj5NxNHmSZlP/ZdgbTsaDsddLoau7/3ln5P7QdDqpP08XThQ7S+fQ6e43mgdN+nGqjUPTaCjzcxLdvv2r6qi+ipSnZipUvpqoFNqB6qkEfl+oQ9VUU+h7qRbQF0NrTOO+Wqo94J0BlQ8UPei9gM8hPL0wvRN4RsyEuD2QEsBfDJyOugs8EdDF0EZpDo3PCBl767AXhIm6ZfDtGqwQelM1gl6OL6eU8qEtqRqo78mGMdg0pR60zjMoM/IKau6UrEoBYQp92O7DeAxtjzhzPzvEk5Dt6Nsejf9NlNiLz56hnal/SMt6+4aEMAVrcIbmlvmp+rCe8jVFQmZobZQJaBmSHRn89Ym2njogX6JtkboHOmfGV2j7QKB7PnP3QP/QyEIexHxNsRQSzQQ0WUD/KfBM4XcJT6eqCz04E3P4XFAbtcnA5znVUoTWF+O9qqVIBdJw3VzSmrNLaxV0UkTeghVmvQ2ukWifgx7gLCmnvKRnjRsXsbAApAaMOgbtrgWrZbzRWtN8Br5e14z3RQK0Q8LCJ4wmp+hFOo4/JZp1buybFT6p58UejHcXqCeE0oHnAH7r58NlNVqh9imbzrd4ajumI0CdUJZe0hqWIEo0TeF7ymqZCuIiRzwr5gRzawDcP1ojOAa+SLAWc/x2Bf8R9PD+sGM/rqA+FvBhHnRN9DyBtk/xd50IQsR0B8S00NI2NyH0ZGsZIKDRIb0Hy6u/Q++yfMRhtK3i2jKLw/ozBosjk1/yp3rbfcEMaYwTIeIJITpUW6T0RvSphvDBqnFRO0zJe4jAS06FktNCMsp2IM4j8qF+91Z10OuWl+8K5bsb8vNVmWyR7dJIShJ5DVpCDVol3+vKNDM1XJ6Zq5q0BPLbQvntNfkJ5FqXKquY5roqu81EXggIedxhm4s6DisVYfF5NCTLJFh8tkL+lgirJcJqi7DagnV/YXx/cS2/D6Gd2zek+je0ZuOWQDPEcXfAlLy/EKe9A2abtTzHk6DY36zaVu+adVXKokgqiBMWhY8Jz0SEDYWPds/414ay8q8N6dhYdczU47xlx8YyDkkS5SOj1ejaWo3oXSPB4iMgr9Y4JD4iR8ZXHJLEVz/TCdB0oz7T1XmH3nyuOaO4oHowr6jwnYZ7pRHR4Q5+nc5uQS716o3L7aqmIG77dB71Qh0ZyQODge1pceKla3GUu6wZOWLnqm9ORSSyXgENJ88hKonUlKTef0NSt/XfL/bsdm+f09nO+g7n3LpvSgWRe24w5ah6vXO5EZHOt+or43RZW+vXvA3fL3Bd8Gd5B2LjmtGp5wK+XymfsSAxK2tZepJyXRU7p9UZShd6c34bQntLfLRZn/EZt02rfFdc2amWPnGNjc2XtDZ0nxTdtmc+NR45ZdAGrA9c2hN7Rs88w35FIz7MEyLpM/6Ydj26LuBQ/TVU3/SXUcp5+mtWR32uu5uOnuLXjKsmpVWDp3QTwTperf7VTcFAmAPOS+eCVW5ZPWI/o4nMzcNkw18dY2WdxwJW+zJ2f2f03fA3Z1uCL5cQlfasXHQeCDFXO6fddA5UOZ/EpI8tp+yaUfgqVqMMdkZOlSQT9sy5sG4hbo+tThJz7tuhO7LEnGpldOaRle4Z++Rp7eNfKt7GvhmtsTHIWK3snMsvuPIYqOOwzQpqeGm0XfdVPsKtWU03t2J0ybY56zmkzgRIGauVvjnsb+RHpMdID1RPZFm/RhdJbp5vVDRzwdssq+jrmVUjtXdutVg2DzlWVZsqlp4Jh70VwRhc34flPl3doS8Ie2ZqrfV7Mw45YbGH5tR5sgXdhj+nW68F67W5iZYrhi4zeNi+UvoWgrMxK0XyekTY65g8D+msqP00Jo9MFHc3pivdfAWU6965kHe2tWbmeP2Sf3Tc+RRtEs7N9a4558ZXNl68EQiL3KOfusV86fek3ieG8IkZekDZXJ9sy7BDBv3bCrIMd8rgfrMNt7vfONz8L5pq4+nRweH9g6OfHjQePjD/YXNLfaG+hPr1UH2nHtIZ2RnodqV+U7+rPxp/Nv5t/Hfnhia9ab7VZ2rt587e//EXLtI=</latexit>

= 0 or 1

hP

= 12 or 11

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hL

<latexit sha1_base64="Quie+/eKOOqVDnqjeLtagvARLUQ=">AAAkUHicrVrNbttGEN6kf6n7Z7e99UJUCdCiqWE7AZJLgaDJIQcdXCtOYliJIFLUT02KKkk5ogW9RJ+g1/ZxeusbtG/QWzszu6QoUdwZwbFhabk7883Pzg5nd+1OglGSHhz8dePmO+++9/4Htz7c+ejjTz79bHfv8+dJNI09/9SLgih+6XYTPxiN/dN0lAb+y0nsd0M38F+4F49x/MWlHyejaPwszSb+q7A7GI/6I6+bQldn98v2sPnDQ6ed+rN07kSxs3AeOJ3dxsH+Af041cahaTSU+TmO9vZ+VW3VU5Hy1FSFyldjlUI7UF2VwO+5OlQHagJ9r9Qc+mJojWjcVwu1A7xToPKBogu9F/A5gKdz0zuGZ8RMiNsDKQH8xcDpqDvAEwFdDG2U5tD4lJCxtw57TpioWwbfrsEKoTdVQ+jl+HJKKR/akqq+ekg2jMCmCfWgdZ5BmZJXUHOnZFUKCBPow3YPxmNoe8SZ+9khnoRsR992afxvosRefPYM7VT9Q1rW2zcghAlYgzM0s8xP1Yf1lG8oEjJDa6NMQMuQ7Mjgr0e09dQB+RJti9Rd0DkzvkLb+wLd85m7C/qHRhbyIOYbiqWQaMagyRz6T4BnAr8LeDpRHejBmZjB55zaqE0GPs+pFiK0nhjvdS1FKpCG6+aS1pxdWrOgkyLyFiwx621wjUT7HHQBZ0E55RU9a9y4iIU5IDVg1DFodyxYTeON5ormU/D1qma8LxKgHRAWPmE0OUUv0nH8KdGscmPftPBJPS/2YLy7QD0mlDY8B/BbPx8uq9EStUfZdLbBU5sxHQHqmLL0gtawBFGiaQrfE1bLVBAXOeJpMSeYWwPg/tEawTHwRYK1mOO3KviPoYf3hx37SQX1iYAP86BroucZtH2Kv+tEECKmWyCmhZa2uQmhJ1vJAAGNDug9WF79bXqX5SMOo20V15ZZHNafMVgcmfySP9Xb7gtmSGMcCxGPCdGh2iKlN6JPNYQPVo2K2mFC3kMEXnIqlJwWklG2A3EekQ/1u7eqg163vHxXKN9dk5+vymSDbJdGUpLIa9AUatAs+V5Xppmp4fLMXNWkKZDfEspvrchPINe6VFnFNNdV2S0m8kJAyOMO21zUcVipCIvPoyFZJsHisxXyN0VYTRFWS4TVEqz7C+P7i2v5fQDt3L4B1b+hNRs3BZohjrsFpuT9hTitLTBbrOU5ngTF/mbVtnrXrKtSFkVSQRyzKHxMeCYibCh8tHvGvzaUpX9tSE+NVU+Zepy37KmxjEOSRPnQaDW8tlZDetdIsPgIyKs1DomPyKHxFYck8dXPdAI0WavPdHXepjefa84oLqgezCsqfKfhXmlIdLiDX6WzW5BLvXrrcjvqQBC3PTqPOldHRnLfYGB7Upx46Voc5S5qRo7YueqZUxGJrNdAw8lziEoiNSWp996S1E3994o9u93bZ3S2s7rDObPum1JB5J4ZTDmqXu9cbkSks436yjhd1tb6NW/D9wtcF/xZ3oHYuKZ06jmH79fKZyxIzMpalJ6kXFfFzml5htKB3pzfhtDaEB8t1md8xm3RKt8WV3aqpU9cY2PzJa0N3SdFt+2ZT4xHThi0PusDl/bEntEzz7Df0IgP84RI+ow/pl2Prgs4VH8F1Tf9ZZRynv6W1VGf626no6f4NeOqcWnV4CndWLCOl6t/eVPQF+aAs9K5YJVbVo/Yz2gic/MwXvNX21hZ57GA1b6M3dsafTv89dmW4MslRKU9Kxed+0LM5c5pO50DVc4nMeljyynbZhS+itUo/a2RUyXJhF1zLqxbiNtlq5PEnPu26Y4sMadaGZ15ZKV7xh55Wvv4l4q3sW9Ka2wEMpYrO+fyC648Buo4bLOCGl4abVd9lY9wa1bTzawYHbJtxnoOqTMBUsZqpW8Oe2v5Eekx0gPVFVnWq9FFkptnaxXNTPA2yyr6embVSO2dWS2WzUOOVdWmiqVnwmFvRTAGV/dhuU+Xd+hzwp6aWmv13oxDTljsgTl1Hm9At+HP6NZrznptZqLliqHLDB62r5S+heBszEqRvBoR9jomz0M6K2o/jcgjY8XdjelKN18B5bp3JuSdbqyZOV6/5B8ddz5Fm4Rzfb1rzpnxlY0XbwTCIvfop04xX/o9qfeJIXxihu5TNtcn2zLskEH/voIsw50wuN9twu3sNg7X/4um2nh+tH94b//op/uNR/fNf9jcUl+pr6F+PVQP1CM6IzsF3a7Ub+p39Ufjz8a/jf9u39CkN823+kKt/Nze+R/RWC7K</latexit>

= 8 or 7

Figure 4.

Configuration of the array. The total thickness H

P

=

h

P

+

h

b

=

12 of the array and the total

thickness H

L

=

h

L

+

h

t

=

8 of the layer are kept constant;

` =

1 and ϕ

b

=

1, ϕ

t

=

ϕ

P

=

0.5. When the

heterogeneities are considered, h

b

=

h

t

=

1.

3.1. Two Reference Solutions

To begin with, we establish two families of reference solutions that will be useful to analyze our

problem. The first is that of Love waves supported by a guiding layer on the top of a substrate with

c

L

<

c

S

, which can be affected by the presence of the bottom heterogeneities. The second family is

that of the Spoof Plasmon Polaritons (SPPs) in the plates, which can be affected by the presence of

heterogeneities at the bottom of the plates.

3.1.1. Love Waves and Modified Love Waves

If we remove the array (Figure

5

), the problem is reduced to a guiding layer sandwiched between

air and the semi-infinite substrate (classical Love wave), and its modified version when a thin hard

layer is added. The exact solutions of these problems are easily obtained. For classical Love waves,

the solution of the scattering problem reads as:

u

(

x, z

) =

e

iβx

×







A cos

(

γ

L

z

)

,

z

∈ (−

h

L

, 0

)

,

e

iγ

S

(z+h

L

)

+

_R

Love

e

−iγ

S

(z+h

L

)

,

z

∈ (−

∞,

−

h

L

)

,

(13)

and using the continuity of the displacement and of the normal stress provides

(

A, R

Love

)

, in particular:

R

Love

=

−

tan

(

γ

L

h

L

)

−

iY

tan

(

γ

L

h

L

) +

iY

,

with Y

=

µ

S

γ

S

µ

L

γ

L

.

(14)

We recover the usual dispersion relation of Love waves for Y imaginary (γ

S

imaginary with a

positive imaginary part) and

|

R

Love

| =

∞, which guaranties a family of Love wave dispersion branches

in ω/c

S

<

β

<

ω

/c

L

; see Figure

5

.

If we add a layer of thickness h

b

in the guiding layer, the exact solution reads as:

u

(

x, z

) =

e

iβx

×























A cos

(

γ

b

z

)

,

z

∈ (−

h

b

, 0

)

,

B cos

(

γ

L

z

) +

C sin

(

γ

L

z

)

,

z

∈ (−

h

L

,

−

h

b

)

,

e

iγ

S

(z+h

L

)

+

_R

b

Love

e

−iγ

S

(z+h

L

)

,

z

∈ (−

∞,

−

h

L

)

.

(15)

Again, applying the continuity of the displacement and of the normal stress at z

=

−

h

L

,

−

h

b

provides

(

A, B, C, R

b

Love

)

and, in particular:

R

b

Love

=

−

tan

(

γ

L

h

L

+

Θ

b

)

−

iY

tan

(

γ

L

h

L

+

Θ

b

) +

iY

,

Θ

b

=

tan

−1



µ

b

γ

b

µ

L

γ

L

tan

(

γ

b

h

b

)



,

(16)