# Lamb modes and Born approximation for small shape defects inversion in
elastic plates

Éric Bonnetier  Institut Fourier, Université Grenoble Alpes, France  Angèle
Niclas  CMAP, École Polytechnique, France  Corresponding author:
<EMAIL_ADDRESS>Laurent Seppecher  Institut Camille Jordan,
École Centrale Lyon, France

###### Abstract

The aim of this work is to present theoretical tools to study wave propagation
in elastic waveguides and perform multi-frequency scattering inversion to
reconstruct small shape defects in a 2D and 3D elastic plate. Given surface
multi-frequency wavefield measurements, we use a Born approximation to
reconstruct localized defect in the geometry of the plate. To justify this
approximation, we introduce a rigorous framework to study the propagation of
elastic wavefield generated by arbitrary sources. By studying the decreasing
rate of the series of inhomogeneous Lamb mode, we prove the well-posedness of
the PDE that model elastic wave propagation in 2D and 3D planar waveguides. We
also characterize the critical frequencies for which the Lamb decomposition is
not valid. Using these results, we generalize the shape reconstruction method
already developed for acoustic waveguide to 2D elastic waveguides and provide
a stable reconstruction method based on a mode-by-mode spacial Fourier
inversion given by the scattered field.

##  1 Introduction

This work is devoted to the reconstruction of small shape defects in a
waveguide using multi-frequency scattering data. It is an extension of the
method exposed in [11] int the case of acoustic waveguide to the case of
elastic plates. If the scalar Helmholtz case is relevant to the non
destructive testing of pipes or optical fibers (see [17]), applications in the
elastic case concern the monitoring of structural parts, airplane, ship,
offshore wind energy plants or bridges for instance (see [34]).

The main common point between acoustic and elastic waveguides is the existence
of a modal decomposition of the wavefield in a sum of explicit guided modes.
The acoustic modes form an orthonormal basis, a property not satisfied by
their elastic counter-parts, called Lamb modes. Several authors have looked
into this feature. The books [28, 1] provide analytic expressions of Lamb
modes as well as dispersion relations for their wavenumbers. In [22, 25, 26] a
new formulation is introduced, the $\displaystyle\bm{X}/\bm{Y}$ formulation,
under which the family of Lamb mode turns out to be complete [3, 18, 8]. The
associated bi-orthogonality relations [14] thus allow the use of the Lamb
basis to decompose any wavefield that propagates in an elastic waveguide as a
sum of Lamb modes.

However, a rigorous mathematical framework is still missing to study the
propagation of an elastic wavefield generated by an arbitrary source term (see
however [5, 6] in 2D). One main goal of the present article is to prove well-
posedness of the system of PDE’s, that models 2D or 3D planar elastic
waveguides with internal and boundary source terms. To this end, we adapt the
strategy developed for acoustic waveguides in [11], which differs from [5].
Under stronger assumptions on the regularity of the source terms than those in
[5], we present in Theorem 2 a constructive proof of existence and regularity
of a wavefield propagating in a two dimensional elastic waveguide.

As it turns out, this result is not valid at some particular frequencies,
which we call critical frequencies, and that are characterized in the proof of
Theorem 2. In particular, we establish in Corollary 1 that the critical
frequencies, for which the Lamb family is no longer complete, coincide with
the vanishing of the bi-orthogonality relation established by [14]. This
result, up to our knowledge, has not been proven before and may help
understanding the mathematical analysis of elastic waveguides.

Concerning the study of wave propagation in three-dimensional plates, most of
the work that we are aware of consists in adapting the 2D framework to
situations with radial or axial symmetry (see for instance [19, 28, 2, 33]).
In [31], arbitrary source terms are considered, without mathematical
justification however. Introducing the Helmholtz-Hodge decomposition of the
wavefield [9], we split the three dimensional system of elasticity into a
system of two independent equations. One of them fits into the scalar wave
framework developed in [11], while the other can be rewritten using the
$\displaystyle\bm{X}/\bm{Y}$ formulation. This provides a full expression for
the decomposition of the wavefield generated by arbitrary source terms in
dimension 3, see Theorem 3.

Equipped with these results, we can generalize the shape reconstruction method
presented in [11] to the case of elastic plates, so as to determine possible
defects (bumps or dips) in the geometry of a plate, from multi-frequency
measurements. We use the very same procedure as in the acoustic case : after
mapping the perturbed plate to a straight configuration, we simplify the
resulting system of equations using the Born approximation. The scattered
wavefield generated by a known incident wavefield in the original geometry,
gives rise in the straightened plate to a boundary source term, that depends
on the shape defect. Using measurements of the scattered field on the surface
of the plate at different frequencies, we can reconstruct in a stable way the
shape defect (provided the latter is small enough). Numerical reconstructions
are presented in the last part of the article, which show the efficiency of
the method.

The paper is organized as follows. In section 2, we study the forward source
problem in a two dimensional waveguide and introduce all the tools needed to
use Lamb waves as a modal basis. In section 3, we generalize the results of
section 2 to the forward source problem in three dimensional plates. Section 4
is devoted to the reconstruction of shape defects in two dimensional plates,
generalizing the method presented in [11]. Finally, in section 5 we show
numerical illustrations of the propagation of waves in two and three
dimensional plates as well as reconstructions of different shape defects.

##  2 Forward source problem in a regular 2D waveguide

In this section, we present a complete study of the forward elastic source
problem in a two-dimensional regular waveguide. We use the
$\displaystyle\bm{X}/\bm{Y}$ formulation developed in [25, 26] which allows a
modal decomposition of any elastic wavefield using Lamb modes. Most of the
results presented here are already known, and can be found in [26, 28, 1]. Our
main contribution is to provide a rigorous proof of well-posedness for the
direct problem and of the fact that its solutions can be represented in terms
of Lamb modes (Theorem 2). We also follow the suggestions in [18] to define
the set of critical frequencies and critical wavenumbers in Definition 3, and
we prove in Corollary 1 that it coincides with the set of frequencies for
which the components $\displaystyle\bm{X}_{n}$ and $\displaystyle\bm{Y}_{n}$
of the eigenmodes are orthogonal for some $\displaystyle n$.

###  2.1 Lamb modes and critical frequencies

We consider a 2D infinite, straight, elastic waveguide
$\displaystyle\Omega=\\{(x,z)\in\mathbb{R}\times(-h,h)\\}$ of width
$\displaystyle 2h>0$. The displacement field is denoted by
$\displaystyle\bm{u}=(u,v)$ . Given a frequency
$\displaystyle\omega\in\mathbb{R}$, and given $\displaystyle(\lambda,\mu)$ the
Lamé parameters of the elastic waveguide, the wavefield $\displaystyle\bm{u}$
satisfies

$\nabla\cdot\bm{\sigma}(\bm{u})+\omega^{2}\bm{u}=-\bm{f}\qquad\text{ in
}\Omega,$ (1)

where $\displaystyle\bm{f}=(f_{1},f_{2})$ is a given source term, and where
the stress tensor $\displaystyle\bm{\sigma}(\bm{u})$ is defined by

$\bm{\sigma}(\bm{u})=\left(\begin{array}[]{cc}(\lambda+2\mu)\partial_{x}u+\lambda\partial_{z}v&\mu\partial_{z}u+\mu\partial_{x}v\\\
\mu\partial_{z}u+\mu\partial_{x}v&\lambda\partial_{x}u+(\lambda+2\mu)\partial_{z}v\end{array}\right):=\left(\begin{array}[]{cc}s&t\\\
t&r\end{array}\right).$ (2)

In this work, we assume that a Neumann boundary condition is imposed on both
sides of the plate

$\bm{\sigma}(\bm{u})\cdot\nu=\bm{b}^{\text{top}}\quad\text{ on
}\partial\Omega_{\text{top}},\qquad\bm{\sigma}(\bm{u})\cdot\nu=\bm{b}^{\text{bot}}\quad\text{
on }\partial\Omega_{\text{bot}},$ (3)

where
$\displaystyle\bm{b}^{\text{top}}=(b_{1}^{\text{top}},b_{2}^{\text{bot}})$ and
$\displaystyle\bm{b}^{\text{bot}}=(b_{1}^{\text{bot}},b_{2}^{\text{bot}})$ are
given boundary source terms. This condition could easily be replaced by a
Dirichlet or a Robin condition without much changes in the following analysis.
The setting is represented in Figure 1.

$\displaystyle\bm{f}$$\displaystyle\Omega$$\displaystyle
h$$\displaystyle-h$$\displaystyle e_{x}$$\displaystyle
e_{z}$$\displaystyle\bm{b}^{\text{top}}$$\displaystyle\bm{b}^{\text{bot}}$
Figure 1:  Parametrization of a two dimensional plate $\displaystyle\Omega$.
Elastic wavefields are generated using an internal source term
$\displaystyle\bm{f}$, and boundary source terms
$\displaystyle\bm{b}^{\text{top}}$ and $\displaystyle\bm{b}^{\text{bot}}$.

In [22] this equation is analyzed in an operator form
$\displaystyle\bm{Z}=\mathcal{L}(\bm{Z})$ where
$\displaystyle\bm{Z}=(u,t,s,v)$. This idea was then adapted in [25] to
formalize the so-called $\displaystyle\bm{X}/\bm{Y}$ formulation. We introduce
the variables

$\bm{X}=(u,t),\qquad\bm{Y}=(-s,v),$ (4)

with which the elasticity equation can be rewritten as follows:

######  Proposition 1.

The system (1), with the Neumann boundary conditions (3), is equivalent to

$\partial_{x}\left(\begin{array}[]{c}\bm{X}\\\
\bm{Y}\end{array}\right)=\mathcal{L}(\bm{X},\bm{Y})+\left(\begin{array}[]{c}0\\\
-f_{2}-b_{2}^{\text{top}}\delta_{z=h}-b_{2}^{\text{bot}}\delta_{z=-h}\\\
f_{1}+b_{1}^{\text{top}}\delta_{z=h}+b_{2}^{\text{top}}\delta_{z=-h}\\\
0\end{array}\right)\quad\text{ in }\Omega,$ (5)

with the boundary condition $\displaystyle B_{1}(\bm{X})=B_{2}(\bm{Y})=0$,
where $\displaystyle\mathcal{L}(\bm{X},\bm{Y})=(F(\bm{Y});G(\bm{X}))$ and
$\displaystyle F$, $\displaystyle G$, $\displaystyle B_{1}$ and $\displaystyle
B_{2}$ are differential matrix operators defined by

$F=\left(\begin{array}[]{cc}-\displaystyle\frac{1}{\lambda+2\mu}&-\displaystyle\frac{\lambda}{\lambda+2\mu}\partial_{z}\\\
\displaystyle\frac{\lambda}{\lambda+2\mu}\partial_{z}&-\omega^{2}-\displaystyle\frac{4\mu(\lambda+\mu)}{\lambda+2\mu}\partial^{2}_{zz}\end{array}\right),\qquad
G=\left(\begin{array}[]{cc}\omega^{2}&\partial_{z}\\\
-\partial_{z}&\displaystyle\frac{1}{\mu}\end{array}\right),$ (6)
$B_{1}(\bm{X})=\bm{X}\cdot e_{z},\qquad
B_{2}(\bm{Y})=-\frac{\lambda}{\lambda+2\mu}\bm{Y}\cdot
e_{x}+\frac{4\mu(\lambda+\mu)}{\lambda+2\mu}\partial_{z}\bm{Y}\cdot e_{z}.$
(7)

The proof of this proposition follows the same steps as that presented in
Appendix A of [26]. In this formulation, the operators $\displaystyle F$ and
$\displaystyle G$ only depend on $\displaystyle z$, and are defined on one
section of the waveguide, while derivatives with respect to $\displaystyle x$
only appear in the left-hand side of (5). We consider the space

$H_{0}:=\left\\{(\bm{X},\bm{Y})\in(\text{H}^{2}(-h,h))^{4}\,|\,B_{1}(\bm{X})(\pm
h)=B_{2}(\bm{Y})(\pm h)=0\right\\},$ (8)

and the operator

$\mathcal{L}:\begin{array}[]{rcl}H_{0}&\rightarrow&(\text{L}^{2}(-h,h))^{4}\\\
(\bm{X},\bm{Y})&\mapsto&(F(\bm{Y}),G(\bm{X}))\end{array}.$ (9)

Our goal is to diagonalize this operator and, to this end, we introduce the
Lamb modes:

######  Definition 1.

A Lamb mode $\displaystyle(\bm{X},\bm{Y})\in H_{0}$, associated to the
wavenumber $\displaystyle k\in\mathbb{C}$, is a non-trivial solution of
$\displaystyle\mathcal{L}(\bm{X},\bm{Y})=ik(\bm{X},\bm{Y})$.

The next Proposition provides the analytical expressions of these modes. The
proof can be found in [1, 28].

######  Proposition 2.

The set of wavenumbers $\displaystyle k\in\mathbb{C}$ associated to Lamb modes
is countable, and every such wavenumber $\displaystyle k$ satisfies the
symmetric Rayleigh-Lamb equation

$p^{2}=\frac{\omega^{2}}{\lambda+2\mu}-k^{2},\qquad
q^{2}=\frac{\omega^{2}}{\mu}-k^{2},\qquad\left(q^{2}-k^{2}\right)^{2}=-4k^{2}pq\frac{\tan(ph)}{\tan(qh)},$
(10)

or the antisymmetric Rayleigh-Lamb equation

$p^{2}=\frac{\omega^{2}}{\lambda+2\mu}-k^{2},\qquad
q^{2}=\frac{\omega^{2}}{\mu}-k^{2},\qquad\left(q^{2}-k^{2}\right)^{2}=-4k^{2}pq\frac{\tan(qh)}{\tan(ph)}.$
(11)

If $\displaystyle k$ satisfies (10), the associated Lamb mode is called
symmetric and is proportional to
$\displaystyle(\bm{X}(z),\bm{Y}(z))=$

$\left(\begin{array}[]{c}u(z)\\\ t(z)\\\ -s(z)\\\
v(z)\end{array}\right)=\left(\begin{array}[]{c}ik(q^{2}-k^{2})\sin(qh)\cos(pz)-2ikpq\sin(ph)\cos(qz)\\\
2ik\mu(q^{2}-k^{2})p(-\sin(qh)\sin(pz)+\sin(ph)\sin(qz))\\\
(q^{2}-k^{2})((\lambda+2\mu)k^{2}+\lambda p^{2})\sin(qh)\cos(pz)-4\mu
pqk^{2}\sin(ph)\cos(qz)\\\
-p(q^{2}-k^{2})\sin(qh)\sin(pz)-2k^{2}p\sin(ph)\sin(qz)\end{array}\right).$
(12)

If $\displaystyle k$ satisfies (11), the associated Lamb mode is called anti-
symmetric and is proportional to $\displaystyle(\bm{X}(z),\bm{Y}(z))=$

$\left(\begin{array}[]{c}u(z)\\\ t(z)\\\ -s(z)\\\
v(z)\end{array}\right)=\left(\begin{array}[]{c}ik(q^{2}-k^{2})\cos(qh)\sin(pz)-2ikpq\cos(ph)\sin(qz)\\\
2ik\mu(q^{2}-k^{2})p(\cos(qh)\cos(pz)-\cos(ph)\cos(qz))\\\
(q^{2}-k^{2})((\lambda+2\mu)k^{2}+\lambda p^{2})\cos(qh)\sin(pz)-4\mu
pqk^{2}\cos(ph)\sin(qz)\\\
p(q^{2}-k^{2})\cos(qh)\cos(pz)+2k^{2}p\cos(ph)\cos(qz)\end{array}\right).$
(13)

######  Remark 1.

We see on the above expressions that $\displaystyle p$ and $\displaystyle q$
are defined up to a multiplication by $\displaystyle-1$. However, since Lamb
modes are defined up to a multiplicative constant, the choice of the sign of
$\displaystyle p$ or $\displaystyle q$ does not change the associated value of
$\displaystyle k$ or the associated Lamb mode.

We notice that if $\displaystyle k$ is a solution of the Rayleigh-Lamb
equation then $\displaystyle-k$ and $\displaystyle\bar{k}$ are also solutions.
Figure 2 depicts different wavenumbers $\displaystyle k$ where
$\displaystyle\text{Real}(k)\geq 0$ and $\displaystyle\text{Imag}(k)\geq 0$,
in terms of the frequency $\displaystyle\omega$.

$\displaystyle 0$$\displaystyle 2$$\displaystyle 4$$\displaystyle
6$$\displaystyle 8$$\displaystyle 0$$\displaystyle 2$$\displaystyle
4$$\displaystyle 6$$\displaystyle 8$$\displaystyle 0$$\displaystyle
2$$\displaystyle
4$$\displaystyle\text{Imag}(k)h$$\displaystyle\text{Real}(k)h$$\displaystyle\omega
h$propagativeinhomogeneousevanescentcritical Figure 2:  Solutions of the
symmetric Rayleigh-Lamb equation (12) in the space
$\displaystyle\text{Imag}(k)\geq 0$, $\displaystyle\text{Real}(k)\geq 0$ with
$\displaystyle\mu=0.25$ and $\displaystyle\lambda=0.31$. Solutions on the full
space can be obtained by axial symmetries. Propagative, evanescent and
inhomogeneous modes are represented by different colors. Critical points are
represented by red dots.

We can distinguish three different types of modes (represented in different
colors in the above Figure) as in [19]:

######  Definition 2.

There are three types of Lamb modes:

  * •

If $\displaystyle k\in\mathbb{R}$, the mode oscillates in the waveguide
without energy decay and is called propagative.

  * •

If $\displaystyle k\in i\mathbb{R}$, the mode decays exponentially to zero as
$\displaystyle|x|\to\infty$, and is called evanescent.

  * •

If $\displaystyle\text{Real}(k)\neq 0$ and $\displaystyle\text{Imag}(k)\neq
0$, the mode oscillates quickly toward zero and is called inhomogeneous.

The completeness of Lamb modes depends on whether the frequency
$\displaystyle\omega$ is critical as defined below:

######  Definition 3.

A frequency $\displaystyle\omega$ and a wavenumber $\displaystyle k$ are said
to be critical if they satisfy $\displaystyle k=0$ or condition (10) and
$\displaystyle\Gamma_{S}=0$ (resp. condition (11) and
$\displaystyle\Gamma_{A}=0$) where

$\displaystyle\displaystyle\Gamma_{S}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle
h(q^{2}-k^{2})^{2}\sin(qh)^{2}+4k^{2}p^{2}\sin(ph)^{2}$
$\displaystyle\displaystyle+(q^{2}-k^{2})\sin(ph)\cos(ph)\sin(qh)^{2}\left(\frac{q^{2}-k^{2}}{p}-8p-\frac{2p}{k^{2}}-\frac{p}{q^{2}}\right),$
$\displaystyle\displaystyle\Gamma_{A}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle
h(q^{2}-k^{2})^{2}\cos(qh)^{2}+4k^{2}p^{2}\cos(ph)^{2}$
$\displaystyle\displaystyle-(q^{2}-k^{2})\cos(ph)\sin(ph)\cos(qh)^{2}\left(\frac{q^{2}-k^{2}}{p}-8p-\frac{2p}{k^{2}}-\frac{p}{q^{2}}\right).$

We denote by $\displaystyle\omega_{\text{crit}}$ the set of critical
frequencies, and by $\displaystyle k_{\text{crit}}$ the set of associated
critical wavenumbers.

Critical points solution to $\displaystyle k=0$ or to
$\displaystyle\Gamma_{S}=0$, where $\displaystyle\Gamma_{S}$ is defined in
(3), are represented in Figure 2. We notice that critical points seem to be
located at the junction of branches of modes of different types (see [18, 28]
for more details). Next, we introduce the functional space

$H:=\text{H}^{1}(-h,h)\times\text{L}^{2}(-h,h)\times\text{L}^{2}(-h,h)\times\text{H}^{1}(-h,h).$
(16)

and state the following completeness result:

######  Theorem 1.

At frequency $\displaystyle\omega$, Lamb modes form a complete set of
functions in $\displaystyle H$ if and only if
$\displaystyle\omega\notin\omega_{\text{crit}}$.

###### Proof.

Step 1 : It is shown in [3] (see also [18, 8]) that the operator
$\displaystyle\mathcal{L}$ satisfies the following properties:

  * •

There exists a set of five rays in the complex plane such that the angles
between adjacent rays are less than $\displaystyle\pi/2$,

  * •

Sufficiently far from the origin, all the points on these rays lie in the
resolvent set of $\displaystyle\mathcal{L}$,

  * •

There exits $\displaystyle N\in\mathbb{N}$ such that the resolvent of
$\displaystyle\mathcal{L}$ satisfies

$\|(\mathcal{L}-\ell I)^{-1})\|=\mathcal{O}(|\ell|^{N})\text{ as
}|\ell|\rightarrow+\infty\text{ along each ray.}$ (17)

Invoking Theorem 6.2 in [20], one may then infer that the family of Lamb modes
forms a complete set of functions if and only if for every associated
wavenumber $\displaystyle k$,

$\text{Ker}(\mathcal{L}-ikI)=\text{Ker}(\mathcal{L}-ikI)^{2}.$ (18)

Step 2 : The above condition is however implicit and does not allow an
effective determination of the frequencies for which the Lamb modes form a
complete set. In [18], a simpler condition than (18) is derived (although not
proved) with a reference to [30]. Our goal is to derive an equivalent
condition, that only depends on the parameters of the problem. To this end, we
generalize the approach in [28] and in view of (18), we seek to characterize
under which conditions generalized eigenvalues exist. We present the argument
in the case of a symmetric Lamb mode, the antisymmetric situation can be
handled in the same manner.

Assume that $\displaystyle(u_{0},t_{0},-s_{0},v_{0})$ is a symmetric Lamb mode
(given by (12)) associated with a wavenumber $\displaystyle k\in\mathbb{C}$ so
that (18) is not satisfied. We look for $\displaystyle(u,t,-s,v)\in H_{0}$
that satisfies

$\mathcal{L}(u,t,-s,v)=ik\mathcal{L}(u,t,-s,v)+(u_{0},t_{0},-s_{0},v_{0}).$
(19)

Defining

$f_{1}=-(\lambda+\mu)\partial_{z}u_{0}-2ik\mu v_{0},\qquad
f_{2}=-2(\lambda+2\mu)iku_{0}-(\lambda+\mu)\partial_{z}v_{0},$ (20)

we notice that $\displaystyle(u,v)$ satisfies the equation

$\left\\{\begin{array}[]{cl}(\lambda+2\mu)\partial_{zz}v+(\lambda+\mu)ik\partial_{z}u-\mu
q^{2}v=f_{1}&\text{ in }(-h,h),\\\
\mu\partial_{zz}u+(\lambda+\mu)ik\partial_{z}v-p^{2}(\lambda+2\mu)u=f_{2}&\text{
in }(-h,h),\\\ \partial_{z}u(\pm h)+ikv(\pm h)=-v_{0}(\pm h),&\\\ \lambda
iku(\pm h)+(\lambda+2\mu)\partial_{z}v(\pm h)=-\lambda u_{0}(\pm
h).&\end{array}\right.$ (21)

We introduce the auxiliary functions

$\phi=-\frac{\lambda+2\mu}{\omega^{2}}(iku+\partial_{z}v),\qquad\psi=\frac{\mu}{\omega^{2}}(\partial_{z}u-ikv),$
(22)

which turn out to solve the following second order linear ODE’s with constant
coefficients

$\partial_{zz}\phi(z)+p^{2}\phi(z)=-2ik(q^{2}-k^{2})\sin(qh)\cos(pz)+2ikpq\frac{\lambda+\mu}{\mu}\sin(ph)\cos(qz),$
(23)
$\partial_{zz}\psi(z)+q^{2}\psi(z)=-(q^{2}-k^{2})p\sin(qh)\frac{\lambda+\mu}{\lambda+2\mu}\sin(pz)+4k^{2}p\sin(ph)\sin(qz).$
(24)

The solutions of the above ODE’s are explicit. Using the fact that
$\displaystyle u=ik\phi-\partial_{z}\psi+f_{2}/\omega^{2}$ and $\displaystyle
v=\partial_{z}\phi+ik\psi+f_{1}/\omega^{2}$, we find that

$\displaystyle\displaystyle u(z)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle
A_{1}ik\cos(pz)+A_{2}ik\sin(pz)+A^{\prime}_{1}q\sin(qz)-A^{\prime}_{2}q\cos(qz)$
$\displaystyle\displaystyle+\frac{k^{2}(q^{2}-k^{2})}{p}\sin(qh)z\sin(pz)-2k^{2}p\sin(ph)z\sin(qz)+\frac{2k^{4}\mu
p}{q\omega^{2}}\sin(ph)\cos(qz)$
$\displaystyle\displaystyle+\left(q^{2}-k^{2}+\frac{k^{2}(q^{2}-k^{2})(\lambda+2\mu)}{\omega^{2}}\right)\sin(qh)\cos(pz),$
$\displaystyle\displaystyle v(z)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-
A_{1}p\sin(pz)+A_{2}p\cos(pz)+A^{\prime}_{1}ik\cos(qz)+A^{\prime}_{2}ik\sin(qz)$
$\displaystyle\displaystyle-
ik(q^{2}-k^{2})\sin(qh)z\cos(pz)-\frac{2k^{2}p}{q}\sin(ph)z\cos(qz)+\left(2ikp+\frac{2ik^{3}\mu
p}{\omega^{2}}\right)\sin(ph)\sin(qz)$
$\displaystyle\displaystyle-\frac{ik^{3}(q^{2}-k^{2})(\lambda+2\mu)}{\omega^{2}p}\sin(qh)\sin(pz),$

for some $\displaystyle
A_{1},A_{2},A^{\prime}_{1},A^{\prime}_{2}\in\mathbb{C}$. Expressing the
boundary conditions, we obtain

$M\left(\begin{array}[]{c}A_{1}\\\
A^{\prime}_{2}\end{array}\right)=-\left(\begin{array}[]{c}b_{1}\\\
b_{2}\end{array}\right),\quad\text{with}\quad\left(\begin{array}[]{cc}-2ikp\sin(ph)&(q^{2}-k^{2})\sin(qh)\\\
-\mu(q^{2}-k^{2})\cos(ph)&2ik\mu q\cos(qh)\end{array}\right),$ (27)
$b_{1}=2k^{2}(q^{2}-k^{2})h\cos(ph)\sin(qh)-\frac{2(q^{2}-k^{2})k^{2}p}{q}h\cos(qh)\sin(ph)\\\
+\sin(ph)\sin(qh)\left(-9pk^{2}-pq^{2}+\frac{4k^{2}pq^{2}}{\omega^{2}}+\frac{(q^{2}-k^{2})(k^{2}-p^{2})(\omega^{2}+(\lambda+2\mu)k^{2})}{p\omega^{2}}\right),$
$b_{2}=\left(4\mu
ik^{3}p+\frac{ik\mu(q^{2}-k^{2})^{2}}{p}\right)h\sin(ph)\sin(qh)+\cos(qh)\sin(ph)\left(4ik\mu
qp-\frac{4ik^{5}p\mu^{2}}{q\omega^{2}}\right)\\\
+\cos(ph)\sin(qh)\left(-\frac{2ik\mu(q^{2}-k^{2})}{\omega^{2}}(\omega^{2}+k^{2}(\lambda+2\mu))+\lambda
ik(q^{2}-k^{2})\right).$

Since $\displaystyle k$ satisfies the symmetric Rayleigh-Lamb equation (10),
the determinant of the matrix $\displaystyle M$ vanishes.

If the first column of the matrix $\displaystyle M$ is non zero, then (27) has
a solution if and only if

$\left|\begin{array}[]{cc}2ikp\sin(ph)&b_{1}\\\
\mu(q^{2}-k^{2})\cos(ph)&b_{2}\end{array}\right|=0.$ (28)

Computing this determinant and using the relation (10) leads to

$-2k^{2}\mu\Bigg{[}h(q^{2}-k^{2})^{2}\sin(qh)^{2}+4k^{2}p^{2}\sin(ph)^{2}\\\
+(q^{2}-k^{2})\sin(ph)\cos(ph)\sin(qh)^{2}\left(\frac{q^{2}-k^{2}}{p}-8p-\frac{2p}{k^{2}}-\frac{p}{q^{2}}\right)\Bigg{]}=0,$
(29)

in other words, $\displaystyle\Gamma_{S}=0$ and
$\displaystyle\omega\in\omega_{\text{crit}}$.

Assume now that $\displaystyle M_{11}=M_{21}=0$. Then either $\displaystyle
k=0$ and $\displaystyle q^{2}\cos(ph)=0$ (the condition $\displaystyle q=0$ is
excluded as it yields to a trivial eigenfunction
$\displaystyle(u_{0},t_{0},-s_{0},v_{0})$) and the system takes the form

$\left(\begin{array}[]{cc}0&q^{2}\sin(qh)\\\
0&0\end{array}\right)\left(\begin{array}[]{c}A_{1}\\\
B_{2}\end{array}\right)=\left(\begin{array}[]{c}2pq^{2}\sin(qh)\\\
0\end{array}\right),$ (30)

and has non trivial solutions (for instance $\displaystyle B_{2}=2p$ and
$\displaystyle A_{1}=0$). Or $\displaystyle k\neq 0$, $\displaystyle
p\in\pi\mathbb{Z}$ and $\displaystyle q^{2}-k^{2}=0$, in which case
$\displaystyle(b_{1},b_{2})=(0,0)$ and the system also has nontrivial
solutions. In this latter case, one can check that $\displaystyle\Gamma_{S}=0$
as well.

Conversely, if (29) holds, then using (2.1)-(2.1) one can construct a solution
$\displaystyle(u,t,-s,v)$ to (19). This shows that
$\displaystyle\omega\in\omega_{\text{crit}}$ if an only if (18) is not
satisfied, and concludes the proof of the Theorem. ∎

######  Remark 2.

When $\displaystyle\omega\in\omega_{\text{crit}}$, one needs to add
generalized eigenmodes to the Lamb modes to obtain a complete family [18, 3].
Our proof can be useful if ones want to find the expression of such
generalized modes: finding $\displaystyle(A_{1},B_{2})$ solution to (27) and
replacing it in (2.1) and (2.1) gives the expression of the generalized modes.

######  Remark 3.

If we derive equation (10) (resp. (11)) with respect to $\displaystyle k$, we
notice that $\displaystyle\partial_{k}\omega=0$ if and only if (3) (resp. (3))
is satisfied. This shows that critical points are exactly located where
$\displaystyle\partial_{k}\omega=0$. When $\displaystyle k\neq 0$, these
points are called zero velocity group points (ZGV points) and have been
extensively studied (see for instance [4]).

###  2.2 Solution of the 2D elasticity problem

In the rest of this section, we assume that
$\displaystyle\omega\notin\omega_{\text{crit}}$, so that Lamb modes form a
complete family, however they do not necessarily yield an orthonormal basis.
In order to identify the decomposition of a given function of $\displaystyle
H$ on the Lamb basis, we split the set of wavenumbers $\displaystyle k$ in two
parts:

######  Definition 4.

Let $\displaystyle\omega\notin\omega_{\text{crit}}$.

  * •

We say that a Lamb mode with wavenumber $\displaystyle k$ is right-going if
$\displaystyle\text{Imag}(k)>0$ or $\displaystyle\text{Imag}(k)=0$ and
$\displaystyle\partial_{k}\omega>0$,

  * •

We say that a Lamb mode is left-going if $\displaystyle\text{Imag}(k)<0$ or
$\displaystyle\text{Imag}(k)=0$ and $\displaystyle\partial_{k}\omega<0$.

We index the right-going modes by $\displaystyle n\in\mathbb{N}^{*}$, and sort
them by ascending order of imaginary part and descending order of real part.

We illustrate this classification in Figure 3, where right and left-going
wavenumbers are represented at the frequency $\displaystyle\omega=1.37$.

$\displaystyle-4$$\displaystyle-3$$\displaystyle-2$$\displaystyle-1$$\displaystyle
0$$\displaystyle 1$$\displaystyle 2$$\displaystyle 3$$\displaystyle
4$$\displaystyle-10$$\displaystyle 0$$\displaystyle
10$$\displaystyle\text{Real}(k)$$\displaystyle\text{Imag}(k)$right goingleft
going$\displaystyle\text{Imag}(k)=0$$\displaystyle-4$$\displaystyle-3$$\displaystyle-2$$\displaystyle-1$$\displaystyle
0$$\displaystyle 1$$\displaystyle 2$$\displaystyle 3$$\displaystyle
4$$\displaystyle 1.3$$\displaystyle 1.35$$\displaystyle
1.4$$\displaystyle\text{Real}(k)$$\displaystyle\omega$Rayleigh eq.right
goingleft going Figure 3:  Representation of right and left-going wavenumbers
at the frequency $\displaystyle\omega=1.37$ with $\displaystyle\mu=0.25$ and
$\displaystyle\lambda=0.31$. Right-going modes are represented by circles and
left-going modes by triangles. Top: classification of the wavenumbers with
$\displaystyle\text{Imag}(k)\neq 0$. Bottom : classification of the
wavenumbers with $\displaystyle\text{Imag}(k)=0$ by looking at
$\displaystyle\partial_{k}\omega$.

As mentioned previously, if $\displaystyle k_{n}$ is a right-going mode,
$\displaystyle-k_{n}$ is also solution of the Rayleigh-Lamb equation and is
then a left-going mode. We also notice, using (12), that if
$\displaystyle(\bm{X}_{n},\bm{Y}_{n})$ is (up to a multiplicative constant)
the right-going Lamb mode associated to $\displaystyle k_{n}$ and
$\displaystyle(\widetilde{\bm{X}}_{n},\widetilde{\bm{Y}_{n}})$ is the left-
going Lamb mode associated to $\displaystyle-k_{n}$, then

$u_{n}=-\widetilde{u}_{n},\quad v_{n}=\widetilde{v}_{n},\quad
s_{n}=\widetilde{s}_{n},\quad t_{n}=-\widetilde{t}_{n}.$ (31)

It follows that for any $\displaystyle(\bm{X},\bm{Y})\in H$, there exist
$\displaystyle(A_{n})_{n\in\mathbb{N}^{*}},(B_{n})_{n\in\mathbb{N}^{*}}$ such
that

$(\bm{X},\bm{Y})=\sum_{n>0}A_{n}(\bm{X}_{n},\bm{Y}_{n})+\sum_{n>0}B_{n}(\widetilde{\bm{X}}_{n},\widetilde{\bm{Y}_{n}})=\sum_{n>0}A_{n}(\bm{X}_{n},\bm{Y}_{n})+\sum_{n>0}B_{n}(-\bm{X}_{n},\bm{Y}_{n}).$
(32)

Defining $\displaystyle a_{n}=A_{n}-B_{n}$ and $\displaystyle
b_{n}=A_{n}+B_{n}$,

$\bm{X}=\sum_{n>0}a_{n}\bm{X}_{n},\qquad\bm{Y}=\sum_{n>0}b_{n}\bm{Y}_{n}.$
(33)

This decomposition with right-going modes is easier to handle than the full
decomposition: We prove below that we can find an explicit expression of
$\displaystyle a_{n}$ and $\displaystyle b_{n}$ given
$\displaystyle(\bm{X},\bm{Y})$. We denote
$\displaystyle\langle\cdot,\cdot\rangle$ the product defined by

$\forall\xi_{i}\in\text{L}^{2}((-h,h),\mathbb{C})\qquad\langle(\xi_{1},\xi_{2}),(\xi_{3},\xi_{4})\rangle=\int_{-h}^{h}\xi_{1}(z)\xi_{3}(z)+\xi_{2}(z)\xi_{4}(z)\mathrm{d}z.$
(34)

Note that this product is not a scalar product since the
$\displaystyle\xi_{i}$’s are complex-valued. The following proposition states
that families $\displaystyle(\bm{X}_{n})_{n\in\mathbb{N}^{\star}}$ and
$\displaystyle(\bm{Y}_{n})_{n\in\mathbb{N}^{\star}}$ are bi-orthogonal:

######  Proposition 3.

For every $\displaystyle n,m>0$,
$\displaystyle\langle\bm{X}_{m},\bm{Y}_{n}\rangle=\delta_{n=m}J_{n}$ where
$\displaystyle J_{n}=i\omega^{2}k\,\Gamma_{S}$ if $\displaystyle n$ is a
symmetric mode and $\displaystyle J_{n}=i\omega^{2}k\,\Gamma_{A}$ if
$\displaystyle n$ is an anti symmetric mode, with $\displaystyle\Gamma_{S}$
and $\displaystyle\Gamma_{A}$ defined in (3) and (3). Especially,
$\displaystyle J_{n}\neq 0$ if and only if
$\displaystyle\omega\notin\omega_{\text{crit}}$.

###### Proof.

The proof that $\displaystyle\langle\bm{X}_{m},\bm{Y}_{n}\rangle=0$ if
$\displaystyle m\neq n$ can be found in [14, 26]. Then, using expressions (12)
and (13), we can compute $\displaystyle\langle\bm{X}_{n},\bm{Y}_{n}\rangle$ as
in [26]. ∎

This provides a new characterization of critical points:

######  Corollary 1.

The three following definitions of the set of critical frequencies are
equivalent:

$\begin{array}[]{rcl}\omega_{\text{crit}}&=&\\{\omega\in\mathbb{R}_{+}\,|\,\exists
n\in\mathbb{N}^{\star}\quad\langle\bm{X}_{n},\bm{Y}_{n}\rangle=0\\}\\\
&=&\\{\omega\in\mathbb{R}_{+}\,|\,(\bm{X}_{n},\bm{Y}_{n})_{n>0}\text{ does not
form a complete set of functions in }H\\}\\\
&=&\\{\omega\in\mathbb{R}_{+}\,|\,\exists
n\in\mathbb{N}^{\star}\quad\partial_{k_{n}}\omega=0\\}.\end{array}$ (35)

Proposition 3 allows us to compute the coefficients in a decomposition (33),
as

$a_{n}=\frac{\langle\bm{X},\bm{Y}_{n}\rangle}{J_{n}},\qquad
b_{n}=\frac{\langle\bm{Y},\bm{X}_{n}\rangle}{J_{n}}.$ (36)

We use the modal decomposition to provide an outgoing condition for elastic
waveguides. For acoustic waves, one may ask that each modal component should
satisfy a one dimensional Sommerfeld radiation condition (see [11] for
instance). In the same spirit, we consider the following

######  Definition 5.

A wavefield $\displaystyle\bm{u}\in\text{H}^{2}_{\text{loc}}(\Omega)$ is said
to be outgoing if $\displaystyle\bm{X}$ and $\displaystyle\bm{Y}$ defined in
(4) satisfy

$\left|\langle\bm{Y},\bm{X}_{n}\rangle^{\prime}(x)\frac{x}{|x|}-ik_{n}\langle\bm{Y},\bm{X}_{n}\rangle(x)\right|\,,\,\left|\langle\bm{X},\bm{Y}_{n}\rangle^{\prime}(x)\frac{x}{|x|}-ik_{n}\langle\bm{X},\bm{Y}_{n}\rangle(x)\right|\underset{|x|\rightarrow+\infty}{\longrightarrow}0\quad\forall
n\in\mathbb{N}^{*}.$ (37)

This condition guarantees existence and uniqueness for the source problem (1)
as we prove below. For every $\displaystyle r>0$, we set
$\displaystyle\Omega_{r}:=(-r,r)\times(-h,h)$. We consider that any source
defined on $\displaystyle\Omega_{r}$ (resp. $\displaystyle(-r,r)$) is extended
by $\displaystyle 0$ on $\displaystyle\Omega$ (resp.
$\displaystyle\mathbb{R}$).

######  Theorem 2.

Let $\displaystyle r>0$. For every
$\displaystyle\omega\notin\omega_{\text{crit}}$,
$\displaystyle\bm{f}=(f_{1},f_{2})\in\text{H}^{1}(\Omega_{r})$ and
${}^{\text{top}}=(b^{\text{top}}_{1},b^{\text{top}}_{2}),^{\text{bot}}=(b^{\text{bot}}_{1},b^{\text{bot}}_{2})\in\widetilde{\text{H}}^{3/2}(-r,r)$,
the system

$\left\\{\begin{array}[]{cl}\nabla\cdot\bm{\sigma}(\bm{u})+\omega^{2}\bm{u}=-\bm{f}&\text{
in }\Omega,\\\ \bm{\sigma}(\bm{u})\cdot\nu=^{\text{top}/\text{bot}}&\text{ on
}\Omega_{\text{top}/\text{bot}},\\\ \bm{u}\text{ is
outgoing,}&\end{array}\right.$ (38)

has a unique solution
$\displaystyle\bm{u}\in\text{H}^{3}_{\text{loc}}(\Omega)$. This solution
admits a Lamb-mode decomposition

$u(x,z)=\sum_{n>0}a_{n}(x)u_{n}(z),\qquad v(x,z)=\sum_{n>0}b_{n}(x)v_{n}(z),$
(39)

where $\displaystyle a_{n},b_{n}$ are solutions to the decoupled Helmholtz
system

$\left\\{\begin{array}[]{c}a_{n}^{\prime\prime}+k_{n}^{2}a_{n}=ik_{n}F_{1}^{n}-{F_{2}^{n}}^{\prime},\\\
b_{n}^{\prime\prime}+k_{n}^{2}b_{n}={F_{1}^{n}}^{\prime}-ik_{n}F_{2},\end{array}\right.$
(40) $\text{where}\qquad
F_{i}^{n}(x)=\frac{1}{J_{n}}\left(\int_{-h}^{h}f_{i}(x,z)u_{n}(z)\mathrm{d}z+b_{i}^{\text{top}}(x)u_{n}(h)+b_{i}^{\text{bot}}(x)u_{n}(-h)\right),\quad
i\in\\{1,2\\}.\,\,\,$ (41)

Equivalently, $\displaystyle a_{n}=G_{1}^{n}\ast F_{1}^{n}-G_{2}^{n}\ast
F_{2}^{n}$ and $\displaystyle b_{n}=G_{2}^{n}\ast F_{1}^{n}-G_{1}^{n}\ast
F_{2}^{n}$ with

$G_{1}^{n}(x)=\frac{1}{2}e^{ik_{n}|x|},\qquad
G_{2}^{n}(x)=\frac{x}{2|x|}e^{ik_{n}|x|}.$ (42)

Moreover, there exists a constant $\displaystyle C>0$, which only depends on
$\displaystyle h$, $\displaystyle\omega$ and $\displaystyle r$, such that

$\|\bm{u}\|_{\text{H}^{3}(\Omega_{r})}\leq
C\left(\|\bm{f}\|_{\text{H}^{1}(\Omega)}+\|^{\text{top}}\|_{\text{H}^{3/2}(\mathbb{R})}+\|^{\text{bot}}\|_{\text{H}^{3/2}(\mathbb{R})}\right).$
(43)

###### Proof.

This proof is an adaptation of the proof presented in Appendix A of [11].

Step 1. We first show uniqueness of the solution. Assume that $\displaystyle
u$ solves (38) with $\displaystyle\bm{f}=0$,
${}^{\text{top}}=^{\text{bot}}=0$. The associated fields
$\displaystyle\bm{X},\bm{Y}$ defined in (4) can be decomposed as

$\bm{X}(x,z)=\sum_{n>0}a_{n}(x)\bm{X}_{n}(z),\qquad\bm{Y}(x,z)=\sum_{n>0}b_{n}(x)\bm{Y}_{n}(z).$
(44)

As proved in [26], the operators $\displaystyle F$ and $\displaystyle G$
defined in (5) are self adjoint on $\displaystyle H_{0}$. By projecting (4)
with the product $\displaystyle\langle\cdot,\cdot\rangle$ on
$\displaystyle\bm{X}_{n}$ and $\displaystyle\bm{Y}_{n}$, we see that

$a_{n}^{\prime}=ik_{n}b_{n},\qquad b_{n}^{\prime}=ik_{n}a_{n}.$ (45)

Solving this system of ODE’s and using the outgoing condition shows that
$\displaystyle a_{n}=b_{n}=0$, leading to $\displaystyle\bm{u}=0$.

Step 2. Assume that the functions $\displaystyle(u,v)$ defined in (39) is well
defined. A quick computation shows that the associated fields
$\displaystyle(\bm{X},\bm{Y})$ satisfy (5) for every mode since

${G^{n}_{1}}^{\prime}=ik_{n}G^{n}_{2},\qquad{G^{n}_{2}}^{\prime}=ik_{n}G^{n}_{1}+\delta_{0}.$
(46)

Assuming that $\displaystyle|x|>r$, we also see that

$\left|\langle\bm{X},\bm{Y}_{n}\rangle^{\prime}(x)\frac{x}{|x|}-ik_{n}\langle\bm{X},\bm{Y}_{n}\rangle(x)\right|\\\
=\left|\left(\frac{x}{|x|}{G_{1}^{n}}^{\prime}-ik_{n}G_{1}^{n}\right)\ast
F_{1}^{n}(x)-\left(\frac{x}{|x|}{G_{2}^{n}}^{\prime}-ik_{n}G_{2}^{n}\right)\ast
F_{2}^{n}(x)\right|=0.$ (47)

Repeating this computation for $\displaystyle\langle\bm{Y},\bm{X}_{n}\rangle$,
shows that $\displaystyle(u,v)$ satisfies the outgoing condition.

Step 3. We prove that the functions given by (39) are well-defined, in other
words that the series in (39) converge. We know from [28] that the number of
evanescent and propagative modes is finite, so we only need to study the
convergence of the inhomogeneous modes. We begin by noticing that if
$\displaystyle k_{n}$ is an inhomogeneous mode, then $\displaystyle
k_{m}=-\overline{k_{n}}$ also satisfies the dispersion relation, and
$\displaystyle X_{m}=\overline{X_{n}}$ and $\displaystyle
Y_{m}=\overline{Y_{n}}$. We index the subset of inhomogeneous wavenumbers with
positive real part by $\displaystyle j\in\mathbb{N}^{*}$, and all the
asymptotic comparison are now meant when $\displaystyle j\to+\infty$. Let
$\displaystyle N=2j-1/2$ (resp. $\displaystyle N=2j+1/2$) if $\displaystyle
k_{j}$ is associated to a symmetric (resp. antisymmetric) mode. Using [23], we
know that

$hk_{j}=\frac{1}{2}\ln(2\pi N)+i\frac{\pi N}{2}-i\frac{\ln(2\pi N)}{2\pi
N}+O\left(\frac{1}{N}\right).$ (48)

Let $\displaystyle p_{j}$ and $\displaystyle q_{j}$ be the quantities defined
in (10). We notice that $\displaystyle p_{j},q_{j}\sim-ik_{j}$. Since
$\displaystyle(p_{j}-ik_{j})(p_{j}+ik_{j})=\omega^{2}/(\lambda+2\mu)$ and
$\displaystyle(q_{j}-ik_{j})(q_{j}-ik_{j})=\omega^{2}/\mu$ it follows that

$p_{j}=-ik_{j}+\underbrace{\frac{\omega^{2}}{(\lambda+2\mu)\pi}}_{c_{p}}\frac{1}{N}+o\left(\frac{1}{N}\right),\quad
q_{j}=-ik_{j}+\underbrace{\frac{\omega^{2}}{\mu\pi}}_{c_{q}}\frac{1}{N}+o\left(\frac{1}{N}\right).$
(49)

We notice that

$\sin(ik_{j}z)=\frac{iz}{2|z|}\exp\left(\frac{|z|}{h}\left(\frac{1}{2}\ln(2\pi
N)+i\frac{\pi N}{2}-i\frac{\ln(2\pi N)}{2\pi
N}\right)\right)+O\left(\frac{1}{N^{1-|z|/2h}}\right),$ (50)
$\cos(ik_{j}z)=\frac{1}{2}\exp\left(\frac{|z|}{h}\left(\frac{1}{2}\ln(2\pi
N)+i\frac{\pi N}{2}-i\frac{\ln(2\pi N)}{2\pi
N}\right)\right)+O\left(\frac{1}{N^{1-|z|/2h}}\right),$ (51)

and if $\displaystyle\alpha$ stands for $\displaystyle p$ or $\displaystyle
q$,

$\cos(\alpha_{j}z)\sim\cos(ik_{j}z)+\frac{c_{\alpha}}{N}z\sin(ik_{j}z),\quad\sin(\alpha_{j}z)\sim-\sin(ik_{j}z)+\frac{c_{\alpha}}{N}z\cos(ik_{j}z).$
(52)

Using the definition of the symmetric modes (12), we find that
$\displaystyle(\bm{X}_{j}(z);\bm{Y}_{j}(z))\sim$

$\left(\begin{array}[]{c}h-|z|\\\ -i\mu\pi N(z-hz/|z|)\\\ \mu\pi N(h-|z|)\\\
i(z-hz/|z|)\end{array}\right)\frac{\pi^{3}N^{2}(c_{p}-c_{q})}{16}\exp\left(\left(\frac{|z|}{h}+1\right)\left(\frac{\ln(2\pi
N)}{2}+\frac{i\pi N}{2}-\frac{i\ln(2\pi N)}{2\pi N}\right)\right).$

and it follows that

$\|u_{j}\|_{\text{L}^{2}(-h,h)},\|v_{j}\|_{\text{L}^{2}(-h,h)}\sim\frac{\pi^{4}h^{3/2}N^{3}|c_{p}-c_{q}|}{4\ln(2\pi
N)^{3/2}},\quad\|u_{j}\|_{\text{L}^{\infty}(-h,h)},\|v_{j}\|_{\text{L}^{\infty}(-h,h)}\sim\frac{\pi^{5}hN^{2}|c_{p}-c_{q}|}{8},$
$\|t_{j}\|_{\text{L}^{2}(-h,h)},\|s_{j}\|_{\text{L}^{2}(-h,h)}\sim\frac{\pi^{5}h^{3/2}\mu
N^{4}|c_{p}-c_{q}|}{4\ln(2\pi
N)^{3/2}},\quad|J_{j}|\sim\frac{h^{2}\pi^{6}N^{5}\omega^{2}|c_{p}-c_{q}|}{8}.$

Similar estimates can be derived for the antisymmetric modes, which yield the
same asymptotic behaviors. Defining

$a_{j}=G_{1}^{j}\ast F_{1}^{j}-G_{2}^{j}\ast F_{2}^{j},\qquad
b_{j}=G_{2}^{j}\ast F_{1}^{j}-G_{1}^{j}\ast F_{2}^{j},$ (53)

we see using Young’s inequality that

$\|a_{j}\|_{\text{L}^{2}(-r,r)}\leq\|G_{1}^{j}\|_{\text{L}^{1}(-r,r)}\|F_{1}^{j}\|_{\text{L}^{2}(-r,r)}+\|G_{2}^{j}\|_{\text{L}^{1}(-r,r)}\|F_{2}^{j}\|_{\text{L}^{2}(-r,r)}.$
(54)

From the asymptotics of $\displaystyle k_{j}$ it follows that
$\displaystyle\|G_{1}^{j}\|_{\text{L}^{1}(\mathbb{R})},\|G_{2}^{j}\|_{\text{L}^{1}(\mathbb{R})}\leq
1/N$. Thus if $\displaystyle u^{p}_{j}$ denotes a primitive of $\displaystyle
u_{j}$, we see that

$\int_{-h}^{h}f_{1}(x,z)u_{j}(z)\mathrm{d}y=\left[f_{1}(x,z)u_{j}^{p}(z)\right]^{z=h}_{z=-h}-\int_{-h}^{h}\partial_{z}f_{1}(x,z)u_{j}^{p1}(z)\mathrm{d}z.$
(55)

Using the previous estimates, we find that $\displaystyle
u^{p}_{j}(z)\sim\tfrac{2}{i\pi N}u_{j}(z)$ and so there exists a constant
$\displaystyle c_{1}>0$, that depends on $\displaystyle\omega$ and
$\displaystyle h$, such that

$\|F_{1}^{j}\|_{\text{L}^{2}(-r,r)}\leq\frac{c_{1}}{N^{3}}\left(\|f_{1}\|_{\text{H}^{1}(\Omega)}+\|b_{1}^{\text{top}}\|_{\text{L}^{2}(\mathbb{R})}+\|b_{1}^{\text{bot}}\|_{\text{L}^{2}(\mathbb{R})}\right).$
(56)

We obtain a similar estimate for $\displaystyle F_{2}$. Finally, using the
triangular inequality,

$\|u\|_{\text{L}^{2}(\Omega_{r})}\leq\sum_{n|k_{n}\in\mathbb{R},i\mathbb{R}}\|a_{n}\|_{\text{L}^{2}(-r,r)}\|u_{n}\|_{\text{L}^{2}(-h,h)}+2\sum_{j\in\mathbb{N}^{*}}\|a_{j}\|_{\text{L}^{2}(-r,r)}\|u_{j}\|_{\text{L}^{2}(-h,h)},$
(57)

which leads to

$\|u\|_{\text{L}^{2}(\Omega)}\leq\Bigg{[}\sum_{n|k_{n}\in\mathbb{R},i\mathbb{R}}\frac{2r}{|J_{n}|}\left(\|u_{n}\|_{\text{L}^{2}(-h,h)}+\|v_{n}\|_{\text{L}^{2}(-h,h)}\right)\|u_{n}\|_{\text{L}^{2}(-h,h)}\\\
+2c_{1}\sum_{j\in\mathbb{N}^{*}}\frac{1}{(2j\pm 1/2)\ln(2j\pm
1/2)^{3/2}}\Bigg{]}\left(\|\bm{f}\|_{\text{H}^{1}(\mathbb{R})}+\|^{\text{top}}\|_{\text{L}^{2}(\mathbb{R})}+\|^{\text{bot}}\|_{\text{L}^{2}(\mathbb{R})}\right),$
(58)

A similar control holds for $\displaystyle v$. Elliptic regularity results
(see e.g. [15]) show that there exists a constant $\displaystyle c_{3}$
depending on $\displaystyle r$ such that

$\|\bm{u}\|_{\text{H}^{3}(\Omega_{r})}\leq
c_{3}\left(\|\bm{u}\|_{\text{L}^{2}(\Omega_{r})}+\|\bm{f}\|_{\text{H}^{1}(\mathbb{R})}+\|^{\text{top}}\|_{\text{H}^{3/2}(\mathbb{R})}+\|^{\text{bot}}\|_{\text{H}^{3/2}(\mathbb{R})}\right),$
(59)

which together with (58) conclude the proof. ∎

######  Remark 4.

This result is probably not optimal: indeed, in the scalar case one can merely
assume that the source term lies in $\displaystyle\text{L}^{2}(\Omega_{r})$
and obtain a solution in $\displaystyle\text{H}^{2}_{\text{loc}}(\Omega)$ (see
[11]). However, in the present case, the Lamb modes are not orthogonal and
Parseval equality does not hold, so that in the above proof, we controlled
terms using the triangular inequality, which may lead to a loss of accuracy.
We can see in the proof that the extra regularity of the source terms is
needed to derive (56), which in turn yields the convergence of the series
(57). Providing adaptation to elastic waveguides, the theory developed in [24]
may be better adapted to treat source terms with lower regularity.

To conclude, in this section we have constructed an explicit solution of the
elasticity problem in a regular waveguide, and have shown that its norm is
controlled by that of the source terms. Such estimates will be useful in the
following to perform the Born approximation.

##  3 Forward source problem in a regular 3D plate

In this section, we are interested in the forward source problem in a three-
dimensional regular waveguide with two infinite dimensions. Our motivation
comes from the experiments reported in [4], where the authors try to
reconstruct width defects in thin elastic plates. The propagation of waves in
three-dimension waveguides with one infinite dimension such as pipes or air
ducts is a direct generalization of the two dimensional case presented in the
previous section, see for instance [6]. Two main issues are at stake. First,
as mentioned in [19], in addition to longitudinal and transverse modes, one
needs to take into account horizontal shear modes in order to form a complete
modal basis. Second, one would like to generalize the
$\displaystyle(\bm{X},\bm{Y})$ formulation of Definition 1 to 3D.

Given an elastic wavefield $\displaystyle{\bf u}=(u,v,w)$ that propagates in a
three dimension plate, our main contribution consists in introducing two
auxiliary variables $\displaystyle\alpha$ and $\displaystyle\beta$ in (64)
that only depend on $\displaystyle u$ and $\displaystyle v$, which allow the
decoupling of the equations of elasticity. We show that
$\displaystyle(\alpha,w)$ can be decomposed using Lamb modes, while
$\displaystyle\beta$ represents the horizontal shear modes. This allows us to
obtain a generalization of Theorem 2 to three dimensional plates.

###  3.1 Decoupling of the linear elastic equation

Let us consider a 3D infinite elastic plate
$\displaystyle\Omega=\mathbb{R}^{2}\times(-h,h)$, where $\displaystyle h>0$ is
half of the waveguide thickness. For every $\displaystyle r>0$, we define
$\displaystyle\Omega_{r}=B_{2}(0,r)\times(-h,h)$ where $\displaystyle
B_{2}(0,r)$ is the ball in $\displaystyle\mathbb{R}^{2}$ centered at
$\displaystyle(0,0)$ with radius $\displaystyle r$. A point
$\displaystyle(x,y,z)\in\Omega$ will be denoted by $\displaystyle(\bm{x},z)$,
and the elastic displacement by $\displaystyle\bm{u}=(u,v,w)$. Given a
frequency $\displaystyle\omega\in\mathbb{R}$ and given
$\displaystyle(\lambda,\mu)$ the Lamé coefficients of the elastic waveguide,
the wavefield $\displaystyle\bm{u}$ satisfies

$\nabla\cdot\bm{\sigma}(\bm{u})+\omega^{2}\bm{u}=-\bm{f}\qquad\text{ in
}\Omega,$ (60)

where $\displaystyle\bm{f}=(f_{1},f_{2},f_{3})$ is a source term and
$\displaystyle\bm{\sigma}(\bm{u})$ is the stress tensor defined by

$\bm{\sigma}(\bm{u})=\left(\begin{array}[]{ccc}\begin{array}[]{l}(\lambda+2\mu)\partial_{x}u\\\
\quad+\lambda\partial_{y}v+\lambda\partial_{z}w\end{array}&\mu\partial_{y}u+\mu\partial_{x}v&\mu\partial_{z}u+\mu\partial_{x}w\\\
\mu\partial_{y}u+\mu\partial_{x}v&\begin{array}[]{l}(\lambda+2\mu)\partial_{y}v\\\
\quad+\lambda\partial_{x}u+\lambda\partial_{z}w\end{array}&\mu\partial_{z}v+\mu\partial_{y}w\\\
\mu\partial_{z}u+\mu\partial_{x}w&\mu\partial_{z}v+\mu\partial_{y}w&\begin{array}[]{l}(\lambda+2\mu)\partial_{z}w\\\
\quad+\lambda\partial_{x}u+\lambda\partial_{y}v\end{array}\end{array}\right).$
(61)

In the following, we study the case of Neumann boundary conditions

$\bm{\sigma}(\bm{u})\cdot\nu=^{\text{top}}\quad\text{ on
}\partial\Omega_{\text{top}},\qquad\bm{\sigma}(\bm{u})\cdot\nu=^{\text{bot}}\quad\text{
on }\partial\Omega_{\text{bot}},$ (62)

where
${}^{\text{top}}=(b_{1}^{\text{top}},b_{2}^{\text{bot}},b_{3}^{\text{bot}})$
and
${}^{\text{bot}}=(b_{1}^{\text{bot}},b_{2}^{\text{bot}},b_{3}^{\text{bot}})$
are boundary source terms. However, our analysis applies also to the case of
Dirichlet or Robin boundary conditions. We represent the set-up in Figure 4.

$\displaystyle e_{z}$$\displaystyle e_{y}$$\displaystyle
e_{x}$$\displaystyle\Omega$$\displaystyle-h$$\displaystyle
h$${}^{\text{top}}$${}^{\text{bot}}$$\displaystyle\bm{f}$ Figure 4:  The three
dimensional plate $\displaystyle\Omega$. Elastic wavefields are generated by
an internal source term $\displaystyle\bm{f}$ and by boundary source terms
${}^{\text{top}}$ and ${}^{\text{bot}}$.

To adapt the $\displaystyle\bm{X}/\bm{Y}$ formulation to 3D, we introduce the
following notations : for a vector field
$\displaystyle\bm{g}=(g_{1},g_{2},g_{3})$ and a scalar field $\displaystyle g$
we set

$\text{div}_{2}(\bm{g})=\partial_{x}g_{1}+\partial_{y}g_{2},\quad\text{curl}_{2}(\bm{g})=\partial_{x}g_{2}-\partial_{y}g_{1},\quad\Delta_{2}(g)=\partial_{xx}g+\partial_{yy}g.$
(63)

Given $\displaystyle{\bf u}=(u,v,w)\in H^{1}_{\text{loc}}(\Omega)$ we define

$\alpha=\text{div}_{2}(\bm{u}),\qquad\beta=\text{curl}_{2}(\bm{u}).$ (64)

Note that $\displaystyle\alpha$ and $\displaystyle\beta$ only involve the in-
plane components of $\displaystyle\bm{u}$. The following proposition shows how
these new variables decouple the elasticity system:

######  Proposition 4.

If $\displaystyle\bm{u}$ is solution of (60) with boundary conditions (62),
then $\displaystyle(\alpha,w)$ satisfies

$\left\\{\begin{array}[]{cl}(\lambda+2\mu)\Delta_{2}\alpha+\mu\partial_{zz}\alpha+\omega^{2}\alpha+(\lambda+\mu)\Delta_{2}\partial_{z}w=-\text{div}_{2}(\bm{f})&\text{
in }\Omega,\\\
\displaystyle(\lambda+\mu)\partial_{z}\alpha+(\lambda+2\mu)\partial_{zz}w+\mu\Delta_{2}w+\omega^{2}w=-f_{3}&\text{
in }\Omega,\\\
\displaystyle\partial_{z}\alpha+\mu\Delta_{2}w=\text{div}_{2}(^{\text{top}/\text{bot}})&\text{
on }\partial\Omega_{\text{top}/\text{bot}},\\\
\displaystyle(\lambda+2\mu)\partial_{z}w+\lambda\alpha=b_{3}^{\text{top}/\text{bot}}&\text{
on }\partial\Omega_{\text{top}/\text{bot}},\end{array}\right.$ (65)

while $\displaystyle\beta$ satisfies

$\left\\{\begin{array}[]{cl}\displaystyle\mu\Delta_{2}\beta+\mu\partial_{zz}\beta+\omega^{2}\beta=-\text{curl}_{2}(\bm{f})&\text{
in }\Omega,\\\
\displaystyle\mu\partial_{z}\beta=\text{curl}_{2}(^{\text{top}/\text{bot}})&\text{
on }\partial\Omega_{\text{top}/\text{bot}}.\end{array}\right.$ (66)

###### Proof.

If we denote $\displaystyle L_{i}$ the lines of (60) and $\displaystyle B_{i}$
the lines of (62), we compute
$\displaystyle\partial_{x}L_{1}+\partial_{y}L_{2}$,
$\displaystyle\partial_{x}B_{1}+\partial_{y}B_{2}$ and with $\displaystyle
L_{3}$ and $\displaystyle B_{3}$, we find (65). Then,
$\displaystyle\partial_{x}L_{2}-\partial_{y}L_{1}$ and
$\displaystyle\partial_{x}B_{2}-\partial_{y}B_{1}$ give (66). ∎

We start with the study of equation (66), which is a Helmholtz equation,
similar to that of acoustic waveguides, for the function $\displaystyle\beta$.
Inspired by [11], we introduce a decomposition of $\displaystyle\beta$ as a
sum of horizontal shear modes.

######  Definition 6.

For every $\displaystyle n\in\mathbb{N}$, we define
$\displaystyle\kappa_{n}^{2}=\omega^{2}/\mu-n^{2}\pi^{2}/4h^{2}$ with
$\displaystyle\text{Real}(\kappa_{n})\geq 0$ and
$\displaystyle\text{Imag}(\kappa_{n})\geq 0$. We define the $\displaystyle
n$-th shear horizontal mode (SH mode) $\displaystyle\varphi_{n}$ by

$\varphi_{n}(z):=\left\\{\begin{array}[]{cl}1/\sqrt{2h}&\text{ if }n=0,\\\
\frac{1}{\sqrt{h}}\cos\left(\frac{n\pi(z+h)}{2h}\right)&\text{
else.}\end{array}\right.$ (67)

The sequence $\displaystyle(\varphi_{n})_{n\geq 0}$ defines an orthonormal
basis of $\displaystyle\text{L}^{2}(-h,h)$ for the scalar product

$(g_{1}\,|\,g_{2}):=\int_{-h}^{h}g_{1}(x)g_{2}(x)\mathrm{d}x.$ (68)

Classical results on waveguides (see e.g. [12]) show that if one imposes a
Sommerfeld radiation condition [29], the problem (66) is well-posed except for
the frequencies

$\omega_{n}^{sh}=\frac{\sqrt{\mu}\pi}{2h}n,\quad n\geq 0.$ (69)

More precisely, when
$\displaystyle\omega\notin\omega_{\text{crit}}^{sh}:=\\{\omega_{n}^{sh},n\geq
0\\}$, the following result holds:

######  Proposition 5.

For every $\displaystyle\omega\notin\omega_{\text{crit}}^{sh}$,
$\displaystyle\bm{f}\in\text{H}^{1}(\Omega_{r})$ and
${}^{\text{top}},^{\text{bot}}\in\widetilde{H}^{3/2}(-r,r)$, the problem

$\left\\{\begin{array}[]{cl}\mu\Delta_{2}\beta+\mu\partial_{zz}\beta+\omega^{2}\beta=-\text{curl}_{2}(\bm{f})&\text{
in }\Omega,\\\
\mu\partial_{z}\beta=\text{curl}_{2}(^{\text{bot}/\text{top}})&\text{ on
}\partial\Omega_{\text{bot}/\text{top}},\\\
\sqrt{R}\left[\partial_{r}-i\kappa_{n}\right]\left(\beta\,|\,\varphi_{n}\right)(Re^{i\theta})\underset{R\rightarrow+\infty}{\longrightarrow}0\quad\forall
n\geq 0\quad\forall\theta\in(0,2\pi),\end{array}\right.$ (70)

has a unique solution $\displaystyle\beta\in\text{H}_{\text{loc}}^{2}(\Omega)$
which decomposes as

$\displaystyle\displaystyle\beta(\bm{x},z)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{n\geq 0}-(\Gamma^{n}\ast
F_{sh}^{n})(\bm{x})\varphi_{n}(z),$ (71)

where $\displaystyle\Gamma^{n}$ denotes the Hankel function of the first kind
$\displaystyle\Gamma^{n}(\bm{x})=-\tfrac{i}{4}\text{H}^{(1)}_{0}(\kappa_{n}|\bm{x}|)$,
and where

$\displaystyle\displaystyle F_{sh}^{n}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\mu}\left(\int_{-h}^{h}\text{curl}_{2}(\bm{f})\varphi_{n}+\text{curl}_{2}(^{\text{top}})\varphi_{n}(1)+\text{curl}_{2}(^{\text{bot}})\varphi_{n}(0)\right).$

###### Proof.

We follow the exact same steps as Appendix A in [11]. Since the
$\displaystyle\varphi_{n}$ form an orthonormal basis, any function
$\displaystyle\beta$ can be decomposed as
$\displaystyle\beta=\textstyle\sum\nolimits\beta_{n}\varphi_{n}$. Projecting
on the SH modes, the problem (70) is equivalent to the collection of problems
indexed by $\displaystyle n\in\mathbb{N}$

$\left\\{\begin{array}[]{cl}\Delta_{2}\beta_{n}+\kappa_{n}^{2}b_{n}=g_{sh}^{n}&\text{
in }\mathbb{R}^{2}\\\
\sqrt{r}\left[\partial_{r}-i\kappa_{n}\right]\beta_{n}(Re^{i\theta})\underset{r\rightarrow+\infty}{\longrightarrow}0\quad\forall\theta\in(0,2\pi).\end{array}\right.$
(72)

As $\displaystyle\Delta\Gamma^{n}+\kappa_{n}^{2}\Gamma^{n}=\delta_{0}$ [35],
each function $\displaystyle\beta_{n}$ can be expressed as the convolution
$\displaystyle\Gamma^{n}*F^{n}_{sh}$. The series (71) can be shown to converge
in $\displaystyle\text{L}^{2}_{\text{loc}}(\Omega)$ and provides a solution in
the sense of distributions to (70). Elliptic regularity allows then to show
that $\displaystyle\beta$ is actually in
$\displaystyle\text{H}^{2}_{\text{loc}}(\Omega)$. ∎

Next, we study equation (65), which resembles the two dimensional elasticity
system studied in section 2. We propose to adapt the
$\displaystyle\bm{X}/\bm{Y}$ formulation to this new equation. We define the
variables

$t=\mu\partial_{z}\alpha+\mu\Delta_{2}w,\quad
s=(\lambda+2\mu)\alpha+\lambda\partial_{z}w,\quad\bm{X}=(\alpha,t),\quad\bm{Y}=(-s,w).$
(73)

Then, we adapt Proposition 1:

######  Proposition 6.

The system (65) is equivalent to

$\left(\begin{array}[]{c}\bm{X}\\\
\Delta_{2}\bm{Y}\end{array}\right)={\mathcal{L}}(\bm{X},\bm{Y})+\left(\begin{array}[]{c}0\\\
-f_{3}-\delta_{z=h}b_{3}^{\text{top}}-\delta_{z=-h}b_{3}^{\text{bot}}\\\
\text{div}_{2}(\bm{f})+\delta_{z=h}\text{div}_{2}(^{\text{top}})+\delta_{z=-h}\text{div}_{2}(^{\text{bot}})\\\
0\end{array}\right)\quad\text{ in }\Omega,$ (74)

with $\displaystyle B1(\bm{X})=B2(\bm{Y})=0$, where
$\displaystyle{\mathcal{L}}(\bm{X},\bm{Y})=(F(\bm{Y});G(\bm{X}))$, and
$\displaystyle F$, $\displaystyle G$, $\displaystyle B_{1}$ and $\displaystyle
B_{2}$ are the same matrix operators as those defined in Proposition 1 in (6)
and (7).

We thus may use Lamb modes to diagonalize the operator
$\displaystyle{\mathcal{L}}$ as in the 2D situation, and obtain in this way a
result similar to Theorem 2. Let $\displaystyle u_{n},v_{n},k_{n}$ be defined
as in section 2 and assume that
$\displaystyle\omega\notin\omega_{\text{crit}}$.

######  Proposition 7.

For every $\displaystyle\omega\notin\omega_{\text{crit}}$,
$\displaystyle\bm{f}\in\text{H}^{1}(\Omega_{r})$ and
${}^{\text{top}},^{\text{bot}}\in\widetilde{H}^{3/2}(-r,r)$, the problem

$\left\\{\begin{array}[]{cl}(\lambda+2\mu)\Delta_{2}\alpha+\mu\partial_{zz}\alpha+\omega^{2}\alpha+(\lambda+\mu)\Delta_{2}\partial_{z}w=-\text{div}_{2}(\bm{f})&\text{
in }\Omega,\\\
\displaystyle(\lambda+\mu)\partial_{z}\alpha+(\lambda+2\mu)\partial_{zz}w+\mu\Delta_{2}w+\omega^{2}w=-f_{3}&\text{
in }\Omega,\\\
\displaystyle\partial_{z}\alpha+\mu\Delta_{2}w=\text{div}_{2}(^{\text{top}/\text{bot}})&\text{
on }\partial\Omega_{\text{top}/\text{bot}},\\\
\displaystyle(\lambda+2\mu)\partial_{z}w+\lambda\alpha=b_{3}^{\text{top}/\text{bot}}&\text{
on }\partial\Omega_{\text{top}/\text{bot}},\\\
\sqrt{R}\left[\partial_{r}-ik_{n}\right]\langle\bm{Y},\bm{X}_{n}\rangle(Re^{i\theta})\underset{R\rightarrow+\infty}{\longrightarrow}0\quad\forall
n>0\quad\forall\theta\in(0,2\pi),&\\\
\sqrt{R}\left[\partial_{r}-ik_{n}\right]\langle\bm{X},\bm{Y}_{n}\rangle(Re^{i\theta})\underset{R\rightarrow+\infty}{\longrightarrow}0\quad\forall
n>0\quad\forall\theta\in(0,2\pi),\end{array}\right.$ (75)

has a unique solution
$\displaystyle(\alpha,w)\in\text{H}^{2}_{\text{loc}}(\Omega)\times\text{H}^{3}_{\text{loc}}(\Omega)$
which decomposes as

$\alpha(\bm{x},z)=\sum_{n>0}((k_{n}^{2}F_{3}^{n}+ik_{n}F_{1}^{n})\ast
G^{n})u_{n}(z),\quad w(\bm{x},z)=\sum_{n>0}((-ik_{n}F_{3}^{n}+F_{1}^{n})\ast
G^{n})v_{n}(z),$ (76)

where $\displaystyle
G^{n}(\bm{x})=-\tfrac{i}{4}\text{H}^{(1)}_{0}(k_{n}|\bm{x}|)$ and

$F_{1}^{n}=\frac{1}{J_{n}}\left(\int_{-h}^{h}\text{div}_{2}(\bm{f})u_{n}+\text{div}_{2}(^{\text{top}})u_{n}(h)+\text{div}_{2}(^{\text{bot}})u_{n}(-h)\right),$
(77)
$F_{3}^{n}=\frac{1}{J_{n}}\left(\int_{-h}^{h}f_{3}v_{n}+b_{3}^{\text{top}}v_{n}(h)+b_{3}^{\text{bot}}v_{n}(-h)\right).$
(78)

###### Proof.

The proof is very similar to that of Theorem 2. We start by decomposing
$\displaystyle(\bm{X},\bm{Y})$ in the form

$\displaystyle\displaystyle\bm{X}(x,y,z)\;=\;\sum_{n>0}a_{n}(x,y)\bm{X}_{n}(z),$
$\displaystyle\displaystyle\bm{Y}(x,y,z)\;=\;\sum_{n>0}b_{n}(x,y)\bm{Y}_{n}(z).$
(79)

Injecting in (74), one obtains, instead of (45) in the 2D case,

$a_{n}=ik_{n}b_{n},\quad\Delta_{2}b_{n}=ik_{n}a_{n}.$ (80)

If (76) is well defined, it satisfies (75) since
$\displaystyle\Delta_{2}G^{n}=-k_{n}^{2}G^{n}+\delta_{0}$. The study of the
asymptotic behavior of $\displaystyle a_{n}$ and $\displaystyle b_{n}$ can be
performed as in section 2, using the fact that

$\|k_{n}^{2}G^{n}\|_{\text{L}^{1}(B(0,r))}=\mathcal{O}(1),\quad\|ik_{n}G^{n}\|_{\text{L}^{1}(B(0,r))}=\mathcal{O}(1/n),\quad\|G^{n}\|_{\text{L}^{1}(B(0,r))}=\mathcal{O}(1/n^{2}).$
(81)

It follows that the sum of series (79) are in
$\displaystyle\text{L}^{2}_{\text{loc}}(\Omega)$ and provide a solution of
(75) in the sense of distributions. Rewriting this system as an elliptic
system

$\displaystyle\left\\{\begin{array}[]{cl}(\lambda+2\mu)\Delta_{2}\alpha+\mu\partial_{zz}(\alpha)=R(w,{\bf
f})&\quad\text{in}\;\Omega,\\\
(\lambda+2\mu)\partial_{zz}w+(\lambda+2\mu)\Delta_{2}w=S(\alpha,f_{3})&\quad\text{in}\;\Omega,\\\
\partial_{z}\alpha=r(\Delta_{2}w,{\bf
b}^{\text{top}/\text{bot}})&\quad\text{on}\;\partial\Omega_{\text{top}/\text{bot}},\\\
\partial_{z}w=s(\alpha,b_{3}^{\text{top}/\text{bot}})&\quad\text{on}\;\partial\Omega_{\text{top}/\text{bot}},\end{array}\right.$

using elliptic regularity [15] and a bootstrap argument, one further infers
that $\displaystyle\alpha,w\in\text{H}^{2}_{loc}(\Omega)$, which concludes the
proof of the Proposition. ∎

###  3.2 Helmholtz-Hodge decomposition

Now that equations (65) and (66) are solved, we return to equations (60) and
(62). We need to ensure that given the expressions of $\displaystyle\alpha$
and $\displaystyle\beta$, we can recover a unique expression for
$\displaystyle u$ and $\displaystyle v$. To this end, we use the Helmholtz-
Hodge decomposition, which states that under certain conditions, a vector
field can be decomposed in a unique way as the sum of a curl-free and
divergence-free fields. This decomposition is mostly used in fluid mechanics
to analyze three dimensional vector fields (see for instance [9, 13]). In our
case, we apply it to two dimensional vector fields, since we are only
interested in finding a link between $\displaystyle(u,v)$ and
$\displaystyle(\alpha,\beta)$. We give the corresponding statement below, the
proof of which and be found in [27] concerning existence, while uniqueness is
addressed in [32].

######  Proposition 8.

Every vector field
$\displaystyle\bm{\xi}\in\text{H}^{1}_{\text{loc}}(\mathbb{R}^{3},\mathbb{C}^{2})$,
vanishing at infinity, can be uniquely decomposed as
$\displaystyle\bm{\xi}=\bm{d}+\bm{c}$ where
$\displaystyle\text{curl}_{2}(\bm{d})=0$ and
$\displaystyle\text{div}_{2}(\bm{c})=0$. The couple
$\displaystyle(\bm{d},\bm{c})$ is called the Helmholtz-Hodge decomposition
(HHD) of $\displaystyle\bm{\xi}$. Moreover, $\displaystyle\bm{\xi}$ is
uniquely determined by $\displaystyle\text{div}_{2}(\bm{\xi})$ and
$\displaystyle\text{curl}_{2}(\bm{\xi})$.

Providing enough regularity on $\displaystyle\alpha$ and $\displaystyle\beta$,
one can compute $\displaystyle(u,v)$ using the formula [27]

$(u,v)=-\nabla(\mathcal{G}(\alpha))+\nabla\times(\mathcal{G}(\beta)),$ (82)

where $\displaystyle\mathcal{G}$ represents the Newtonian potential operator
which convolves each function with $\displaystyle
x\mapsto\log(|\bm{x}|)/(2\pi)$. However, in the following, we will not need to
use this formula. Indeed, we exhibit expressions of $\displaystyle u$ and
$\displaystyle v$ that satisfy $\displaystyle\text{div}_{2}(\bm{u})=\alpha$
and $\displaystyle\text{curl}_{2}(\bm{u})=\beta$ and thus are the ones we look
for thanks to the previous uniqueness result.

We introduce an outgoing radiation condition for 3D wavefields:

######  Definition 7.

A wavefield $\displaystyle\bm{u}\in\text{H}^{2}_{\text{loc}}(\Omega)$ is said
to be outgoing if it vanishes at infinity, and if $\displaystyle\bm{X},\bm{Y}$
defined in (73) and $\displaystyle\beta$ defined in (64) satisfy

$\sqrt{R}\left[\partial_{r}-ik_{n}\right]\langle\bm{X},\bm{Y}_{n}\rangle(Re^{i\theta})\,,\,\sqrt{R}\left[\partial_{r}-ik_{n}\right]\langle\bm{Y},\bm{X}_{n}\rangle(Re^{i\theta})\underset{R\rightarrow+\infty}{\longrightarrow}0\quad\forall
n>0\quad\forall\theta\in(0,2\pi),$ (83)
$\sqrt{R}\left[\partial_{r}-i\kappa_{n}\right]\left(\beta\,|\,\varphi_{n}\right)(Re^{i\theta})\underset{R\rightarrow+\infty}{\longrightarrow}0\quad\forall
n\geq 0\quad\forall\theta\in(0,2\pi).$ (84)

Under this condition, uniqueness of solutions to the source problem in 3D will
be guaranteed, as stated in the next Theorem. We first introduce some
notations. We define the scalar convolution by

$\bm{g_{1}}\ast\cdot\bm{g_{2}}(x)=\int_{\mathbb{R}^{2}}\bm{g_{1}}(\bm{x}-\bm{y})\cdot\bm{g_{2}}(\bm{y})\mathrm{d}\bm{y},$
(85)

and introduce the Green functions

$G_{1}^{n}(\bm{x})=-\frac{i}{4}\text{H}^{(1)}_{0}(k_{n}|\bm{x}|),\quad\bm{G_{2}}^{n}(\bm{x})=\nabla
G_{1}^{n}(\bm{x}),\quad
G_{3}^{n}(\bm{x})=-\frac{i}{4}\text{H}^{(1)}_{0}(\kappa_{n}|\bm{x}|).$ (86)

Let $\displaystyle(\bm{f}^{L},\bm{f}^{sh})$,
$\displaystyle(\bm{b_{\text{top}}}^{L},\bm{b_{\text{top}}}^{sh})$,
$\displaystyle(\bm{b_{\text{bot}}}^{L},\bm{b_{\text{bot}}}^{sh})$ denote the
HHD of $\displaystyle(f_{1},f_{2})$,
$\displaystyle(b_{1}^{\text{top}},b_{2}^{\text{top}})$,
$\displaystyle(b_{1}^{\text{bot}},b_{2}^{\text{bot}})$ respectively and set

$g_{n}^{z}(\bm{x})=\frac{1}{J_{n}}\left(\int_{-h}^{h}f_{3}(\bm{x},z)v_{n}(z)\mathrm{d}z+b_{3}^{\text{top}}(\bm{x})v_{n}(h)+b_{3}^{\text{bot}}(\bm{x})v_{n}(-h)\right),$
(87)
$\bm{g}_{n}^{\ell}=\frac{1}{J_{n}}\left(\int_{-h}^{h}\bm{f}^{\ell}(\bm{x},z)u_{n}(z)\mathrm{d}z+\bm{b}_{\text{top}}^{\ell}u_{n}(h)+\bm{b}_{\text{bot}}^{\ell}u_{n}(-h)\right),\quad\ell\in\\{L,sh\\}.$
(88)

######  Theorem 3.

Let $\displaystyle r>0$. For every
$\displaystyle\omega\notin\omega_{\text{crit}}\cup\omega_{\text{crit}}^{sh}$,
$\displaystyle\bm{f}\in\text{H}^{3}(\Omega_{r})$ and
${}^{\text{top}},^{\text{bot}}\in\widetilde{H}^{3/2}(-r,r)$, the problem

$\left\\{\begin{array}[]{cl}\nabla\cdot\bm{\sigma}(\bm{u})+\omega^{2}\bm{u}=-\bm{f}&\text{
in }\Omega,\\\ \bm{\sigma}(\bm{u})\cdot\nu=^{\text{top}/\text{bot}}&\text{ on
}\Omega_{\text{top}/\text{bot}},\\\ \bm{u}\text{ is
outgoing,}&\end{array}\right.$ (89)

has a unique solution
$\displaystyle\bm{u}\in\text{H}^{3}_{\text{loc}}(\Omega)$. This solution
admits a decomposition $\displaystyle\bm{u}=\bm{u}^{L}+\bm{u}^{sh}$ with

$\bm{u}^{L}(\bm{x},z)=\left(\begin{array}[]{c}\sum_{n>0}\bm{A}_{n}(\bm{x})u_{n}(z)\\\
\sum_{n>0}b_{n}(\bm{x})v_{n}(z)\end{array}\right),\quad\bm{u}^{Sh}(\bm{x},z)=\left(\begin{array}[]{c}\sum_{n\geq
0}\bm{C}_{n}(\bm{x})\varphi_{n}(z)\\\ 0\end{array}\right),$ (90)

where $\displaystyle\bm{A}_{n},b_{n},\bm{C}_{n}$ satisfy the equations

$\left\\{\begin{array}[]{c}\Delta_{2}\bm{A}_{n}+k_{n}^{2}\bm{A}_{n}=-\nabla
g_{n}^{z}+ik_{n}\bm{g}^{L}_{n},\\\
\Delta_{2}\bm{C}_{n}+\kappa_{n}^{2}\bm{C}_{n}=-\bm{g}^{sh}_{n},\\\
\Delta_{2}b_{n}+k_{n}^{2}b_{n}=-ik_{n}g_{n}^{z}+\text{div}_{2}(\bm{g}^{L}_{n}).\end{array}\right.$
(91)

Equivalently,
$\displaystyle\bm{A}_{n}=-g_{n}^{z}\ast\bm{G_{2}}^{n}+ik_{n}\bm{g}_{n}^{L}\ast
G_{1}^{n}$, $\displaystyle\bm{C}_{n}=-\bm{g}_{n}^{sh}/\mu\ast G_{3}^{n}$ and
$\displaystyle b_{n}=-ik_{n}g_{n}^{z}\ast
G^{n}_{1}+\bm{g}^{L}_{n}\ast\cdot\,\bm{G_{2}}^{n}$. Moreover, there exists a
constant $\displaystyle C>0$ depending only on $\displaystyle h$,
$\displaystyle\omega$ and $\displaystyle r$ such that

$\|\bm{u}\|_{\text{H}^{3}(\Omega_{r})}\leq
C\left(\|\bm{f}\|_{\text{H}^{1}(\Omega)}+\|^{\text{top}}\|_{\text{H}^{3/2}(\mathbb{R})}+\|^{\text{bot}}\|_{\text{H}^{3/2}(\mathbb{R})}\right).$
(92)

###### Proof.

If $\displaystyle\bm{f}$, ${}^{\text{top}}$ and ${}^{\text{bot}}$ vanish,
uniqueness in Propositions 7 and 5 show that $\displaystyle\alpha=\beta=w=0$.
Since $\displaystyle u$ and $\displaystyle v$ are uniquely determined by
$\displaystyle\text{div}_{2}(u,v)$ and $\displaystyle\text{curl}_{2}(u,v)$, it
follows that $\displaystyle u=v=0$ and the uniqueness of a solution is
established. Moreover, Propositions 7 and 5 provide expressions of
$\displaystyle\alpha$, $\displaystyle\beta$ and $\displaystyle w$. Using the
HHD, there exists a unique wavefield $\displaystyle\bm{u}$ determined by
$\displaystyle(\alpha,\beta)$ and we can check that expressions provided in
(90) indeed provide a solution. Finally, the control of the wavefield with
respect to source terms is obtained in the same manner as in the proof of
Theorem 2, using the following estimates on the Green functions:

$\|G_{1}^{n}\|_{\text{L}^{1}(B(0,r))}=\mathcal{O}(1/n^{2}),\quad\|\bm{G_{2}}^{n}\|_{\text{L}^{1}(B(0,r))}=\mathcal{O}(1/n).$
(93)

∎

##  4 Reconstruction of small shape defects from multi-frequency measurements
in 2D

In this section, we consider the inverse problem of reconstructing of small
shape defects in an elastic plate from multi-frequency surface measurements.
We first detail the method used in the two dimensional case. Its 3D
generalization is discussed at the end of the section. We follow the method
developed in [11] for acoustic waveguides. In the present case, given current
experimental setups [19], we assume that the measurements consist in surface
measurements of the displacement fields, rather than measurements in a section
of the waveguide, which were considered in the acoustic case.

We consider a plate $\displaystyle\widetilde{\Omega}$ that contains localized
bumps, defined by

$\displaystyle\displaystyle\widetilde{\Omega}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\\{(x,z)\in\mathbb{R}^{2}\,|\,h(-1+2g_{2}(x))<z<h(1+2g_{1}(x))\\},$

where $\displaystyle g_{1},g_{2}$ are $\displaystyle{\mathcal{C}^{2}}$
functions with compact support, such that
$\displaystyle-1+2g_{2}(x)<1+2g_{1}(x)$, see Figure 5. Note that
$\displaystyle g_{1}$ and $\displaystyle g_{2}$ are not required to have a
constant sign. Hereafter $\displaystyle\widetilde{\Omega}$ is called the
perturbed plate.

$\displaystyle-h$$\displaystyle
h$$\displaystyle\bm{u}^{\text{inc}}$$\displaystyle 2hg_{1}(x)$$\displaystyle
2hg_{2}(x)$$\displaystyle\widetilde{\Omega}$ Figure 5:  Representation of
shape defects in a plate of width $\displaystyle h$.

An incident wavefield $\displaystyle\bm{u}^{\text{inc}}$ is send in the
waveguide, and we denote by $\displaystyle\widetilde{\bm{u}}$ the total
wavefield and
$\displaystyle\widetilde{\bm{u}}^{s}:=\widetilde{\bm{u}}-\bm{u}^{\text{inc}}$
the scattered wavefield. Our goal is to reconstruct functions $\displaystyle
g_{1}$ and $\displaystyle g_{2}$ from the wavefields scattered by the defects,
and to solve the inverse problem

$\text{Find }\quad(g_{1},g_{2})\quad\text{from
}\quad\widetilde{\bm{u}}^{s}(\omega,x,h(x))\qquad\forall
x\in\mathbb{R}\quad\forall\omega\in(0,\omega_{\max}).$ (94)

###  4.1 Born approximation

As incident wave, we use the function corresponding to the first symmetric
Lamb mode of a straight guide, which we denote by
$\displaystyle\bm{u}^{\text{inc}}(x,z):=\bm{u}_{1}(z)e^{ik_{1}x}$. The same
analysis could be conducted with any other Lamb mode. However, note that the
first symmetric Lamb mode has the advantage to propagate at any frequency,
which is not the case for the other symmetric Lamb modes. The total wavefield
$\displaystyle\widetilde{\bm{u}}$ solves the equations of elasticity in the
waveguide:

$\left\\{\begin{array}[]{cl}\nabla\cdot\bm{\sigma}(\widetilde{\bm{u}})+\omega^{2}\widetilde{\bm{u}}=0&\text{
in }\widetilde{\Omega},\\\ \bm{\sigma}(\widetilde{\bm{u}})\cdot\nu=0&\text{ on
}\partial\widetilde{\Omega}_{\text{top}/\text{bot}}.\end{array}\right.$ (95)

Then, the scattered wavefield solves

$\left\\{\begin{array}[]{cl}\nabla\cdot\bm{\sigma}(\widetilde{\bm{u}}^{s})+\omega^{2}\widetilde{\bm{u}}^{s}=0&\text{
in }\widetilde{\Omega},\\\
\bm{\sigma}(\widetilde{\bm{u}}^{s})\cdot\nu=-\bm{\sigma}(\bm{u}^{\text{inc}})\cdot\nu&\text{
on }\partial\widetilde{\Omega}_{\text{top}/\text{bot}},\\\ \bm{u}^{s}\text{ is
outgoing.}\end{array}\right.$ (96)

For every $\displaystyle(x,z)\in\partial\widetilde{\Omega}$, we know, using
the definition of Lamb modes, that

$\bm{\sigma}(\bm{u}^{\text{inc}})=\left(\begin{array}[]{cc}s_{1}(z)&t_{1}(z)\\\
t_{1}(z)&r_{1}(z)\end{array}\right)e^{ik_{1}x},$ (97)

so if $\displaystyle(x,z)\in\partial\widetilde{\Omega}_{\text{top}}$ then
$\displaystyle\bm{\sigma}(\widetilde{\bm{u}}^{s})\cdot\nu$ is equal to

$-\bm{\sigma}({\widetilde{\bm{u}}}^{\text{inc}})\cdot\frac{1}{\sqrt{1+4h^{2}g_{1}^{\prime}(x)^{2}}}\left(\begin{array}[]{c}-2hg_{1}^{\prime}(x)\\\
1\end{array}\right)=-\frac{e^{ik_{1}x}}{\sqrt{1+4h^{2}g_{1}^{\prime}(x)^{2}}}\left(\begin{array}[]{c}-2hg_{1}^{\prime}(x)s_{1}(z)+t_{1}(z)\\\
-2hg_{1}^{\prime}(x)t_{1}(z)+r_{1}(z)\end{array}\right).$ (98)

We can also do the same thing on
$\displaystyle\partial\widetilde{\Omega}_{\text{bot}}$ to explicit the
equation of the scattered wavefield. Then, following the steps of [11], we map
the perturbed waveguide $\displaystyle\widetilde{\Omega}$ to a regular
waveguide $\displaystyle\Omega:=\mathbb{R}\times(-h,h)$ using the mapping

$\phi(x,z)=\left(x,\left(1+g_{1}(x)-g_{2}(x)\right)z+hg_{1}(x)+hg_{2}(x)\right).$
(99)

We define $\displaystyle\bm{u}^{s}=\widetilde{\bm{u}}^{s}\circ\phi$. Equation
(96) in the perturbed waveguide $\displaystyle\widetilde{\Omega}$ is
equivalent to the following equation in the regular waveguide
$\displaystyle\Omega$:

$\left\\{\begin{array}[]{cl}\nabla\cdot\bm{\sigma}(\bm{u}^{s})+\omega^{2}\bm{u}^{s}=-a(\bm{u}^{s})&\text{
in }\Omega,\\\
\bm{\sigma}(\bm{u}^{s})\cdot\nu=\left(\begin{array}[]{c}2hg_{1}^{\prime}(x)s_{1}(\phi(x,z))-t_{1}(\phi(x,z))\\\
2hg_{1}^{\prime}(x)t_{1}(\phi(x,z))-r_{1}(\phi(x,z))\end{array}\right)e^{ik_{1}x}-b_{1}(\bm{u}^{s})&\text{
on }\partial\Omega_{\text{top}},\\\
\bm{\sigma}(\bm{u}^{s})\cdot\nu=\left(\begin{array}[]{c}-2hg_{2}^{\prime}(x)s_{1}(\phi(x,z))-t_{1}(\phi(x,z))\\\
-2hg_{2}^{\prime}(x)t_{1}(\phi(x,z))-r_{1}(\phi(x,z))\end{array}\right)e^{ik_{1}x}-b_{2}(\bm{u}^{s})&\text{
on }\partial\Omega_{\text{bot}},\\\ \bm{u}^{s}\text{ is
outgoing,}\end{array}\right.$ (100)

where we denote $\displaystyle f_{1}(x)=hg_{1}(x)+hg_{2}(x)$, $\displaystyle
f_{2}(x)=1+g_{1}(x)-g_{2}(x)$,

$a(\bm{u})=-\frac{f_{1}^{\prime}f_{2}+f_{2}^{\prime}(z-f_{1})}{{f_{2}}^{2}}\left(\begin{array}[]{c}2(\lambda+2\mu)\partial_{xz}u+(\lambda+\mu)\partial_{zz}v\\\
2\mu\partial_{xz}v+(\lambda+\mu)\partial_{zz}u\end{array}\right)-\frac{f_{2}^{\prime}}{{f_{2}}^{2}}\left(\begin{array}[]{c}(\lambda+2\mu)\partial_{z}v\\\
\mu\partial_{z}u\end{array}\right)\\\
+\frac{(f_{1}^{\prime}f_{2}+f_{2}^{\prime}(z-f_{1}))^{2}}{{f_{2}}^{4}}\left(\begin{array}[]{c}(\lambda+2\mu)\partial_{zz}u\\\
\mu\partial_{zz}v\end{array}\right)+\left(\frac{1}{f_{2}}-1\right)\left(\begin{array}[]{c}(\lambda+\mu)\partial_{xz}v\\\
(\lambda+\mu)\partial_{xz}u\end{array}\right)\\\
-\frac{f_{1}^{\prime\prime}{f_{2}}^{2}+(f_{2}^{\prime\prime}-2{f_{2}^{\prime}}^{2}f_{2})(z-f_{1})-2f_{2}^{\prime}f_{1}^{\prime}f_{2}}{{f_{2}}^{3}}\left(\begin{array}[]{c}(\lambda+2\mu)\partial_{y}u\\\
\mu\partial_{z}v\end{array}\right)+\left(\frac{1}{{f_{2}}^{2}}-1\right)\left(\begin{array}[]{c}\mu\partial_{zz}u\\\
(\lambda+2\mu)\partial_{zz}v\end{array}\right),$ (101)

and

$b_{1}(\bm{u})=-g_{1}^{\prime}(\lambda+2\mu)\partial_{x}u-\lambda\frac{g_{1}^{\prime}}{f_{2}}\partial_{z}v+\mu\left(\frac{1}{f_{2}}-1\right)\partial_{z}u+\frac{f_{1}^{\prime}f_{2}+f_{2}^{\prime}(z-f_{1})}{{f_{2}}^{2}}(g_{1}^{\prime}(\lambda+2\mu)\partial_{z}u-\mu\partial_{z}v),$
(102)
$b_{2}(\bm{u})=g_{2}^{\prime}\mu\partial_{x}v+\mu\frac{g_{2}^{\prime}}{f_{2}}\partial_{z}u-(\lambda+2\mu)\left(\frac{1}{f_{2}}-1\right)\partial_{z}v-\frac{f_{1}^{\prime}f_{2}+f_{2}^{\prime}(z-f_{1})}{{f_{2}}^{2}}(g_{2}^{\prime}\mu\partial_{z}v-\lambda\partial_{z}u).$
(103)

From now on, we only consider small shape defects, i.e. we assume that the
quantity

$\varepsilon=\max\left(\|g_{1}\|_{\text{W}^{2,\infty}(\mathbb{R})},\|g_{2}\|_{\text{W}^{2,\infty}(\mathbb{R})}\right),$
(104)

is small compared to the size of the supports of $\displaystyle g_{1}$ and
$\displaystyle g_{2}$, and compared to the width of the waveguide. A direct
computation leads to the following bounds for operators $\displaystyle a$ and
$\displaystyle b$:

######  Proposition 9.

For every $\displaystyle r>0$, there exist two constants $\displaystyle A,B>0$
depending only on $\displaystyle\omega$, $\displaystyle h$ and $\displaystyle
r$ such that

$\|a(\bm{u})\|_{\text{H}^{1}(\Omega_{r})}\leq
A\varepsilon\|\bm{u}\|_{\text{H}^{3}(\Omega_{r})},\quad\|b(\bm{u})\|_{\text{H}^{3/2}(\Omega_{r})}\leq
B\varepsilon\|\bm{u}\|_{\text{H}^{3}(\Omega_{r})}.$ (105)

Following the steps of [11], we define the Born approximation
$\displaystyle\bm{v}$ of $\displaystyle\bm{u}^{s}$ by

$\left\\{\begin{array}[]{cl}\nabla\cdot\bm{\sigma}(\bm{v})+\omega^{2}\bm{v}=0&\text{
in }\Omega,\\\
\bm{\sigma}(\bm{v})\cdot\nu=\left(\begin{array}[]{c}2hg_{1}^{\prime}(x)s_{1}(\phi(x,z))-t_{1}(\phi(x,z))\\\
2hg_{1}^{\prime}(x)t_{1}(\phi(x,z))-r_{1}(\phi(x,z))\end{array}\right)e^{ik_{1}x}&\text{
on }\partial\Omega_{\text{top}},\\\
\bm{\sigma}(\bm{v})\cdot\nu=\left(\begin{array}[]{c}-2hg_{2}^{\prime}(x)s_{1}(\phi(x,z))-t_{1}(\phi(x,z))\\\
-2hg_{2}^{\prime}(x)t_{1}(\phi(x,z))-r_{1}(\phi(x,z))\end{array}\right)e^{ik_{1}x}&\text{
on }\partial\Omega_{\text{bot}},\\\ \bm{v}\text{ is
outgoing.}\end{array}\right.$ (106)

The following proposition, the proof of which is similar to Propositions 5 and
6 of [11], shows that $\displaystyle\bm{v}$ is a good approximation of
$\displaystyle\bm{u}$ if the defect is small:

######  Proposition 10.

Let $\displaystyle C>0$ be the constant defined in Theorem 2. If
$\displaystyle\varepsilon C(A+B)<1$ then (100) has a unique solution
$\displaystyle\bm{u}^{s}$ and

$\|\bm{u}^{s}-\bm{v}\|_{\text{H}^{3}(\Omega_{r})}\leq\frac{\varepsilon
C(A+B)}{1-\varepsilon
C(A+B)}4rCh\varepsilon(\|s_{1}\|_{\text{H}^{2}}+\|t_{1}\|_{\text{H}^{2}}+\|r_{1}\|_{\text{H}^{2}}).$
(107)

Finally, to simplify the boundary source term and get rid of the dependency on
$\displaystyle\phi$, we notice that

$g_{1}^{\prime}(x)s_{1}(\phi(x,z))=g_{1}^{\prime}(x)s_{1}(h)+\mathcal{O}(\varepsilon^{2}),\quad
t_{1}(\phi(x,z))=(g_{1}^{\prime}(x)-g_{2}^{\prime}(x))\partial_{z}t_{1}(h)+\mathcal{O}(\varepsilon^{2}),$
(108) $g_{1}^{\prime}(x)t_{1}(\phi(x,z))=\mathcal{O}(\varepsilon^{2}),\quad
r_{1}(\phi(x,z))=(g_{1}^{\prime}(x)-g_{2}^{\prime}(x))\partial_{z}r_{1}(h)+\mathcal{O}(\varepsilon^{2}).$
(109)

We define a simpler approximation $\displaystyle\bm{w}$ of
$\displaystyle\bm{v}$ as the solution of

$\left\\{\begin{array}[]{cl}\nabla\cdot\bm{\sigma}(\bm{w})+\omega^{2}\bm{w}=0&\text{
in }\Omega,\\\
\bm{\sigma}(\bm{w})\cdot\nu=\left(\begin{array}[]{c}2hg_{1}^{\prime}(x)s_{1}(h)-(g_{1}^{\prime}(x)-g_{2}^{\prime}(x))\partial_{z}t_{1}(h)\\\
-(g_{1}^{\prime}(x)-g_{2}^{\prime}(x))\partial_{z}r_{1}(h)\end{array}\right)e^{ik_{1}x}&\text{
on }\partial\Omega_{\text{top}},\\\
\bm{\sigma}(\bm{w})\cdot\nu=\left(\begin{array}[]{c}-2hg_{2}^{\prime}(x)s_{1}(h)-(g_{1}^{\prime}(x)-g_{2}^{\prime}(x))\partial_{z}t_{1}(h)\\\
(g_{1}^{\prime}(x)-g_{2}^{\prime}(x))\partial_{z}r_{1}(h)\end{array}\right)e^{ik_{1}x}&\text{
on }\partial\Omega_{\text{bot}},\\\ \bm{w}\text{ is
outgoing.}\end{array}\right.$ (110)

Using the control provided by Theorem 1, $\displaystyle\bm{w}$ is a good
approximation of $\displaystyle\bm{v}$ if $\displaystyle\varepsilon$ is small
enough and there exists a constant $\displaystyle D>0$ such that

$\|\bm{v}-\bm{w}\|_{\text{H}^{3}(\Omega_{r})}\leq\varepsilon^{2}Dhr\left(\|s_{1}\|_{\text{H}^{2}}+\|\partial_{z}t_{1}\|_{\text{H}^{2}}+\|\partial_{z}r_{1}\|_{\text{H}^{2}}\right).$
(111)

###  4.2 Boundary source inversion

From now on, we denote by $\displaystyle\bm{u}$ the solution to (110)
generated with boundary source terms denoted by ${}^{\text{top}}$ and
${}^{\text{bot}}$. Given a maximal frequency $\displaystyle\omega_{\max}$, we
measure the wavefield at the surface of the perturbed plate for every
$\displaystyle\omega\in(0,\omega_{\max})$. Using the previous Born
approximation, we can assume that the wavefield $\displaystyle\bm{u}$ is
measured on the surface $\displaystyle y=h$ and that the measurements may
contain noise. For every frequency $\displaystyle\omega$ and $\displaystyle
x\in\mathbb{R}$, the measured value of $\displaystyle\bm{u}(x,h)$ is denoted
by $\displaystyle\bm{u}_{\omega}(x,h)$. Similarly, the associated wavenumbers
and Lamb modes are denoted by $\displaystyle k_{n}(\omega)$ and
$\displaystyle(u_{n,\omega},v_{n,\omega})$ respectively. Using Theorem 2, we
know that

$u_{\omega}(x,h)=\sum_{n>0}a_{n}(x)u_{n,\omega}(h),\qquad
v_{\omega}(x,h)=\sum_{n>0}b_{n}(x)v_{n,\omega}(h),$ (112)

where $\displaystyle a_{n}=G_{1}^{n}\ast F^{n}_{1}-G_{2}^{n}\ast F_{2}^{n}$,
$\displaystyle b_{n}=G_{2}^{n}\ast F_{1}^{n}-G_{1}^{n}\ast F_{2}^{n}$ and
$\displaystyle G_{1}^{n},G_{2}^{n}$ are defined in (42) and

$F^{n}_{1}(x)=\frac{e^{ik_{1}x}}{J_{n}}\left((2hg_{1}^{\prime}s_{1}(h)-(g_{1}^{\prime}-g_{2}^{\prime})\partial_{z}t_{1}(h))u_{n,\omega}(h)-(2hg_{2}^{\prime}s_{1}(h)+(g_{1}^{\prime}-g_{2}^{\prime})\partial_{z}t_{1}(h))u_{n,\omega}(-h)\right),$
(113)
$F^{n}_{2}=\frac{e^{ik_{1}x}}{J_{n}}(g_{1}^{\prime}-g_{2}^{\prime})\partial_{z}r_{1}(g)\left(-u_{n,\omega}(h)+u_{n,\omega}(-h)\right).$
(114)

Assuming that $\displaystyle x$ is located on the left of the support of the
sources,

$(u_{\omega},v_{\omega})(x,h)=\frac{1}{2}\sum_{n>0}(u_{n,\omega},-v_{n,\omega})(h)e^{-ik_{n}(\omega)x}\int_{\mathbb{R}}e^{ik_{n}(\omega)z}(F^{n}_{1}(z)-F^{n}_{2}(z))\mathrm{d}z,$
(115)

As explained in [21] and illustrated in [19], we can use a spatial Fourier
transform along $\displaystyle x$ to separate each term of the sum. We notice
that up to a multiplicative coefficient, the fields $\displaystyle u_{\omega}$
and $\displaystyle v_{\omega}$ contain the same information about the source,
so that only measurements of one component of the displacement are needed.

Further, since noise is likely to pollute the response of evanescent and
inhomogeneous modes in real-life experiments, we only consider the propagative
modes and for these modes $\displaystyle n$ we have access to

$\int_{\mathbb{R}}e^{i(k_{n}(\omega)+k_{1}(\omega))z}(g_{1}^{\prime}(z)c^{1}_{n}+g_{2}^{\prime}(z)c^{2}_{n})\mathrm{d}z\qquad\forall\omega\in\mathbb{R}_{+},$
(116)

where $\displaystyle c^{1}_{n}$ and $\displaystyle c^{2}_{n}$ are known
coefficients depending on the mode $\displaystyle n$. We use the following
definition for the Fourier transform

$\mathcal{F}(g)(\xi)=\int_{\mathbb{R}}g(z)e^{-i\xi z}\mathrm{d}z.$ (117)

From now on, we consider that $\displaystyle n=1$ is the first propagative
symmetric Lamb mode and $\displaystyle n=2$ is the first propagative
antisymmetric Lamb mode. Both modes exist at any frequency
$\displaystyle\omega$, and $\displaystyle\omega\mapsto k_{1}(\omega)$ or
$\displaystyle\omega\mapsto k_{2}(\omega)$ are increasing functions that map
$\displaystyle\mathbb{R}_{+}$ to $\displaystyle\mathbb{R}_{+}$ (see an
illustration in Figure 2 for the symmetric case, and for more details we refer
to [28]). In particular, if we set $\displaystyle\xi=2k_{1}$, the available
information amounts to knowing
$\displaystyle\mathcal{F}(c_{1}^{1}g_{1}^{\prime}+c_{1}^{2}g_{2}^{\prime})(\xi)$
for every $\displaystyle\xi\in(0,2k_{1}(\omega_{\max}))$. Similarly, if
$\displaystyle\xi=k_{1}+k_{2}$, we have knowledge of
$\displaystyle\mathcal{F}(c_{2}^{1}g_{1}^{\prime}+c_{2}^{2}g_{2}^{\prime})$
for every $\displaystyle\xi\in(0,k_{1}(\omega_{\max})+k_{2}(\omega_{\max}))$.
We define

$\xi_{\max}=\min\left(k_{1}(\omega_{\max})+k_{2}(\omega_{\max}),2k_{1}(\omega_{\max})\right).$
(118)

Looking at expressions (113)-(114), we notice that the linear combinations
$\displaystyle c_{1}^{1}g_{1}^{\prime}+c_{1}^{2}g_{2}^{\prime}$ and
$\displaystyle c_{2}^{1}g_{1}^{\prime}+c_{2}^{2}g_{2}^{\prime}$ are
independent so the functions $\displaystyle g_{1}^{\prime}$ and $\displaystyle
g_{2}^{\prime}$ can be reconstructed using the inverse Fourier transform, in a
stable way as the next Proposition shows (its proof is the same as Proposition
12 in [11]).

######  Proposition 11.

Let $\displaystyle g,g^{\text{app}}\in\mathcal{C}^{2}(-r,r)$ and their Fourier
transform $\displaystyle d=\mathcal{F}(g)$ and $\displaystyle
d^{\text{app}}=\mathcal{F}(g^{\text{app}})$ defined on
$\displaystyle(0,\xi_{\max})$. Assume that there exists $\displaystyle M>0$
such that
$\displaystyle\|g\|_{\text{H}^{1}(-r,r)},\|g^{\text{app}}\|_{\text{H}^{1}(-r,r)}\leq
M$, then

$\|g-g^{\text{app}}\|^{2}_{\text{L}^{2}(-r,r)}\leq\frac{4}{\pi}\|\mathcal{F}(g)-\mathcal{F}(g^{\text{app}})\|_{\text{L}^{2}(0,\xi_{\max})}+\frac{2\pi}{\xi_{\max}^{2}}M^{2}.$
(119)

######  Remark 5.

We notice that the above estimate is actually better than the one presented in
the acoustic case in Proposition 12 of [11]. Indeed, in the acoustic case,
there is only one propagative mode at every frequency, and the function
$\displaystyle k_{1}+k_{2}$ is not one-to-one from
$\displaystyle\mathbb{R}_{+}$ to $\displaystyle\mathbb{R}_{+}$. In the elastic
case however, we take advantage of the existence of two different Lamb modes
propagating at every frequency.

Given the reconstructions of $\displaystyle g_{1}^{\prime}$ and $\displaystyle
g_{2}^{\prime}$, we can integrate these functions, using the fact that
$\displaystyle g_{1}$ and $\displaystyle g_{2}$ have compact support, and
obtain an approximation of the shapes of the defects. In the previous
estimate, the error $\displaystyle\|d-d^{\text{app}}\|_{\text{L}^{2}(-r,r)}$
contains both the measurement error, as well as the error caused by the Born
approximation (107). It follows that the reconstruction error decreases when
the size of the defects gets smaller and when $\displaystyle\omega_{\max}$
increases. We present examples of numerical reconstructions in the next
section.

We conclude this section by discussing possible extensions of this work.
First, the method presented here could be implemented in a similar fashion in
3D. Indeed, using a Born approximation, one can show that the measurements are
close to those emanating from a wavefield generated by two boundary source
terms that depend on $\displaystyle\nabla g_{1}$ and $\displaystyle\nabla
g_{2}$ in a regular waveguide. In 3D, a Hankel transform plays the role of the
Fourier transform, and one obtains thus a reconstruction.

Second, by the same method, one can also reconstruct bends in an elastic
waveguide, in a similar manner as in the acoustic case described in section
3.2 of [11]. However, the detection of homogeneities seems more difficult.
Following section 3.4 of [11], one could use a Born approximation to
approximate the measurements by a wavefield generated by an internal source
term $\displaystyle\bm{f}$, that depends on a transformed inhomogeneity in a
regular waveguide. However, it does not seem easy to extract, from the
measurements, something like the Fourier transform of a function, that would
characterize the inhomogeneity, as in (116).

##  5 Numerical results

In this last section, we illustrate the results of Theorems 2 and 3, and
present numerical reconstructions of small shape defects.

Concerning Theorems 2 and 3, we compare the modal expressions of
$\displaystyle\bm{u}$ given in (39) and (90) to the wavefields generated using
Matlab in 2D, and Freefem++ [16] in 3D, respectively used to solve (38) and
(89). In the following, we assume that sources are supported in
$\displaystyle\Omega_{r}$ where $\displaystyle r=3$ in 2D and $\displaystyle
r=1$ in 3D. To solve the elastic equation, we use the finite element method
with a perfectly matched layer (PML) [7] placed in
$\displaystyle\Omega_{8}\setminus\Omega_{4}$ in 2D, and
$\displaystyle\Omega_{2.5}\setminus\Omega_{1.3}$. Since PML’s do not handle
the presence of right-going propagating modes correctly when the wavenumber is
negative (see an example of such wavenumbers in Figure 3), we use the strategy
presented in [10] which modify the PML to provide a correct approximation of
the wavefield, for every non critical frequency. The coefficient of absorption
in the PML is defined by $\displaystyle\alpha=-((|x|-4)\textbf{1}_{|x|\geq 4}$
in 2D or $\displaystyle\alpha=-((|\bm{x}|-1.3)\textbf{1}_{|\bm{x}|\geq 1.3}$
in 3D. The structured mesh is built with a stepsize of $\displaystyle 10^{-4}$
in 2D and $\displaystyle 10^{-2}$ in 3D.

We first illustrate the two dimensional case, and the modal decomposition (39)
solution to (38). Numerical representations of the wavefield
$\displaystyle\bm{u}$, obtained using the modal decomposition (39) are
presented in Figure 6 as well as the wavefields generated by the finite
element method, showing good visual agreement. Their computed relative error
in $\displaystyle\text{L}^{\infty}(\Omega_{r})$ and
$\displaystyle\text{L}^{2}(\Omega_{r})$ is smaller than $\displaystyle 2\%$.

$\displaystyle-2$$\displaystyle 0$$\displaystyle 2$$\displaystyle-0.1$$\displaystyle 0$$\displaystyle 0.1$$\displaystyle\text{Real}(u^{\text{mod}})$ | $\displaystyle-2$$\displaystyle 0$$\displaystyle 2$$\displaystyle-0.1$$\displaystyle 0$$\displaystyle 0.1$$\displaystyle\text{Real}(u^{\text{FEM}})$$\displaystyle-2$$\displaystyle-1$$\displaystyle 0$$\displaystyle 1$$\displaystyle 2$
---|---
$\displaystyle-2$$\displaystyle 0$$\displaystyle 2$$\displaystyle-0.1$$\displaystyle 0$$\displaystyle 0.1$$\displaystyle\text{Imag}(v^{\text{mod}})$ | $\displaystyle-2$$\displaystyle 0$$\displaystyle 2$$\displaystyle-0.1$$\displaystyle 0$$\displaystyle 0.1$$\displaystyle\text{Imag}(v^{\text{FEM}})$$\displaystyle-2$$\displaystyle 0$$\displaystyle 2$
Figure 6:  Comparison between a wavefield $\displaystyle\bm{u}$ computed using
the modal solution (39) or using a finite element method. Top: comparison
between real parts of $\displaystyle u$. Bottom: comparison between imaginary
parts of $\displaystyle v$. Similar results could also be obtained for
$\displaystyle\text{Imag}(u)$ and $\displaystyle\text{Real}(v)$. The
parameters of the problem are $\displaystyle\lambda=0.31$,
$\displaystyle\mu=0.25$, $\displaystyle h=0.1$, $\displaystyle\omega=13.7$.
The sum in the modal decomposition of $\displaystyle\bm{u}^{\text{mod}}$ is
cut at $\displaystyle N=20$ modes. Wavefields are generated using an internal
source term $\displaystyle\bm{f}$ defined in (E1) and boundary source terms
${}^{\text{top}}$ and ${}^{\text{bot}}$ defined in (E2). Their support is
represented in red. Here, the relative
$\displaystyle\text{L}^{\infty}(\Omega_{r})$-error is $\displaystyle 1.7\%$
and the $\displaystyle\text{L}^{2}(\Omega_{r})$-error is $\displaystyle
1.4\%$.

A similar comparison is carried out in 3D for the modal decomposition (90)
solution to (89). To visualize the decomposition
$\displaystyle\bm{u}=\bm{u}^{L}+\bm{u}^{sh}$, we first choose a curl-free
internal source given by (E3). The modal simulation is compared with the
fields obtained from a finite element approximation in Figure 7. Again, both
approximations of the true wavefield are visually similar, even if that
produced by the finite element discretization seems to propagate at a higher
velocity. This could be caused by the fact that the step size of the
discretization may not be sufficiently small. We point out that the
calculation times of these simulations are not the same: while the finite
element method takes around eight hours to run, the modal decomposition
produces a result in less than two minutes. This underlines the interest of
using the modal solution to do computations in three-dimensional perfect
plates. Next, we choose a divergence-free boundary source term given by (E4).
Comparisons are presented in Figure 8, and similar conclusions can be drawn.

$\displaystyle\text{Real}(w^{\text{mod}})$ | $\displaystyle\text{Real}(w^{\text{FEM}})$ |
---|---|---
|  | $\displaystyle-2$$\displaystyle-1$$\displaystyle 0$$\displaystyle\cdot 10^{-2}$
$\displaystyle\text{Real}(u^{\text{mod}})$ | $\displaystyle\text{Real}(u^{\text{FEM}})$ |
|  | $\displaystyle-1$$\displaystyle 0$$\displaystyle 1$$\displaystyle\cdot 10^{-2}$
Figure 7:  Comparison between $\displaystyle\bm{u}$ computed using the modal solution (90) and using the finite element method with a curl-free internal source (E3) represented in red. Top: comparison of $\displaystyle\text{Real}(w)$. Bottom: comparison of $\displaystyle\text{Real}(u)$ (plots of $\displaystyle\text{Imag}(u),\text{Real}(v),\text{Imag}(v),\text{Imag}(w)$ look similar). The parameters are $\displaystyle\lambda=0.31$, $\displaystyle\mu=0.25$, $\displaystyle h=0.2$, $\displaystyle\omega=10$. $\displaystyle N=20$ modes are used in the decomposition of $\displaystyle\bm{u}^{\text{mod}}$. The relative $\displaystyle\text{L}^{\infty}(\Omega_{r})$-error is $\displaystyle 7.2\%$ and the $\displaystyle\text{L}^{2}(\Omega_{r})$-error is $\displaystyle 9.4\%$. $\displaystyle\text{Real}(v^{\text{mod}})$ | $\displaystyle\text{Real}(v^{\text{FEM}})$ |
---|---|---
|  | $\displaystyle-5$$\displaystyle 0$$\displaystyle 5$$\displaystyle\cdot 10^{-2}$
Figure 8: Comparison between a wavefield $\displaystyle\bm{u}$ computed using
the modal solution (90) or using a finite element method with a divergence
free boundary source (E4) represented in red. The real parts of $\displaystyle
v$ is shown (plots of
$\displaystyle\text{Imag}(v),\text{Real}(u),\text{Imag}(u),\text{Real}(w),\text{Imag}(w)$
would have similar aspect). The parameters are $\displaystyle\lambda=0.31$,
$\displaystyle\mu=0.25$, $\displaystyle h=0.2$, $\displaystyle\omega=10$.
$\displaystyle N=10$ modes are used in the decomposition of
$\displaystyle\bm{u}^{\text{mod}}$. The relative
$\displaystyle\text{L}^{\infty}(\Omega_{r})$-error is $\displaystyle 5.3\%$
and the $\displaystyle\text{L}^{2}(\Omega_{r})$-error is $\displaystyle
4.1\%$.

Finally, we illustrate in Figure 9 two numerical reconstructions of small
defects. Synthetic surface measurements are generated using the finite element
method described above for different frequencies. Then, we reconstruct the
derivative of defects profiles $\displaystyle g_{1}$ and $\displaystyle g_{2}$
using the penalized least square algorithm described in [11]. We get
reconstructions as good or even better than the one presented in the acoustic
case (see Figure 11 in [11]), and we notice that the reconstruction seems more
robust than the acoustic one then the size of the defect increases. Table 1
illustrates this point as it depicts the relative error on a reconstruction of
$\displaystyle g_{1}$ when its amplitude increases. This table can be compared
to Table 2 in [11] where the same relative error in the acoustic case turns
out to be bigger.

Figure 9:  Reconstruction of two shape defects. In black, the initial shape of $\displaystyle\Omega$, and in red the reconstruction, slightly shifted for comparison purposes. In both cases, $\displaystyle h=0.1$, $\displaystyle\omega_{\max}=17$ and the interval $\displaystyle(0,\omega_{\max})$ is discretize with $\displaystyle 170$ points. The relative $\displaystyle\text{L}^{2}$-error is $\displaystyle 4.7\%$ on the left and $\displaystyle 5.1\%$ on the right. Functions $\displaystyle g_{1}$ and $\displaystyle g_{2}$ are given in (E5) and (E6). $\displaystyle A$ | $\displaystyle 0.1$ | $\displaystyle 0.2$ | $\displaystyle 0.3$ | $\displaystyle 0.5$
---|---|---|---|---
$\displaystyle\|g_{1}-g_{1}^{\text{app}}\|_{\text{L}^{2}(\mathbb{R})}/\|g_{1}\|_{\text{L}^{2}(\mathbb{R})}$ | $\displaystyle 4.5\%$ | $\displaystyle 6.4\%$ | $\displaystyle 9.2\%$ | $\displaystyle 18.3\%$
Table 1:  Relative errors on the reconstruction of $\displaystyle h$ for
different amplitudes $\displaystyle A$ for the shape defects $\displaystyle
g_{1}(x)=A\textbf{1}_{3\leq x\leq 5}(x-3)^{2}(5-x)^{2}$ and $\displaystyle
g_{2}(x)=0$. In every reconstruction, $\displaystyle h=0.1$,
$\displaystyle\omega_{\max}=17$ and the interval
$\displaystyle(0,\omega_{\max})$ is discretize with $\displaystyle 170$
points.

## Appendix: Expressions for numerical simulations

$\bm{f}(x,y)=-100\,\textbf{1}_{(x-0.5)^{2}+\frac{(y-0.06)^{2}}{0.015^{2}}<1}(x,y)\left(1-(x-0.5)^{2}+\frac{(y-0.06)^{2}}{0.015^{2}}\right)(x+2y;1),\hfill$
(E1)
$^{\text{top}}(x)=\frac{10}{\sqrt{2\pi}}e^{-\frac{(x+0.5)^{2}}{200}}(1;x),\quad^{\text{bot}}(x)=20\,\textbf{1}_{[2,2.5]}(x)(x-2)(x-2.5)(1;\sin(x)),\hfill$
(E2)
$\bm{f}(x,y,z)=z\frac{50}{\pi}e^{-\frac{(x^{2}+y^{2})}{200}}\left(-x;-y;1\right),$
(E3)
$^{\text{top}}(x,y)=\frac{25}{\pi}e^{-\frac{(x^{2}+y^{2})}{200}}(-y;x;0),\quad^{\text{bot}}(x,y)=0.$
(E4) $g_{1}(x)=\frac{5}{16}\textbf{1}_{3.2\leq x\leq
4.2}(x-3.2)^{2}(4.2-x)^{2},\quad g_{2}(x)=-\frac{35}{16}\textbf{1}_{3.4\leq
x\leq 4}(x-3.4)^{2}(4-x)^{2}.$ (E5)
$g_{1}(x)=\frac{125}{16}\textbf{1}_{3.7\leq x\leq
4.2}(x-3.7)^{2}(4.2-x)^{2},\quad g_{2}(x)=\frac{125}{16}\textbf{1}_{3.4\leq
x\leq 4}(x-3.4)^{2}(4-x)^{2}.$ (E6)

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