# A footnote to a theorem of Kawamata

Margarida Mendes Lopes  CAMGSD/Departamento de Matemática, Instituto Superior
Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
<EMAIL_ADDRESS>,  Rita Pardini  Dipartimento di Matematica,
Università degli studi di Pisa, Largo Pontecorvo 5, 56127 Pisa (PI), Italy
<EMAIL_ADDRESS>and  Sofia Tirabassi  Department of Mathematics,
Stockholm University, Albano Campus, Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract.

Kawamata has shown that the quasi-Albanese map of a quasi-projective variety
with log-irregularity equal to the dimension and log-Kodaira dimension 0 is
birational. In this note we show that under these hypotheses the quasi-
Albanese map is proper in codimension 1 as conjectured by Iitaka.

###### Key words and phrases:

quasi-abelian varieties, semi-abelian varieties, open varieties, quasi-
Albanese morphism, logarithmic Kodaira dimension, birational geometry of log-
varieties, WWPB-equivalence, affine varieties

###### 2020 Mathematics Subject Classification:

Primary 14E05, Secondary 14J10, 14K99, 14L40, 14R05

## Introduction

Given a smooth complex quasi-projective variety $V$, by Hironaka’s theorem on
the resolution of singularities we can embedd $V$ into a smooth projective
variety $X$ such that the complement $D:=X\backslash V$ is a reduced divisor
on $X$ with simple normal crossings.

One can use the compactification $X$ to define the logarithmic invariants of
$V$. In particular, if $K_{X}$ denotes the canonical divisor of $X$, then:

  * -

the _log-Kodaira dimension_ of $V$ is
$\overline{\kappa}(V):=\kappa(X,K_{X}+D);$

  * -

the _log-irregularity_ of $V$ is $\overline{q}(V):=h^{0}(X,\Omega^{1}_{X}(\log
D))$.

It easy to show that these invariants do not depend on the choice of the
compactification $X$.

In a similar way to projective varieties, we can associate to $V$ a quasi-
abelian variety (i.e., an algebraic group which does not contain
$\mathbb{G}_{a}$) $A(V)$, called the _quasi-Albanese variety_ of $V$ and a
morphism $a_{V}\colon V\rightarrow A(V)$ which is called the _quasi-Albanese
morphism_.

Kawamata, in his celebrated work [Ka81], provides a numerical criterion for
the birationality of the quasi-Albanese map:

######  Theorem (Kawamata, Theorem 28 of [Ka81]).

Let $V$ be a smooth complex algebraic variety of dimension $n$. If
$\overline{\kappa}(V)=0$ and $\overline{q}(V)=n$ then the quasi-Albanese
morphism $a_{V}\colon V\rightarrow A(V)$ is birational.

In this note, we strengthen Kawamata’s Theorem in a way that was conjectured
by Iitaka in [Ii77, p. 501]. Our main result is the following:

######  Theorem A.

Let $V$ be a smooth complex algebraic variety of dimension $n$ with
$\overline{\kappa}(V)=0$ and $\overline{q}(V)=n$. Denote by $a_{V}\colon
V\rightarrow A(V)$ the quasi-Albanese morphism (which is birational).

Then there is a closed subset $Z\subset A(V)$ of codimension $>1$ such that,
setting $V^{0}:=V\backslash a_{V}^{-1}(Z)$, the restriction
$a_{V|_{V^{0}}}\colon{V^{0}}\rightarrow A(V)\backslash Z$ is proper.

Using the notion of WWPB equivalence introduced in [Ii77], we can rewrite our
statement in an alternative way:

######  Theorem A*.

Let $V$ be a smooth complex algebraic variety of dimension $n$. Then $V$ is
WWPB equivalent to a quasi-abelian variety if and only if
$\overline{\kappa}(V)=0$ and $\overline{q}(V)=n$.

While the usual Kodaira dimension and irregularity are birational invariants,
this is not the case for logarithmic invariants. As an easy example of this
phenomenon, one can think of $\mathbb{P}^{1}$ and
$\mathbb{P}^{1}\backslash\\{p_{0}+p_{1}\\}$: they are birationally equivalent
but their invariants are completely different. However, WWPB-equivalent
varieties share the same logarithmic invariants. Therefore this seems to be
the right notion of equivalence to consider when studying the birational
geometry of open varieties.

We conclude this section by remarking that the condition of being WWPB-
equivalent is a really strong one: WWPB maps between normal affine varieties
are indeed isomorphisms. Thus we get the following corollary of Theorem A:

######  Corollary B.

A smooth complex affine variety $V$ of dimension $n$ is isomorphic to
$\mathbb{G}_{m}^{n}$ if and only if $\overline{\kappa}(V)=0$ and
$\overline{q}(V)=n$.

Notation. We work over the complex numbers. If $X$ is a smooth projective
variety, we denote by $K_{X}$ the canonical class, and by
$q(X):=h^{0}(X,\Omega^{1}_{X})=h^{1}(X,\mathcal{O}_{X})$ its _irregularity_.

We identify invertible sheaves and Cartier divisors and we use the additive
and multiplicative notation interchangeably. Linear equivalence is denoted by
$\simeq$. Given two divisors $D_{1}$ and $D_{2}$ on $X$, we write
$D_{1}\succeq D_{2}$ (respectively $D_{1}\succ D_{2}$) if the divisor
$D_{1}-D_{2}$ is (strictly) effective.

Acknowledgments. M. Mendes Lopes was partially supported by FCT/Portugal
through Centro de Análise Matemática, Geometria e Sistemas Dinâmicos (CAMGSD),
IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020. R. Pardini was partially
supported by the project PRIN 2017SSNZAW$\\_$004 “Moduli Theory and Birational
Classification" of Italian MIUR and is a member of GNSAGA of INDAM.
S.Tirabassi was partially supported by the Knut and Alice Wallenberg
foundation under grant no. KAW 2019.0493.

##  1\. Preliminaries

In this section, we provide the background material and preliminary lemmas
which are essential for our main result. In particular, we review the
logarithmic ramification formula in 1.1, and give a quick introduction to
quasi-abelian varieties and quasi-Albanese morphism in 1.2. For the sake of
brevity, we do not cover here the general theory of log-varieties and WWPB
maps. An exhaustive introduction to the theory of logarithmic forms can be
found in [Ii82]. For a quick overview, with also an introduction to WWPB
equivalence, we refer to our own [MLPT2022].

###  1.1. Logarithmic ramification formula

Let $V$, $W$ be smooth varieties of dimension $n$ and let $h\colon V\to W$ be
a dominant morphism. Let $g\colon X\to Y$ be a morphism extending $h$, where
$X$, $Y$ are smooth compactifications of $V$, respectively $W$, such that
$D:=X\setminus V$ and $\Delta:=Y\setminus W$ are simple normal crossings (snc
for short) divisors. Then the pull back of a logarithmic $n$-form on $Y$ is a
logarithmic $n$-form on $X$, and a local computation shows that there is an
effective divisor $\overline{R}_{g}$ of $X$ \- the logarithmic ramification
divisor \- such that the following linear equivalence holds:

(1) $K_{X}+D\simeq g^{*}(K_{Y}+\Delta)+\overline{R}_{g}.$

Equation (1) is called the logarithmic ramification formula (cf. [Ii82,
§11.4]).

In [MLPT2022, Lemma 1.8] we observed the following useful fact:

######  Lemma 1.1.

In the above set-up, denote by $R_{g}$ the (usual) ramification divisor of
$g$. Let $\Gamma$ be an irreducible divisor such that
$g(\Gamma)\not\subseteq\Delta$. Then $\Gamma$ is a component of
$\overline{R}_{g}$ if and only if $\Gamma$ is a component of $D+R_{g}$.

Next we characterize the crepant exceptional divisors, namely those that do
not contribute to $\overline{R}_{g}$:

######  Lemma 1.2.

Let $g:X\rightarrow Y$ be a birational morphism of smooth projective
varieties, let $D_{X}\subseteq X$ and $D_{Y}\subseteq Y$ be snc divisors such
that $g(X\backslash D_{X})\subseteq Y\backslash D_{Y}$, and consider $E$ an
irreducible $g$-exceptional divisor. If $E$ does not appear in
$\overline{R}_{g}$, then $g(E)$ is an irreducible component of the
intersection of some components of $D_{Y}$.

###### Proof.

Let $p\in E$ general and consider $q:=g(p)\in g(E)$.

If $g(E)$ is not contained in $D_{Y}$, then $q$ is not in $D_{Y}$ and so, by
Lemma 1.1, $E\preceq\overline{R}_{g}$.

Thus, we can assume that $g(E)\subseteq D_{Y}$ and, therefore, $E\preceq
D_{X}$. Denote by $D_{1},\ldots,D_{k}$ the components of $D_{Y}$ containing
$g(E)$ and let $x_{i}$ be local equations for $D_{i}$ near $q$.

Assume by contradiction that $g(E)$ is a proper subset of $D_{1}\cap\ldots\cap
D_{k}$. Since $q$ is general, $g(E)$ is smooth near $q$, and hence locally a
complete intersection. Since $D_{Y}$ is snc, we can complete
$x_{1},\ldots,x_{k}$ to a system of local coordinates
$x_{1},\ldots,x_{k},x_{k+1},\ldots x_{n}$ such that around $q$ the set $g(E)$
is locally defined by the vanishing of $x_{1},\ldots,x_{k+s}$ for some $s\geq
1$. In this setting, locally around $q$ the line bundle
$\mathcal{O}_{Y}(K_{Y}+D_{Y})$ is generated by

$\displaystyle\sigma$
$\displaystyle:=\frac{dx_{1}}{x_{1}}\wedge\cdots\wedge\frac{dx_{k}}{x_{k}}\wedge
dx_{k+1}\wedge\cdots\wedge dx_{n}$ $\displaystyle\>=(x_{k+1}\cdots
x_{k+s})\frac{dx_{1}}{x_{1}}\wedge\cdots\wedge\frac{dx_{k+s}}{x_{k+s}}\wedge
dx_{k+s+1}\wedge\cdots\wedge dx_{n}.$

Choose a local equation $t$ for $E$ near $p$. Then we can write

$g^{*}x_{i}=t^{\alpha_{i}}h_{i},$

with $h_{i}$ regular and not divisible by $t$. Observe that $\alpha_{i}>0$ if
$i\leq k+s$, while $\alpha_{i}=0$ for $i>k+s$. Thus we get

$g^{*}\sigma=ht^{\alpha_{k+1}+\cdots+\alpha_{k+s}}\cdot\frac{dt}{t}\wedge\tau,$

where $h$ is a regular function and $\tau$ is a $(n-1)$-form without poles
along $E$. We conclude that $E$ appears in $\overline{R}_{g}$ with
multiplicity at least $\alpha_{k+1}+\cdots+\alpha_{k+s}>0$. This is a
contradiction because we are assuming that $E$ does not appear in
$\overline{R}_{g}$. ∎

We close the section by proving a partial converse of the previous lemma:

######  Lemma 1.3.

Let $W$ be a quasi-projective variety with compactification $Y$ and snc
boundary $\Delta$. Let $C\subseteq\Delta$ be an irreducible component of the
intersection of some components of $\Delta$, and denote by
$\epsilon_{C}:Y_{C}\rightarrow Y$ the blow-up of $C$. If
$\Delta_{C}:=\epsilon_{C}^{-1}\Delta$ is the set-theoretic pre-image of
$\Delta$, then $Y_{C}$ is also a compactification of $W$ with snc boundary
$\Delta_{C}$ and the logarithmic ramification divisor
$\overline{R}_{\epsilon_{C}}$ is trivial.

###### Proof.

It is clear that $W\simeq Y_{C}\backslash\Delta_{C}$ and that $\Delta_{C}$ has
snc support. Thus we need just to verify the statement about the logarithmic
ramification divisor. Let $k$ be the codimension of $C$ in $Y$, and denote by
$E$ the exceptional divisor of the blow-up. Then

$\displaystyle K_{Y_{C}}+\Delta_{C}\simeq
f^{*}(K_{Y})+(k-1)E+f^{*}(\Delta)-(k-1)E=f^{*}(K_{Y}+\Delta).$

We conclude directly by the logarithmic ramification formula (1) that
$\overline{R}_{\epsilon_{C}}=0$. ∎

###  1.2. Quasi-abelian varieties and their compactifications

A _quasi-abelian variety_ \- in some sources also called a _semiabelian
variety_ \- is a connected algebraic group $G$ that is an extension of an
abelian variety $A$ by an algebraic torus. More precisely, $G$ sits in the
middle of an exact sequence of the form

(2) $1\rightarrow\mathbb{G}_{m}^{r}\longrightarrow G\longrightarrow
A\rightarrow 0.$

We call $A$ the compact part and $\mathbb{G}_{m}^{r}$ the linear part of $G$.

We recall the following from [III, §10]:

######  Proposition 1.4.

Let $G$ be a quasi-abelian variety, let $A$ be its compact part and let
$r:=\dim G-\dim A$. Then there exists a compactification $G\subset Y$ such
that:

  * (a)

$Y$ is a $\mathbb{P}^{r}$-bundle over $A$;

  * (b)

$\Delta:=Y\setminus G$ is a simple normal crossing divisor and
$K_{Y}+\Delta\simeq 0$;

  * (c)

the natural $\mathbb{G}_{m}^{r}$-action on $G$ extends to $Y$.

###### Proof.

As explained in [III, §10], the variety $G$ is a principal
$\mathbb{G}_{m}^{r}$-bundle over $A$. We let
$\rho\colon\mathbb{G}_{m}^{r}\to\operatorname{Aut}({\mathbb{P}}^{r})$ be the
inclusion that maps $(\lambda_{1},\dots\lambda_{r})$ to the automorphism
represented by $\operatorname{Diag}(1,\lambda_{1},\dots\lambda_{r})$ and take
$Y:=G\times_{\rho}{\mathbb{P}}^{r}$. By construction,
$Y=\operatorname{Proj}_{A}({\mathcal{O}}_{A}\oplus L_{1}\oplus\dots\oplus
L_{r})$, for some $L_{i}\in\operatorname{Pic}(A)$, the boundary $\Delta$ is
the sum of the relative hyperplanes $\Delta_{0},\Delta_{1},\dots\Delta_{r}$
given by the inclusions of ${\mathcal{O}}_{Y},L_{1},\dots L_{r}$ into
${\mathcal{O}}_{A}\oplus L_{1}\oplus.\oplus L_{r}$ and the
$\mathbb{G}_{m}^{r}$-action clearly extends to $Y$. Since $K_{A}$ is trivial,
the formula for the canonical class of a projective bundle reads
$K_{Y}\simeq-(\Delta_{0}+\dots+\Delta_{r})$. ∎

####  1.2.1. The quasi-Albanese map

The classical construction of the Albanese variety of a projective variety can
be extended to the non projective case, by replacing regular 1-forms by
logarithmic ones and abelian varieties by quasi-abelian ones. The key fact is
that by Deligne [De71] logarithmic 1-forms are closed (for the details of the
construction see [III], [Fu14, Section 2]). As a consequence, for every $V$
smooth quasi projective variety, there exists a quasi-abelian variety $A(V)$
and a morphism $a_{V}\colon V\to A(V)$ such that any morphism $h\colon
V\rightarrow G$ to a quasi-abelian variety factors through $a_{V}$ in a unique
way up to translation.

##  2\. Proof of theorem A

Let $V$ be a smooth complex quasi-projective variety of dimension $n\geq 2$
satisfying $\overline{\kappa}(V)=0$ and $\overline{q}(V)=n$. Let $A(V)$ be the
quasi-Albanese variety of $V$, and $a_{V}\colon V\rightarrow A(V)$ the quasi-
Albanese morphism.

Pick a compactification $(Y,\Delta)$ of $A(V)$ as in Proposition 1.4. Choose
then a compactification $(X,D)$ of $V$ such that the quasi-Albanese morphism
$a_{V}\colon V\to A(V)$ extends to a morphism $f\colon X\to Y$. Recall that
$f$ is birational by Kawamata’s theorem ([Ka81, Thm. 28]).

We observe that proving Theorem A is equivalent to showing that every
irreducible component of $D$ not contracted by $f$ is mapped to $\Delta$. In
fact, let $D^{\prime}$ be the divisor formed by the irreducible components of
$D$ which are not mapped to $\Delta$, assume that $D^{\prime}$ is contracted
by $f$ and set $Z^{\prime}:=f(D^{\prime})$. This is a closed subset of
codimension at least 2 in $Y$ and thus $Z:=Z^{\prime}\cap A(V)$ is a closed
subset of codimension at least 2. Denoting by $V^{0}$ the open set
$V\backslash a_{V}^{-1}(Z)$, we see that the restriction

${a_{V}}_{|V^{0}}\colon V^{0}\rightarrow A(V)\backslash Z$

is proper.

So we argue by contradiction supposing that there is an irreducible component
$H$ of $D$ such that $f(H)=:\bar{H}$ is a divisor not contained in $\Delta$.
Then

(3) $H=f^{*}(\bar{H})-\sum m_{i}E_{i},$

where the divisors $E_{i}$ are $f$-exceptional divisors. By the logarithmic
ramification formula we have

$K_{X}+D\simeq f^{*}(K_{Y}+\Delta)+\overline{R}_{f}\simeq\overline{R}_{f},$

where the last equality holds because $K_{Y}+\Delta\simeq 0$ (see Proposition
1.4).

We split the proof in two main steps: first, we give an argument in the
special case in which all the $E_{i}$ appearing in (3) belong to
$\overline{R}_{f}$; afterwards, we show that the general situation can always
be reduced to the special case.

###  2.1. Special case: all the $E_{i}$’s are components of $\overline{R}_{f}$

Since $H$ is also a component of $\overline{R}_{f}$ by Lemma 1.1, we have that

$K_{X}+D\simeq\overline{R}_{f}\succeq H+\sum
E_{i}=f^{*}\overline{H}-\sum(m_{i}-1)E_{i}.$

Choose $\alpha$ a rational number, $\alpha\in(0,1)$ such that
$\alpha\geq\frac{m_{i}-1}{m_{i}}$ for every $i$. Then we can write

$\displaystyle f^{*}\overline{H}-\sum(m_{i}-1)E_{i}$ $\displaystyle=\alpha
f^{*}\overline{H}-\sum(m_{i}-1)E_{i}+(1-\alpha)f^{*}\overline{H}$
$\displaystyle\succeq\alpha\left(f^{*}\overline{H}-\sum
m_{i}E_{i}\right)+(1-\alpha)f^{*}\overline{H}$ $\displaystyle=\alpha
H+(1-\alpha)f^{*}\overline{H}.$

It follows that $\kappa(Y,\overline{H})\leq\overline{\kappa}(V)$. We are going
to show that $\kappa(Y,\overline{H})>0$, against the hypothesis
$\overline{\kappa}(V)=0$.

If $\overline{H}$ is the pullback of an effective divisor $D$ via the map
$q\colon Y\rightarrow A$, then $\kappa(Y,\overline{H})\geq\kappa(A,D)>0$,
because an effective divisor on an abelian variety is the pull back of an
ample divisor on a quotient abelian variety.

Thus we may assume that this is not the case, namely that $\overline{H}$
restricts to a divisor on the general fiber $F$ of the projective bundle
$q\colon Y\rightarrow A$. Since $\overline{H}$ is not contained in $\Delta$,
its restriction to the general fiber of $q$ is not contained in $\Delta_{|F}$.
In particular $\overline{H}_{|F}$ is not invariant with respect to the
$\mathbb{G}_{m}^{r}$-action on $F$ (see Proposition 1.4). This means that we
can find a $g\in\mathbb{G}_{m}^{r}$ such that
$g^{*}\overline{H}_{|F}\neq\overline{H}_{|F}$. Since $g^{*}\overline{H}$ and
$\overline{H}$ are linearly equivalent, we obtain $h^{0}(Y,\overline{H})\geq
2$ and we are done.

###  2.2. Reduction to the special case

Up to reordering, we can assume that $E_{1}\npreceq\overline{R}_{f}$. By Lemma
1.2 we have that $f(E_{1})$ is a component of a complete intersection of
components of $\Delta$. Let $\epsilon\colon Y_{1}\rightarrow Y$ be the blow up
of $f(E_{1})$. Observe that $Y_{1}$ is smooth, since $f(E_{1})$ is so. By
Lemma 1.3, $(Y_{1},\Delta_{1})$ is a compactification of $A(V)$ with
$K_{Y_{1}}+\Delta_{1}\simeq 0$. In addition the $\mathbb{G}_{m}^{r}$-action on
$A(V)$ extends to $Y_{1}$ and preserves the components of $\Delta_{1}$.

By the universal property of the blow-up we can factor $f$ as follows:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{f}$$\textstyle{Y}$
$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{1}}$

Let $\overline{H}_{1}:=f_{1}(H)$. This is a divisor not contained in
$\Delta_{1}$, otherwise $f(H)$ would be contained in $\Delta$. Then

$f_{1}^{*}\overline{H}_{1}=H+\sum_{i\geq 2}m_{i}E^{\prime}_{i},$

where the $E^{\prime}_{i}$ are $f_{1}$-exceptional divisors. If all the
$E^{\prime}_{i}$ are appear in the logarithmic ramification divisor
$\overline{R}_{f_{1}}$, we have reduced the situation to the special case and
we are done. Otherwise, we can repeat the above construction. As at every
stage we provide a blow-up $Y_{n}$ of $Y_{n-1}$ and a factorization
$f=\epsilon_{1}\circ\cdots\circ\epsilon_{n}\circ f_{n}$, after finitely many
iterations this process must stop, and we will be in the situation of the
special case.

## References


[1]Matt R. Tranter

1]Department of Physics and Mathematics, Nottingham Trent University, NG11
8NS, United Kingdom

# Detecting Delamination via Nonlinear Wave Scattering in a Bonded Elastic Bar

Jagdeep S. Tamber  David J. Chappell  Jack C. Poore<EMAIL_ADDRESS>[

###### Abstract

In this paper we examine the effect of delamination on wave scattering, with
the aim of creating a control measure for layered waveguides of various
bonding types. Previous works have considered specific widths of solitary
waves for the simulations, without analysing the effect of changing the
soliton parameters. We consider two multi-layered structures: one containing
delamination ‘sandwiched’ by perfect bonding and one containing delamination
but ‘sandwiched’ by soft bonding. These structures are modelled by coupled
Boussinesq-type equations. Matched asymptotic multiple-scale expansions lead
to coupled Ostrovsky equations in soft bonded regions and Korteweg–De Vries
equations in the perfectly bonded and delaminated region. We use the Inverse
Scattering Transform to predict the behaviour in the delaminated regions.

In both cases, numerical analysis shows that we can predict the delamination
length by changes in the wave structure, and that these changes depend upon
the Full Width at Half Magnitude (FWHM) of the incident soliton. In the case
of perfect bonding, we derive a theoretical prediction for the change and
confirm this numerically. For the soft bonding case, we numerically identify a
similar relationship using the change in amplitude. Therefore we only need to
compute one curve to determine the behaviour for any incident solitary wave,
creating a framework for designing measurement campaigns for rigorously
testing the integrity of layered structures.

###### keywords:

nonlinear waves, wave scattering, solitons, Inverse Scattering Transform

##  1 Introduction

Solitary waves are of significant interest from both a mathematical
perspective as well as in physical and engineering applications. They often
arise as solutions to nonlinear equations such as the KdV equation (and its
extensions) in shallow water [1, 2, 3, 4], the Benjamin-Ono equation for
internal waves of stratified fluids [5, 6], the nonlinear Schrödinger equation
in optics [7, 8], and flexural waves in the beam equation [9], to name a few.
From a purely mathematical perspective, there are many studies into the
existence and behaviour of solitons, for example as solutions to the
Boussinesq equation [10, 11]. Boussinesq-type equations are of interest in
this study, in the context of solid mechanics. It is well-known that they can
describe long longitudinal bulk strain waves in elastic waveguides, such as
rods and metal plates (see e.g. [12, 13, 14, 15, 16, 17]). Practical
experiments have confirmed that longitudinal bulk strain solitons exist in
these waveguides, which validated theoretical findings [18, 19, 20, 21].

Indeed, layered waveguides with bonding between the layers can be modelled by
the so-called “doubly dispersive equation” (DDE), which can be derived using
nonlinear elasticity theory for long longitudinal waves in a bar of
rectangular cross-section [12, 13]. The DDE for a bar of rectangular cross-
section $\sigma=2a\times 2b$ has the form

$f_{tt}-c^{2}f_{xx}=\frac{\beta}{2\rho}(f^{2})_{xx}+\frac{Jv^{2}}{\sigma}(f_{tt}-c_{1}^{2}f_{xx})_{xx},$
(1)

where

$\displaystyle\beta$
$\displaystyle=3E+2l(1-2v)^{3}+4m(1+v)^{2}(1-2v)+6nv^{2},$ $\displaystyle c$
$\displaystyle=\sqrt{\frac{E}{\rho}},\quad c_{1}=\frac{c}{\sqrt{2(1+v)}},\quad
J=\frac{4ab(a^{2}+b^{2})}{3},$ (2)

$\rho$ is the density, $E$ is the Young’s modulus, $v$ is the Poisson’s ratio,
while $l$, $m$, $n$ are the Murnaghan’s moduli, and $a$ and $b$ are geometric
and physical parameters.

The case when the interlayer bonding is missing over part of the structure,
known as delamination, is important for a wide range of applications in non-
destructive testing for structural damage in the multi-layer beam-like
structures found throughout civil and mechanical engineering. The governing
mathematical model then takes the form of a scattering problem and, for a
perfectly bonded waveguide (represented in experiments by cyanoacrylate), we
find a series of Boussinesq equations, with continuity conditions on the
interface between sections [22]. Incident solitons fission into multiple
solitons with dispersive radiation, agreeing with theoretical predictions
[22], numerical simulations [23, 24] and experimental observations [20, 25,
26].

In the case of an imperfect “soft” bonding (represented in experiments by
polychloroprene), a model based upon a series of anharmonic coupled dipoles
can be used to derive coupled regularised Boussinesq (cRB) equations to model
long nonlinear longitudinal bulk strain waves in a bi-layer, assuming
sufficiently soft bonding [27]. In this case, when the materials in the layers
are sufficiently close, an incident solitary wave evolves in the bonded region
into a solitary wave with a one-sided, co-propagating oscillatory tail, known
as a radiating solitary wave. In the delaminated regions, the solitary wave
detaches from its tail and this can be used as a measure of delamination [28].
More recently, in the limiting case of a semi-infinite delamination, we find
the emergence of Ostrovsky wave packets in bonded regions [29]. The Ostrovsky
equation originally arose in the context of shallow-water waves, where the
rotation of the Earth is considered [30], and the evolution of wave packets
generated from an initial pulse has been extensively studied [31].

In this paper we aim to use theoretical predictions and numerical simulations
to establish a prediction for the delamination length based upon changes in
the wave during its propagation. We will consider a range of initial
conditions by varying the Full Width at Half Magnitude (FWHM) of the incident
wave, whereas previous studies have only considered a single fixed incident
soliton [23, 28]. Our aim is to find a relationship between the generated
delamination curves for different values of FWHM, so that only one curve needs
to be computed, significantly reducing the computation time. This wider range
of predictions allows for the design of measurement campaigns for detecting
and measuring delamination in layered waveguides. We will consider a multi-
layered symmetric structure with perfect bonding, as well as a two-layered
structure with soft bonding. In both cases, we will consider delamination
‘sandwiched’ by bonding. These structures are illustrated in figures 1 and 2,
and are inspired by an existing experimental set-up [20].

The paper is structured as follows. In Section 2 we introduce the equations
describing longitudinal wave propagation in both the perfectly bonded and soft
bonded cases. We also introduce the weakly-nonlinear solution for the
perfectly bonded case so that we can create a measure of the delamination
length using theoretical predictions. In Section 3, we begin by illustrating
the evolution of incident solitary waves in the cases of both a perfectly
bonded waveguide and a soft bonded waveguide. Next, for the perfectly bonded
case, we use theoretical predictions to determine the length of the
delamination region for a variety of incident solitary waves, tested via
numerical simulations and measuring the amplitude of the transmitted wave with
reference to the incident wave. This gives rise to a relationship between FWHM
of the incident soliton and the delamination length, allowing for efficient
computation for other incident waves. A similar result is also found for the
soft bonded case, analysing the decrease in amplitude after the wave
propagates through a delaminated region. The theoretical prediction is
difficult in this case, so we rely on numerical observations and instruction
from the previous case. In Section 4 we conclude our discussions.

##  2 Problem set-up

###  2.1 Perfectly bonded case

Figure 1: Bi-layer structure with an initial perfect bonded region for
$x_{0}<x<x_{1}$, a delaminated region for $x_{1}<x<x_{2}$ and a perfect bonded
region for $x_{2}<x<x_{3}$. We assume that the materials in both layers are
identical.

We consider the scattering of a long longitudinal strain solitary wave in a
perfectly bonded layered bar with delamination in the centre, as illustrated
in Figure 1. Note that we only illustrate two layers in this figure, but the
setup can accommodate any number of symmetric layers, as we assume that the
materials of the layers are identical and that the bonding is the same between
all layers. This problem is described by the regularized non-dimensional
Boussinesq equations [22]

$\displaystyle u^{(i)}_{tt}-c_{i}^{2}u^{(i)}_{xx}$
$\displaystyle=\varepsilon\left[-12\alpha_{i}u^{(i)}_{x}u^{(i)}_{xx}+2\beta_{i}u^{(i)}_{ttxx}\right],$
(3)

where $i=1,3$ represent the perfect bonded regions and $i=2$ represents the
delaminated region. We have the coefficients $\alpha_{i}$, $\beta_{i}$ and
$c_{i}$, which can theoretically vary between sections (representing a
waveguide with different materials in each section), but for our purposes we
will assume that the sections are of one and the same material. The parameter
$\varepsilon$ is the small wave parameter. The Boussinesq equations are
complemented with continuity conditions, namely continuity of longitudinal
displacement

$\left.u^{(i)}\right|_{x=x_{i}}=\left.u^{(i+1)}\right|_{x=x_{i}},$ (4)

and continuity of normal stress

$\sigma^{(i)}|_{x=x_{i}}=\sigma^{(i+1)}|_{x=x_{i}},$ (5)

where $\sigma^{(i)}$ is defined by our original equation (3) when written in
the form

$u_{tt}^{(i)}=\frac{\mathrm{d}\sigma^{(i)}}{\mathrm{d}x}.$

We consider $\alpha_{i}=1$ for all $i$, $\beta_{1,3}=1$ and

$\beta_{2}(n,k)=\frac{n^{2}+k^{2}}{n^{2}(1+k^{2})},$ (6)

where $n$ represents the number of layers in the structure and $k$ is defined
by the geometry of the waveguide. Referring to Figure 1, the cross section has
width $2a$ and the height of each layer is $2b/n$. In terms of these values,
as there are two layers, $n=2$ and $k=b/a$. In our numerical simulations we
will consider various $n$ and $k$ values.

####  2.1.1 Weakly-nonlinear solution

In order to find theoretical predictions for the evolution of the solitary
waves, we construct a weakly-nonlinear solution and use theoretical results
for the derived equations. For brevity, we only provide a summary of the
results below, more details can be found in Refs. [23, 24, 28, 29]. We seek a
weakly-nonlinear solution for the strains $f^{(i)}=u_{x}^{(i)}$ of the form

$\displaystyle f^{(i)}$
$\displaystyle=T^{(i)}(\xi,X)+R^{(i)}(\eta,X)+\varepsilon P^{(i)}(\xi,\eta,X)$
$\displaystyle~{}~{}~{}+\mathcal{O}\left(\varepsilon^{2}\right),$ (7)

where $\xi=x-c_{i}t$, $\eta=x+c_{i}t$ and $X=\varepsilon x$. Substituting the
respective weakly-nonlinear solutions into the differentiated form of (3),
then applying space-averaging (see [23, 28, 29]) yields the leading order
solutions

$\displaystyle
T^{(i)}_{X}-6\frac{\alpha_{i}}{c_{i}^{2}}T^{(i)}T^{(i)}_{\xi}+\beta_{i}T^{(i)}_{\xi\xi\xi}$
$\displaystyle=0,$ (8) $\displaystyle
R^{(i)}_{X}-6\frac{\alpha_{i}}{c_{i}^{2}}R^{(i)}R^{(i)}_{\eta}+\beta_{i}R^{(i)}_{\eta\eta\eta}$
$\displaystyle=0.$ (9)

To determine “initial conditions” for the equations derived in each section,
we substitute (7) into the continuity conditions (4) – (5) to find values for
$T$ and $R$ at the interface, in terms of the previous transmitted wave. This
gives rise to transmission and reflection coefficients in terms of $c_{i}$. As
we have assumed that the waveguide is one and the same material (so $c_{i}=1$
for all $i$) we have full transmission and no reflection.

####  2.1.2 Theoretical predictions

We can rewrite the transmitted wave equation (8) in the form

$U_{\tau}-6UU_{\chi}+U_{\chi\chi\chi}=0,\quad U|_{\tau=0}=U_{0}(\chi).$ (10)

For a sufficiently rapidly decaying initial condition $U_{0}(\chi)$ on the
infinite line, the solution to (10) is related to the spectral problem for the
Schrödinger equation

$\Psi_{\chi\chi}+[\lambda-U_{0}(\chi)]\Psi=0,$ (11)

where $\lambda$ is the spectral parameter. Finding the evolution of the
scattering data for the discrete and continuous spectra and using these to
reconstruct the solution to the KdV equation is known as the Inverse
Scattering Transform (IST) [32]. We can use the results from the IST to create
theoretical predictions for the solitons in the delaminated region, as well as
in the second bonded region.

We assume that there is an incident soliton in the first region, which is a
travelling wave solution and thus will move in time but retains its shape. To
illustrate the theoretical predictions we consider the second region, where we
have $\beta_{2}$ defined as in (6) and the initial condition for (10) in this
region then takes the form

$U_{0}(\chi)=-A\mathrm{sech}^{2}\left(\frac{\chi}{l}\right),\quad
A=\frac{v}{2\beta_{2}},\quad l=\frac{2}{\sqrt{v}}.$ (12)

In this case the solution will consist of either one soliton, or a series of
solitons, characterised by eigenvalues in the discrete spectrum, and
accompanying dispersive radiation determined by the continuous spectrum. In
some cases we may see the fission of the initial soliton, which is when more
than one soliton is generated, in particular when $\beta_{2}\neq 1$.

The discrete eigenvalues of (11) take the form $\lambda=-k_{n}^{2}$, where

$k_{n}=\frac{1}{2l}\left[\sqrt{1+4Al^{2}}-(2n-1)\right],$ (13)

for $n=1,2,\dots,N$. Recalling (6), the number of solitons, $N$, generated in
the delaminated region is given by the largest integer satisfying the
inequality

$N<\frac{1}{2}\left(\sqrt{1+\frac{8}{\beta_{2}}}+1\right).$ (14)

We can see from (14) that, for $\beta=1$ we will have one soliton, while for
$\beta<1$, we will have more than one soliton and as $\beta$ becomes smaller,
more solitons will be generated. This corresponds to either an increase in
layers in the waveguide, or a change in geometry. As $\tau\rightarrow\infty$,
the solution will evolve into a train of solitary waves, ordered by their
heights, propagating to the right and some dispersive radiation (a dispersive
wave train) propagating to the left (in the moving reference frame), i.e.

$\displaystyle U(\chi,\tau)$
$\displaystyle\sim-\sum^{N}_{n=1}2k_{n}^{2}\textrm{sech}^{2}(k_{n}(\chi-4k_{n}^{2}\tau-\chi_{n}))$
$\displaystyle~{}~{}~{}+\textrm{radiation},$ (15)

where $\chi_{n}$ is the phase shift. In the context of our problem, if there
is an infinite delamination then the solitons will separate and rank order,
while for finite delamination the solitons will only separate for a large
delamination. This allows us to create a measure of the delamination length,
by comparing the measured signal at the end of the bar to the theoretical
prediction.

We introduce the incident solitary wave for $T^{(1)}$, the exact travelling
wave solution of (8), as

$T^{(1)}(\xi,X)=-\frac{v}{2}\mathrm{sech}^{2}\left(\frac{\sqrt{v}}{2}\left(\xi-
vX\right)\right),$ (16)

where $v$ is the phase speed. Solitary waves are often measured in experiments
in terms of their Full Width at Half Magnitude (FWHM), so we rewrite this as

$-\frac{v}{2}\mathrm{sech}^{2}\left(\frac{\sqrt{v}}{4}\mathrm{FWHM}\right)=-\frac{v}{4},$
(17)

and hence we obtain

$v=\left(\frac{4}{\mathrm{FWHM}}\cosh^{-1}(\sqrt{2})\right)^{2}.$ (18)

This allows us to generalise the FWHM based measure to any size of incident
solitary wave.

###  2.2 Imperfect bonding case

The second case we consider is when we have a two layered waveguide with
“soft” bonding between the layers. This is illustrated in Figure 2.

Figure 2: Bi-layer structure with an initial soft bonded region for
$x_{0}<x<x_{1}$, a delaminated region for $x_{1}<x<x_{2}$ and a soft bonded
region for $x_{2}<x<x_{3}$. We assume that the materials in both layers are
similar, that is, their material properties differ by
$\mathcal{O}(\varepsilon)$.

The longitudinal displacement in the bonded regions is described by the
regularized non-dimensional equations

$\displaystyle u^{(i)}_{tt}-u^{(i)}_{xx}$
$\displaystyle=\varepsilon\left[-12u^{(i)}_{x}u^{(i)}_{xx}+2u^{(i)}_{ttxx}\right]$
$\displaystyle~{}~{}~{}-\varepsilon\delta(u^{(i)}-w^{(i)}),$ (19)
$\displaystyle w^{(i)}_{tt}-c^{2}w^{(i)}_{xx}$
$\displaystyle=\varepsilon\left[-12\alpha w^{(i)}_{x}w^{(i)}_{xx}+2\beta
w^{(i)}_{ttxx}\right]$
$\displaystyle~{}~{}~{}+\varepsilon\gamma(u^{(i)}-w^{(i)}),$ (20)

for $x_{i-1}<x<x_{i}$, while in the delaminated regions we have Boussinesq
equations

$\displaystyle u^{(i)}_{tt}-u^{(i)}_{xx}$
$\displaystyle=\varepsilon\left[-12u^{(i)}_{x}u^{(i)}_{xx}+2u^{(i)}_{ttxx}\right],$
(21) $\displaystyle w^{(i)}_{tt}-c^{2}w^{(i)}_{xx}$
$\displaystyle=\varepsilon\left[-12\alpha w^{(i)}_{x}w^{(i)}_{xx}+2\beta
w^{(i)}_{ttxx}\right].$ (22)

As with the perfectly bonded case, these equations are complemented with
continuity conditions at the interfaces between the sections. We have
continuity of longitudinal displacement

$\displaystyle u^{(i)}|_{x=x_{i}}=u^{(i+1)}|_{x=x_{i}},$ $\displaystyle
w^{(i)}|_{x=x_{i}}=w^{(i+1)}|_{x=x_{i}},$ (23)

and continuity of normal stress

$\displaystyle\sigma_{u}^{(i)}|_{x=x_{i}}=\sigma_{u}^{(i+1)}|_{x=x_{i}},$
$\displaystyle\sigma_{w}^{(i)}|_{x=x_{i}}=\sigma_{w}^{(i+1)}|_{x=x_{i}},$ (24)

for $i=1,2$, where $\sigma_{u}$ and $\sigma_{w}$ are defined by (19) and (20)
as

$\displaystyle u_{tt}^{(i)}$
$\displaystyle=\frac{\mathrm{d}\sigma_{u}^{(i)}}{\mathrm{d}x}-\delta(u^{(i)}-w^{(i)}),$
$\displaystyle w_{tt}^{(i)}$
$\displaystyle=\frac{\mathrm{d}\sigma_{w}^{(i)}}{\mathrm{d}x}+\gamma(u^{(i)}-w^{(i)}),$

respectively. We will consider the case here where the materials in the layers
are similar, namely $c-1=\mathcal{O}(\varepsilon)$. We can construct a weakly-
nonlinear solution to this system of equations, as was done in [28], however
we cannot obtain any direct theoretical predictions from this approach as the
derived coupled Ostrovsky equations are not solvable via the Inverse
Scattering Transform. Therefore, we will explore this case numerically to
determine a measure of delamination.

##  3 Numerical results

We now aim to use the derived weakly-nonlinear solution and the theoretical
predictions of Section 2 to introduce a measure of the delamination length in
terms of the change in wave structure. In this section, we first demonstrate
the effect of delamination on the transmitted soliton for both the perfectly
bonded and soft bonded waveguides in Section 3.1. We then introduce a measure
of the delamination length for the perfectly bonded case in Section 3.2, and
for the soft bonded case in Section 3.3. In both cases we consider how these
measures scale with respect to the incident soliton in order to rapidly
recompute results for a wide range of initial conditions. We will use the
finite difference scheme from Ref. [24] to solve the original Boussinesq
system and a semi-analytical method using a pseudospectral scheme for the
derived KdV equations for the perfectly bonded case, similar to the one used
for coupled Ostrovsky equations in [28]. In all cases our simulations will use
a grid spacing of $\Delta x=0.01$ and a time step of $\Delta t=0.01$ for the
finite-difference scheme. For the semi-analytical method we take $\Delta
X=5\times 10^{-4}$ and $\Delta\xi=0.1$, corresponding to $N=131,072$.

###  3.1 Examples of scattering

(a) $D=0$

(b) $D=50$

(c) $D=300$

(d) Comparison of schemes for $D=300$.

Figure 3: The solution at $t=900$ in the final section of the perfectly bonded
waveguide, for various delamination lengths, and comparison of the direct
numerical (blue, solid line) and semi-analytical (red, dotted line)
simulations. Parameters are $\varepsilon=0.1$, $\textrm{FWHM}=5.0$, $n=2$ and
$k=2$. The finite-difference method uses a computational domain of
$[-100,1000]$ and for the semi-analytical pseudospectral method we have
$N=131,072$.

Firstly, we demonstrate the effect of delamination on the propagation of an
incident solitary wave, in both scenarios described in Section 2. For the
perfectly bonded case, let us assume a spatial domain $x\in[-100,1000]$ and we
have a delamination starting at $x=0$ of length $D$. We present three cases:
no delamination, a delamination of length $D=50$ and a delamination of length
$D=300$. These results, as well as a comparison between the weakly-nonlinear
solution and the directly computed solution, are shown in Figure 3. We can see
from Figure 3(a) that, in the case of no delamination, the soliton propagates
without change in shape or structure. When delamination is introduced in
figures 3(b) and 3(c), the soliton fissions into two solitons with
accompanying dispersive radiation. Comparing the case of $D=50$ to $D=300$, we
can see that the second soliton has become more distinct from the radiation
and the first soliton has tended towards its theoretically predicted
amplitude. Figure 3(d) shows that there is a reasonable agreement between the
direct numerical simulation and the semi-analytical method when $D=300$, with
a slight phase shift and amplitude change. This will be reduced for smaller
values of $\varepsilon$ (as we have constructed a series in increasing powers
of $\varepsilon$), however we have qualitatively the same structure.

(a) $D=0$

(b) $D=100$

(c) $D=300$

Figure 4: The solution at $t=900$ in the final section of the upper layer of
the soft bonded waveguide, for various delamination lengths. Parameters are
$\varepsilon=0.05$, $\textrm{FWHM}=5.0$, $c=1.025$, $\alpha=\beta=1.05$,
$\delta=\gamma=1$. The finite-difference method uses a computational domain of
$[-500,1000]$.

Similarly, we consider the case of soft bonding and an incident soliton. We
assume a spatial domain $x\in[-500,1000]$, with a homogeneous (delaminated)
section for $x\in[-500,-400]$ to generate the same wave in both layers, bonded
sections $x\in[-400,0]$ and $x\in[D,1000]$, with a delaminated region for
$x\in[0,D]$. We again present three cases including when there is no
delamination, a shorter delamination length of $D=100$ and a larger
delamination with $D=300$. The results are shown in Figure 4 for the upper
layer, where the lower layer is qualitatively similar. In the case of no
delamination shown in Figure 4(a) we have a solitary wave with a one-sided
oscillatory tail, known as a _radiating solitary wave_. As the delamination
length increases, the solitary wave begins to lose amplitude and expel energy
into its tail through an exchange of energy between the layers. These are
clear signs of the presence of delamination in figures 4(b) and 4(c), with
structural changes that can be detected and the decay in amplitude can be
quantified to give a measure of the length of delamination. Note that in this
case there is no comparison between the simulation schemes since the semi-
analytical scheme is not applicable for the soft bonded case.

###  3.2 Measure of delamination length for perfect bonding

We now expand upon the observations from the previous subsection by
introducing a measure of delamination based upon the theoretical predictions
from Section 2. We then generalise this measure to apply for different
incident soliton widths. If we assume that the solitons are well-separated,
representing the case of infinite delamination, the amplitude of the soliton
can be found from the IST. We have

$\displaystyle A_{3}$ $\displaystyle=A_{1}k_{2}^{2}k_{3}^{2},\quad
k_{2}=\frac{1}{2}\left(\sqrt{1+\frac{8}{\beta_{2}}}-1\right),$ $\displaystyle
k_{3}$ $\displaystyle=\frac{1}{2}\left(\sqrt{1+8\beta_{2}}-1\right),$ (25)

where $A_{1}$ is the amplitude of the incident soliton, $A_{3}$ is the
amplitude of the lead soliton in the second bonded region, and $k_{2}$,
$k_{3}$ are the eigenvalues corresponding to the lead soliton amplitude in the
second and third regions, as determined by the IST. As the delamination length
is reduced, the amplitude in the third region will tend towards the initial
amplitude, $A_{1}$.

Denoting the calculated numerical solution as $A_{\mathrm{num}}$ from the
simulation, we introduce a measure of the amplitude of the lead soliton in the
third section of the bar in comparison to the incident soliton as

$\sigma=\frac{A_{\mathrm{num}}-A_{1}}{A_{3}-A_{1}}\times 100.$ (26)

This corresponds with the measure used in [24], where it was assumed that
$\mathrm{FWHM}=5$. We now compute the solution using the semi-analytical
pseudospectral scheme for a wide range of values of FWHM with the aim of
determining a general rule for the delamination length. The delamination
length is chosen to be $D\in[0,300]$ and we measure the delamination in
multiples of FWHM.

The results are plotted in Figure 5(a) for $n=3$, $k=3$, and in Figure 5(b)
for $n=4$ and $k=3$. We can see that, as the delamination length increases,
the measure $\sigma$ increases until it tends to a value of $1$, and this
behaviour is replicated for different values of FWHM. For larger FWHM it may
not reach this limit for the chosen delamination length. We can also see a
similar behaviour for different values of $n$ and $k$.

(a) Delamination measure for $n=3$, $k=3$.

(b) Delamination measure for $n=4$, $k=3$.

Figure 5: Plots of the change in amplitude of the lead transmitted soliton, in
comparison to the incident soliton and theoretical predictions, as measured by
$\sigma$, for various values of FWHM. Here we take $\varepsilon=0.1$.

To generalise this approach, we consider the equation for the phase speed $v$
in terms of FWHM (18). We can see that $v$ in inversely proportional to the
square of FWHM. Thus, fixing our reference as $\mathrm{FWHM}=5$, we let
$\sigma$ be a function of delamination length, parametrised by FWHM, and we
introduce the scaling

$\tilde{\sigma}(D;\mathrm{FWHM})=\frac{\mathrm{FWHM}^{2}}{25}\sigma(D;\mathrm{FWHM}).$
(27)

The resulting plots are shown in Figure 6(a) for $n=3$, $k=3$, and in Figure
6(b) for $n=4$, $k=3$. We see that the scaled versions are very closely
aligned, with the only disagreement stemming from the restriction on
delamination length for larger values of FWHM. Therefore, after computing one
curve for the smallest value of FWHM, we can reproduce all subsequent curves
for any value of FWHM. This allows for the highly efficient computation of the
delamination curves and for a wide range of experiments with incident waves of
different amplitude.

(a) Scaled delamination curve $n=3$, $k=3$.

(b) Scaled delamination curve $n=4$, $k=3$.

Figure 6: Plots of the scaled delamination curves for the change in amplitude
of the lead transmitted soliton, in comparison to the incident soliton and
theoretical predictions, as measured by $\sigma$. Here we take
$\varepsilon=0.1$.

###  3.3 Finite delamination with soft bonding

We now consider the soft bonded case outlined in Section 2.2. This case was
also studied in Ref. [28], but for only one value of FWHM. In this work we
compute the solution for a wide range of FWHM, using the finite-difference
scheme in [24]. The constructed weakly-nonlinear solution consists of coupled
Ostrovsky equations, in contrast to the KdV equations in the previous case
[28]. Therefore we cannot use the IST to predict the amplitude of the waves in
the bonded regions as coupled Ostrovsky equations are not integrable via the
IST. The incident soliton in this case evolves into a radiating solitary wave,
that is a solitary wave with a one-sided oscillatory tail.

To determine the change in amplitude, as a measure of the delamination length,
we denote the amplitude of the soliton or wave packet in each region as
$A_{L}$, where $L$ is the region index, and we introduce the measure

$\zeta=\frac{|A_{1}-A_{3}|}{A_{1}}\times 100.$ (28)

Figure 7(a) presents the results for $\zeta$, computed over a wide range of
FWHM values, for the upper layer. A similar agreement is seen for the lower
layer, but the results are omitted for brevity. Following the idea from the
perfectly bonded case, we introduce a quadratic scaling. We choose a reference
FWHM value (in this case we choose our lowest value of FWHM, namely
$\mathrm{FWHM}=5$) and then calculate a scaling of the form

$\tilde{\zeta}=\frac{\zeta}{a+b\ \overline{\mathrm{FWHM}}+c\
\overline{\mathrm{FWHM}}^{2}},$ (29)

where we have introduced

$\overline{\mathrm{FWHM}}=\frac{\mathrm{FWHM}}{5},$

and $a$, $b$, $c$ are constants to be found that satisfy the relationship
$a+b+c=1$. The results for $a=0.49$, $b=0.28$, $c=0.23$ are plotted in Figure
7(b) and we can see a good agreement across a range of values of FWHM.
However, this fitting is done by careful choice of parameters rather than
theoretical prediction, as for the previous case. The agreement begins to
worsen slightly after a delamination of 40 units of FWHM, which corresponds to
a minimum of 200 in nondimensional units, however overall the agreement is
still good.

(a) Amplitude change in upper layer

(b) Amplitude change in lower layer

Figure 7: Change in amplitude of the radiating solitary wave in the soft
bonded structure, for various delamination lengths and values of FWHM.
Parameters are $\varepsilon=0.05$, $\delta=\gamma=1$.

We now summarise the scaling used to convert our nondimenesional variables to
dimensional material parameters that can be compared to experimental data, in
order to confirm whether a delamination length of 200 is reasonable. Referring
to the dimensional form of the DDE in (1), with parameters (2), we introduce
the scaling to nondimensional form via

$\tilde{x}=\frac{x}{X},\quad\tilde{f}=\frac{f}{F},\quad\tilde{t}=\frac{t}{T},$
(30)

where

$\displaystyle X$ $\displaystyle=\sqrt{\frac{J\nu^{2}}{2\varepsilon\sigma
c^{2}}\left(c^{2}-c_{1}^{2}\right)},\quad T=\frac{X}{c},$ $\displaystyle F$
$\displaystyle=-\frac{12\varepsilon c^{2}\rho}{\beta}X.$ (31)

We can therefore find the corresponding material length given the
nondimensional length. Let us assume a PMMA bar of 10mm $\times$ 10mm cross-
section, then using the parameters for PMMA from [20] we find that, for
$\varepsilon=0.1$, a delamination length of $x=200$ in nondimensional units
corresponds to a length of approximately $x=520$mm, which is significant given
the experimental materials are usually around $600$mm long in total.
Therefore, restricting our considerations to values of delamination less than
200 nondimensional units is reasonable in the context of practical
applications.

##  4 Conclusion

In this paper we have considered the scattering of a bulk strain solitary wave
in a delaminated bi-layer with either perfect or soft bonding between the
layers. The longitudinal displacements are modelled by either Boussinesq
equations (perfectly bonded or delaminated sections) or coupled Boussinesq
equations (soft bonded sections), with continuity conditions on the interface.
Incident solitary waves undergo fission in delaminated regions in the
perfectly bonded structure, providing a clear indicator of delamination.

We construct theoretical estimates for the amplitude of the solitons after a
delaminated region, using the Inverse Scattering Transform. A measure is
introduced using the theoretical and observed values to predict the
delamination length based upon amplitude changes. This is then extended for
incident waves of different Full Width at Half Magnitude, and a quadratic
scaling is introduced and verified by numerical results. The significance of
this result is that we now only need to compute a single curve in order to
perform a wide range of experiments, which significantly reduces computation
times and allows for further experiments (with different incident solitons) to
be performed rapidly. This was confirmed for various configurations of the
waveguide.

In the case of a soft bonded waveguide with delamination, theoretical
estimates cannot be derived using the Inverse Scattering Transform. In this
case we resort to direct computation of the solution and a comparison between
the amplitude after delamination and the corresponding amplitude for the non-
delaminated case. A similar quadratic scaling can be found, which has a good
agreement up to a delamination length of 200 in nondimensional units or 520mm
in physical units. This is consistent for both layers of the waveguide.

This work facilitates experimentation with a wide range of initial condition
parameters, and provides a framework for detecting delamination in perfectly
bonded and soft bonded waveguides with similar materials in the layers. The
case with distinctly different materials in the layers is more complex, and
some preliminary studies have been conducted into quantifying delamination
[29].

## Declarations

  * •

Funding: Jagdeep S. Tamber would like to thank Nottingham Trent University for
funding through their PhD studentship scheme.

  * •

Competing interests: The authors have no relevant financial or non-financial
interests to disclose.

  * •

Ethics approval: Not applicable.

  * •

Consent to participate: Not applicable.

  * •

Consent for publication: Not applicable.

  * •

Availability of data, code and materials: The datasets generated during this
study can be reproduced using equations throughout the paper and the cited
numerical methods. The codes used to generate the datasets are available from
the corresponding author on reasonable request.

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