NON-DESTRUCTIVE EVALUATION USING GUIDED WAVES IN PIPE-LIKE
STRUCTURES
By
Jingjun Shen

A THESIS
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
MASTER OF SCIENCE
Civil Engineering
2012

ABSTRACT

NON-DESTRUCTIVE EVALUATION USING GUIDED WAVES IN PIPE-LIKE
STRUCTURES
By
Jingjun Shen
For the last hundred years, rapid development in economy and engineering have accelerated
the construction of large amount of different types of civil structures, machines, and airplanes. As
time goes by, those structures and machines have deteriorated and become unable to perform their
designed functions due to the defects generated during their service lives. To test the integrity
of structures and evaluate their health states, in the past several decades, many methods has been
developed and utilized. Ultrasonic guided waves have been utilized for accurate diagnosis of the
structures that have thin structural components because of their good performance in long distance
propagation and sensitivity to defects in the structures. In this thesis, wave propagations excited
by surface-mounted instruments such as PZT and transducer rings attached on an inﬁnite isotropic
hollow cylinder are investigated. A mathematical model of the wave propagation system is studied
both analytically and numerically. A detailed derivation of the characteristic equation for pipes
is conducted, and the development of waves is simulated using the ﬁnite element method (FEM).
Compared with the analytical results, the accuracy of the numerical modeling is veriﬁed, and Lamb
wave propagation in pipes with defects is studied as well. By parametric study, the inﬂuence of the
defect depth on the received signals is investigated.

ACKNOWLEDGMENTS

It is a pleasure to thank those who made this thesis possible. I would like to thank my advisor
Professor Jung-Wuk Hong, who has helped and supported me throughout my master’s study. I am
grateful to his patient guidance, and priceless and insightful advices on every stage of my study,
from the early stage when I chose my research topic to ﬁnishing up this thesis. Without his encouragement and help, I cannot complete this thesis. I am also deeply indebted to Professors Lalita
Udpa and Venkatesh Kodur for their inspirations and encouragements through out my research and
writing of this thesis.
I want to thank Department of Civil and Environmental Engineering and all my colleague for
their help in my research work and my life at MSU. Especially I want to thank Wenyang Liu. We
discussed on my research and he inspired me signiﬁcantly on how to realize my expectations of
my research. His suggestions on my research helped me to complete the thesis.
A special thank goes to my family and my friends, especially Mr. Jun He and Ms. Jiaxun Yu,
for their undivided support and encouragement during my master’s studies. Without their sincere
love and support, I would not be able to complete my thesis.

iii

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . .
List of Figures

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vi

. . . . . . . . . . . . . . . . . . . . vii

List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Structural Health Monitoring and Non-Destructive Evaluation
1.2 PZT and MFC Transducers . . . . . . . . . . . . . . . . . . .
1.2.1 Piezoelectric Effect and PZT Actuator-Sensor . . . . .
1.2.2 MFC Transducers . . . . . . . . . . . . . . . . . . . .
1.3 History of Non-Destructive Evaluation Using Guided Waves .
1.3.1 Theory of Lamb Waves . . . . . . . . . . . . . . . . .
1.4 Lamb Wave Inspection in Industries . . . . . . . . . . . . . .
1.5 Motivations and Objectives of the Research . . . . . . . . . .

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Chapter 2
Lamb Wave in Plates . . . . . . . . . . . . . . . . . . . . . . .
2.1 Characteristic Equations . . . . . . . . . . . . . . . . . . .
2.2 Dispersion Curves . . . . . . . . . . . . . . . . . . . . . . .
2.3 Case study of Lamb wave propagation in an aluminum plate

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Chapter 3
Analytical Calculation of Wave Propagation in Pipe-like Structures
3.1 Derivation of Characteristic Equation . . . . . . . . . . . . . . . .
3.1.1 Mathematical Model . . . . . . . . . . . . . . . . . . . .
3.1.2 Governing Equation . . . . . . . . . . . . . . . . . . . . .
3.1.3 Displacement Field . . . . . . . . . . . . . . . . . . . . .
3.1.4 Stress Fields in Cylindrical Coordinate . . . . . . . . . . .
3.1.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . .
3.1.6 Residue Theorem . . . . . . . . . . . . . . . . . . . . . .

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3.2

3.3

Dispersion Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Name of Dispersion Curves . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Dispersion Curves for Pipes with Different Radius and Wall Thickness
Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4
Numerical Simulation . . . . . . . . . . . . . . . . . . . . .
4.1 Finite Element Formulations . . . . . . . . . . . . . . .
4.2 Comparison of Numerical and Analytical Results . . . .
4.3 Lamb Waves in Pipes with Defects . . . . . . . . . . . .
4.4 Parametric Study of Responses for Different Defect Sizes

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Chapter 5
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Bibliography . . . . . . . . . . . . . . .

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LIST OF TABLES

Table 2.1

Material Properties of the Steel Plate . . . . . . . . . . . . . . . . . . . . 16

Table 2.2

Material Properties of The Aluminum Plate . . . . . . . . . . . . . . . . 16

Table 3.1

Bessel functions in different regions [26] . . . . . . . . . . . . . . . . . . 24

vi

LIST OF FIGURES

Figure 1.1

Possible areas to apply SHM systems: (a) the Golden Gate Bridge [21],
and (b) an airplane [1]. (For interpretation of the references to color in
this and all other ﬁgures, the reader is referred to the electronic version of
this thesis.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Figure 1.2

(a) Angular sweep with angle wedge, and (b) B-scan [13] . . . . . . . . .

5

Figure 1.3

Schematics of PZT actuator. . . . . . . . . . . . . . . . . . . . . . . . .

6

Figure 1.4

Schematic of MFC transducer [33]. . . . . . . . . . . . . . . . . . . . . .

7

Figure 1.5

(a) Cylindrical phased array [35] and (b) mathematical model for longitudinal wave propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 2.1

Schematic of Lamb wave in a plate. . . . . . . . . . . . . . . . . . . . . 13

Figure 2.2

Lamb wave modes in a plate. . . . . . . . . . . . . . . . . . . . . . . . . 14

Figure 2.3

Dispersion curves for (a) wavenumber; (b) phase velocity. . . . . . . . . . 18

Figure 2.4

Input waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 2.5

Comparison of experimental data and numerical simulation. . . . . . . . . 19

Figure 3.1

Contours for complex domain integration: (a) upper semicircle, and (b)
lower semicircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Figure 3.2

Different wave modes in a cylindrical system. . . . . . . . . . . . . . . . 35
vii

Figure 3.3

Dispersion curves for pipes of the outer radius co = 136 mm with the
thicknesses (a) t = 1 mm and (b) t = 2 mm, (c) t = 4 mm and (d)
t = 6 mm, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Figure 3.4

Dispersion curves of pipes of the thickness t =
radii (a) co = 13 mm and (b) co = 25 mm, (c)
co = 136 mm, respectively . . . . . . . . . . . .

Figure 3.5

Dispersion Curve of a pipe with ro = 136 mm and t = 1 mm: (a) vph
versus f and (b) ξ versus f . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Figure 3.6

(a) Tangential traction in time domain and (b) magnitudes of Fourier coefﬁcients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Figure 3.7

Displacements on the outer surface of a pipe with co = 136 mm and
t = 1 mm: (a) uz and (b) ur on the outer surface. . . . . . . . . . . . . . 40

Figure 4.1

(a) An example of discretized FE model, and (b) an element of the model.

Figure 4.2

Comparison of displacement ﬁeld uz on the outside surface of a pipe with
co = 136 mm and t= 1 mm using mesh size in z direction of: (a) 1 mm,
and (b) 0.25 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 4.3

Schematic of a pipe with a defect at z= 0.4 m. . . . . . . . . . . . . . . . 48

Figure 4.4

Comparison of displacements measured in pipes (one is intact, and the
other is damaged): Displacements uz are measured (a) at z= 0.25 m, (b)
at z= 0.45 m, and (c) at z= 0.6 m. . . . . . . . . . . . . . . . . . . . . . . 50

Figure 4.5

Comparison of displacements measured at z = 0.6 m on the outer surface
of pipes with defect width of 1 mm vs 2 mm. . . . . . . . . . . . . . . . . 51

Figure 4.6

Comparison of displacements measured at z = 0.35 m on the outer
surface of pipes with: (a) non-defect vs defect depth of 0.25 mm; (b)
non-defect vs defect depth of 0.5 mm; (c) non-defect vs defect depth of
0.75 mm; (c) non-defect vs defect depth of 1 mm; (e) defect depth of
0.25 mm, 0.5 mm, 0.75 mm and 1 mm. . . . . . . . . . . . . . . . . . . . 52

viii

1 mm with the outer
co = 50 mm and (d)
. . . . . . . . . . . . . 37

45

List of Notations
ǫ

Strain vector

H

Vector potential function in a pipe

B

Strain-displacement matrix

Hu

Shape function

K

Stiffness matrix

M

Mass matrix

u

Displacement vector

∆

Determinant of dispersion equation

λ, µ

Lam´ constants
e

∇

Three dimensional differential operator

ν

Poisson’s ratio

ω

Angular frequency

φ

Scalar potential function

ψ

Vector potential function in a plate

ρ

Mass density

σij

Stress component

ξ

Wavenumber

c1

Longitudinal wave speed

c2

Transverse wave speed

ci

Inner radius of a pipe

co

Outer radius of a pipe

E

Young’s modulus

B
fi

Body force
ix

h

Half of plate thickness

In , Kn Modiﬁed Bessel functions
Jn

Bessel function of ﬁrst kind

t

Time

ur

Radial displacement

uz

Axial displacement

vph

Phase velocity

x, y, z Coordinates
Yn

Bessel function of second kind

x

Chapter 1
Introduction
On August 1st, 2007, during the evening rush hour, I-35W Mississippi River Bridge collapsed
suddenly, and killed 13 people. According to the report from National Transportation Safety Board
[20], the disaster was caused by failure of the undersized gusset plate, which was not able to carry
the increased concrete surface load. Back to history, hundreds of structural failures like I-35W
Bridge collapse happened in the world, and most of them were caused by poor maintenance and
lacking of inspection. On August 1st 1976, a road bridge with trams in Australia failed due to
column fracture; On July 17th 1981, a double-deck suspended footbridge in Hyatte collapsed at
Kansas City, Missouri because of overload and weak joints; On May 12th 2002, the Buran hangar
in Kazakhstan collapsed due to structural failure caused by poor maintenance. All these disasters sounded an alarm that more attention is needed to be paid on the health states of the existing
structures. Structural damages and failures threaten human lives and at the same time cause great
ﬁnancial losses. The United States spends a huge amount of money, more than 200 billion dollars
per year, just for the maintenance of aircrafts, civil structures and mechanical engines. Structural Health Monitoring (SHM) emerges as an important engineering technique since it enables
a cost-effective way to inspect structural health and provides basis for condition-based structural
maintenance.

1

1.1

Structural Health Monitoring and Non-Destructive Evaluation

Before the review of structural health monitoring, it is important to introduce the deﬁnition of
damage. In terms of structural health, damage is deﬁned as the change introduced to the original
system, which can cause adverse effects to its designed function or performance. All damages
begin from the material level and grow to the components or the structure level when the structures
undertake different kinds of loadings [34]. The adverse effects caused by damages might come
immediately or in the future, and even a very small local damage may cause global collapse of
the structures eventually, just as the collapse of I-35W Bridge. Therefore, our goal is to develop
structural health monitoring systems which are sensitive to different kinds of damages, and it will
enable us to avoid sudden failures of the structures by predicting the possible long term failures.

(a)

(b)

Figure 1.1: Possible areas to apply SHM systems: (a) the Golden Gate Bridge [21], and (b) an
airplane [1]. (For interpretation of the references to color in this and all other ﬁgures, the reader is
referred to the electronic version of this thesis.)

Structure Health Monitoring refers to a wide rang of techniques, which include sensors, smart
2

materials, data transmission, signal processing and analysis, and so on. It is an integrated process,
which aims to diagnose damages, defects and ﬂaws in structures by collecting data of structural
responses under certain excitations or surrounding environments and analyzing those data. By
conducting SHM, it is possible to determine the ’state’ of the structural health and to predict the
failure of the structures. There are ﬁve main objectives of SHM [25], which are defect detection
and location, defect identiﬁcation, defect assessment, defect monitoring, and failure prediction. On
the other hand, SHM is usually conducted during the operation of the structures. It provides the
real-time structural behavior under different excitations, which could be valuable information for
better design and management of structures. Figure 1.1 shows the possible areas to apply SHM.
However, there are two main challenges in SHM techniques. One of the most signiﬁcant challenges is signal interpretation. During the vibration-based SHM, the received signals are inﬂuenced
not only by the damages in the structures, but also by the environmental conditions, the operational
variabilities, and so on. Because of the complexity of the surrounding environments, diagnosis of
the defects might be inaccurate. Another challenge is that SHM is often carried out by low frequency global vibration tests. This means typical local-based damages/defects might not be able to
inﬂuence the result of the vibration test. Therefore, immediate structural collapse caused by local
damages might not be predictable using the conventional techniques.
Non-Destructive Evaluation (NDE) is also developed to detect the ﬂaws/faults in structures.
One of the main difference between SHM and NDE is that in an SHM system, the locations of
sensors, and the instrumentations are ﬁxed from measurement to measurement, while in an NDE
system the sensors and instrumentations are movable and the evaluation results of structures vary
[9]. High ﬂexibility makes NDE capable to be conducted at different locations of the structures
and detect damages which might not be detected by SHM. Another difference between SHM and

3

NDE is that for NDE, evaluation of the remaining lifetime is usually not a goal of NDE.
Compared to SHM, NDE is conducted more locally and emphasizes more on the characteristics
of the defects, such as orientation and size, or the severity of the damages [14]. There are mainly
two types of NDE system, passive NDE and active NDE. The passive NDE systems use sensors to
record the loads, stress in the structures and the inﬂuence of surrounding environments to the structures [17]. The passive NDE listens to the behavior of the structures under different load scenarios
and environmental conditions. Active NDE uses transmitters to emit signals to the structures, and
detects the presence and the characteristics of the damages by analyzing the responses given by the
structures.
With the development of new materials and computational capacity, NDE techniques are rapidly
developed in the past several decades. There are several major methodologies to conduct nondestructive evaluation. One of them is ultrasonic testing. For ultrasonic testing, transducers are
applied to excite waves (vibrations) to propagate along the designed path. The signals passing
through the materials or structures will be reﬂected when they meet the cracks, corrosions, or
ﬂaws in the materials, and received by the applied sensors. After that, the collected data will be
analyzed to ﬁnd out the locations and the sizes of the defects. Figure 1.2 shows the schematic of
ultrasonic NDE.
Another commonly used method is performed by using dye penetrant. The method applies a
penetrating liquid over the surface of the structure which needs to be detected. The liquid will
enter to the discontinuities of cracks on the structures. The limitation of this method is that it is
time consuming and usually used to detect surface ﬂaws.
There is another non-destructive evaluation method called radiography method. It is used for
most projects over a century. The industrial radiography uses an X-ray device as a source of

4

Angle
Wedge

θ

(a)

(b)
Figure 1.2: (a) Angular sweep with angle wedge, and (b) B-scan [13]

radiation. When the radiographic image taken by the X-ray device is processed, the image of
varying density in the structure is obtained. By analyzing the image, the location and size of
material imperfection can be identiﬁed.
In this thesis, we investigate NDE using Lamb waves in pipe-like structures. Lamb wave
inspection is recently developed and utilized in non-destructive evaluation for plates and pipe-like
structures due to its good performance in detecting the defects in thin-walled structures.

1.2

PZT and MFC Transducers

In order to excite Lamb waves in thin-walled structures, transducers are needed to be mounted on
the surfaces of the structures and to emit different types of signals. In this section, we introduce two
different transducers, which are the PZT actuator-sensor and the MFC transducer. The mechanisms
of how they work are explained and their advantages and disadvantages are discussed.
5

+ + + + + + + + +

Voltage Voltage
applied not applied

PZT

_ _ _ _ _ _ _ _ _

Figure 1.3: Schematics of PZT actuator.

1.2.1 Piezoelectric Effect and PZT Actuator-Sensor

PZT is the abbreviation of Piezoelectric ceramic Lead Zirconate Titanate. It uses the piezoelectric
effect to measure the stress or strain in a structure by converting them to electrical charge or vice
verse. Figure 1.3 shows the piezoelectric effect of a PZT sensor.
Piezoelectric effect is discovered by Pierre Curie in late nineteenth century, however its ﬁrst
application in sensing was 70 years later in 1950s. Within several decades, PZT has become one of
the most commonly used piezoelectric materials because of its high piezoelectricity and sensitivity
to stress and strain change. However, it is brittle and is not able to work in harsh environments
for long time. Another disadvantage of PZT is that its ﬂexibility is not good enough for curved
surfaces such as surfaces of pipelines. Therefore, a more ﬂexible actuator-sensor which can be
perfectly mounted at the surface of pipe-like structures is developed.
6

Kapton

Acrylic

PZT
Fiber

Copper
Electrodes

Epoxy

Figure 1.4: Schematic of MFC transducer [33].

1.2.2 MFC Transducers
The Macro-Fiber Composite (MFC) is an innovative, low-cost piezoelectric device developed by
NASA in 1999 for controlling vibration and noise. The MFC is an actuator-sensor in the form of a
thin patch. It consists of unidirectionally rectangular piezoceramic rods sandwiched between two
layers of ﬁlms, which containing tiny electrodes that can transfer a voltage directly to and from
ribbon-shaped rods. Just like PZT, the MFC will stretch when it is subjected to a voltage. Figure
1.4 shows the schematic of the components of MFC. Compare to traditional piezoceramics, there
are several advantages of MFC. Firstly, it is ﬂexible and can be perfectly mounted at curved-shape
structures, such as aircraft wings, pipelines, etc. Secondly, ﬁne ceramic ﬁbers provide higher
strength and energy density over monolithic piezoceramics [27]. This makes MFC have higher
durability to the harsh surrounding environments.
7

1.3

History of Non-Destructive Evaluation Using Guided Waves

Lamb waves, different from ultrasonic bulk waves, exist in free plates and pipes, and the Lamb
wave particles move in two-dimensional vibration modes: the direction of wave propagation and
the normal direction of the plate or pipe surface. Lamb waves propagate in plates of thickness less
than 5λ, where λ is the Rayleigh wavelength [2]. When the plate thickness is greater than 5λ, the
waves behave like Rayleigh waves, which are the surface waves existing on the boundary of a free
half space.

1.3.1 Theory of Lamb Waves
Lamb waves are named after a British mathematician, Horace Lamb [22], to acknowledge his
work on investigating the characteristics of vibration of a plate, the thickness of which is much
smaller than the wavelength. His research demonstrated the difference between Lamb waves and
bulk waves, and described the Lamb waves in mathematical equations. There are several reasons
which make Lamb wave to be a relatively ideal tool to detect the defects in thin-walled waveguides. Firstly, Lamb waves can propagate for a longer distance than bulk waves. This makes
lower cost non-destructive evaluation possible, since for longer pipelines fewer devices are needed
for exciting and receiving Lamb wave signals. Secondly, Lamb waves are sensitive to defects in
the waveguides. By selecting different wave modes, defects with different characteristics can be
detected. Thirdly, due to the reﬂection by the boundaries, Lamb waves propagate through the
thickness of the waveguides. It means Lamb wave inspection is able to detect defects not only on
the surfaces of the waveguides but also inside the waveguides [7].
The propagation of guided waves has been studied for over a century. In 1889, Pochhammer
8

and Chree [8] ﬁrst investigated the wave propagation in a free inﬁnite long cylindrical rod. At
the same time, Lord Rayleigh [28] and Horace Lamb considered the problem of wave propagation
in an elastic solid bounded by two parallel surfaces, the distance of which is much smaller than
the wavelength. In the following few decades, theories of wave propagation in solid materials
were fully developed by many researchers, such as Graff [18], Miklowitz [24], etc. Based on
the theory developed by Pochhammer and Chree, Gazis [15] derived the characteristic equation
of the guided waves propagating in a hollow isotropic cylinder, and calculated the solutions of
the equation numerically [16]. Since then, dispersion curves of wave propagation in a hollow
cylinder were developed, and the characteristics of guided wave modes were interpreted. After
Gazis’s work, a great deal of research has been conducted to investigate the wave motions in more
complicated systems such as pipelines with multilayers [26], cylindrical structures surrounded by
different environments [3, 31], and pipelines ﬁlled with ﬂows.

1.4

Lamb Wave Inspection in Industries

Although the theoretical analysis of Lamb wave propagation in pipe-like structures has been well
developed, there were few practical applications of guided waves for the inspection of pipes before
1990 due to the limitation of computational calculations [30]. From 2000, with rapid improvement
of computational capacity, the techniques of guided wave inspection in pipelines were signiﬁcantly
improved, including transducer design and signal analysis. A typical NDE method used for pipelike structures is shown in Figure 1.5 (a) [35]. A transmitter ring, which acts as a receiver as well, is
mounted on a pipe, and different types of signals are emitted from the elements. Once Lamb waves
are excited, the waves travel through the pipeline. When these waves meet defects in it, the waves
change the modes and get reﬂected or diffracted. Therefore, the locations and sizes of defects can
9

be diagnosed by analyzing the received signals. In this thesis, an analytical solution is developed
based on the mathematical model as shown in Figure 1.5 (b), and the solution is compared with
numerical results.

r

θ

z
(a)

(b)

Figure 1.5: (a) Cylindrical phased array [35] and (b) mathematical model for longitudinal wave
propagation.

1.5

Motivations and Objectives of the Research

There are million miles of pipelines across the world, carrying important resources such as water, oil, gas, etc., and meanwhile they are exposed to harsh environments. Those pipelines are
threatened by corrosion, weathering, or mechanical impact, which may cause the leakage of the
transported materials. Therefore, it is urgently required to evaluate the integrity of the pipelines
and ensure safe transportations of the resources.
As mentioned in the previous section, for thin-walled pipelines, the most commonly used nondestructive evaluation tool is Lamb wave. It overcomes the disadvantages of traditional ultrasonic
testing, and the development of industrial techniques also accelerates the application of Lamb wave
inspection.
10

In this thesis, our objectives are to:
• Study Lamb wave propagation in pipe-like structures both analytically and numerically;
• Compare the analytical and numerical results to verify the accuracy of numerical simulation;
• Simulate Lamb wave propagation in pipes with defects;
• Investigate the received signals by parametric study for different defect depths.
The calculated analytical solution is used as a reference signal to verify the accuracy of numerical simulation. On the other hand, once the accuracy of the numerical model is veriﬁed, it is used
to simulate the wave behaviors in pipes with different sizes of defects, and provides a reference
to track the development of the defects and investigate the inﬂuence of defect size on the received
signals.

11

Chapter 2
Lamb Wave in Plates
A thin plate has two paralleled surfaces which form a guide to the Lamb waves and guide
the direction of wave propagation in the plates. For practical application, the materials used in
structures, such as steel and aluminum, have low damping inside the materials and the energy
losses of the Lamb waves are very limited [12]. This is the reason for the long distance propagation
of Lamb waves.
There are two basic types of wave modes existing in a plate, which are the symmetric modes
Si and the antisymmetric modes Ai , respectively. For symmetric wave modes, the particles vibrate
symmetrically with respect to the mid-plane of the plate, while for antisymmetric modes, the wave
modes are antisymmetric to the mid-plane of the plate. The schematics for the two modes are
shown in Figure 2.2. The subscript i denotes the order of the wave mode, and it is assigned
following the order of the cutoff frequency of each wave mode. As the frequency increases, the
number of wave modes in a plate increases as well, thus it is possible that inﬁnite number of wave
modes might exit in a plate.
In order to utilize Lamb waves for plate inspection, a large amount of work has been done
to investigate the characteristics of wave propagation analytically. In this chapter, the derivation
of dispersion equations of Lamb wave propagation in plates is outlined. Detailed investigation of
elastic wave propagation in solid materials can be found in many textbooks [4, 18, 32].
12

z

o

x

2h
y
Figure 2.1: Schematic of Lamb wave in a plate.
A comprehensive solution of Lamb waves was developed by Mindlin in 1950, and Viktorov
gave the detailed investigations of Lamb waves in plates, including dispersion curves and mode
shapes in 1967 [37].

2.1

Characteristic Equations

The mathematical model of Lamb wave propagation in a plate is shown in Figure 2.1. Consider a
plate with inﬁnite extent in x and y directions, and thickness of 2h. It is assumed that the plate is
made of an isotropic material and is placed in vacuum. The wave equations [37] of longitudinal
and shear Lamb wave are expressed as:
∂ 2φ ∂ 2φ ω2
+
+
φ=0
∂x2 ∂z 2
c2
1
2ψ ∂ 2ψ ω2
∂
+
+
ψ=0
∂x2
∂z 2
c2
2

where c1 =

λ+2µ
ρ , c2 =

(2.1)

µ/ρ are the longitudinal and shear wave speeds respectively.

Here, λ and µ are Lam´ ’s constants, ρ is the mass density of the material. φ, ψ are the scalar and
e
13

Antisymmetric

Symmetric

Figure 2.2: Lamb wave modes in a plate.

vector potential functions which are given as:

φ = [A1 sin(pz) + A2 cos(pz)]i(ξx − ωt),

(2.2)

ψ = [B1 sin(qz) + B2 cos(qz)]ei(ξx−ωt) .
where p2 = ω 2 /c2 − ξ 2 , q 2 = ω 2 /c2 − ξ 2 . A1 , A2 , B1 and B2 are the unknown constants which
1
2
are determined by boundary conditions, ω is the angular frequency, and ξ is the wavenumber.
For a free plate, by applying the boundary conditions that the components of stress ﬁelds are
zero at z = h and z = −h. The characteristic equations are obtained as:
4ξ 2 qp
tan(qh)
=−
tan(ph)
(ξ 2 − q 2 )2
tan(qh)
(ξ 2 − q 2 )2
=−
tan(ph)
4ξ 2 qp
Eq. (2.3) are for symmetric and anti-symmetric wave modes, respectively.
14

(2.3)

2.2

Dispersion Curves

Although Lamb waves have unparalleled advantages in long distance structural inspection, signal
interpretation is a major issue which makes the development of the application of Lamb wave
inspection much slower than its theory. As discussed above, with high frequency excitation, a
large number of Lamb wave modes exist in a plate. Those modes are of different amplitudes
and phase velocities and this make it difﬁcult to distinguish the incident signals and the signals
reﬂected/refracted by the defects. On the other hand, Lamb waves are dispersive when they travel
through the waveguides. It means that when the distance the wave traveling is larger, the amplitudes
of the wave modes decreases, which also increases the difﬁculty to ﬁnd the ﬂaws in the structures.
Therefore, one of the goals is to limit the number of wave modes in the plate and ﬁnd a wave mode
which is suitable for the inspection.
Dispersion curves contain the information of the characteristics of each wave mode and thus
is important for selecting proper wave modes for damage detection. In order to calculate the
dispersion curves in a certain plate, the material properties are chosen as shown in Table 2.1.
Figure 2.3 shows the dispersion curves for a plate of 1 mm thickness. It is generated by a free
software PACshare DispersionPlus Curves (Physical Acoustics Corporation, Princeton Junction,
NJ, USA) [10]. From Figure 2.3, we can see two different ways to present dispersion curves.
Figure 2.3 (a) shows the dispersion curves in wavenumber projection. Wavenumber projection
comes from the solutions of dispersion equations, showing the possible wave modes in the wave
guide. Compared to dispersion curves displayed in phase velocity, the wavenumber dispersion
curves have higher linearity and are selected because of easier calculation. Another common way
to display dispersion curves is in the form of phase velocity as shown in Figure 2.3 (b). Phase
15

Table 2.1: Material Properties of the Steel Plate
Young’s Modulus
(GPa)
206

Poisson’s Ratio
0.3

Mass Density
(Kg/m3 )
7850

Table 2.2: Material Properties of The Aluminum Plate
Young’s Modulus
(GPa)
70

Poisson’s Ratio
0.33

Mass Density
(Kg/m3 )
2700

velocity is the velocity at which the phase/crest travels. The relationship between phase velocity
and wavenumber is vph = 2πf /ξ. Actually, Figure 2.3 (b) is most commonly used since it is easy
to read the characteristics of each wave mode.
From Figure 2.3 (b), we can see that for the plate of 1 mm thickness there are two wave modes
existing under frequency of 1 MHz: the S0 and A0 modes. S0 is the fundamental wave mode,
which has almost constant phase velocity through out the frequency range. Because of the stability
of S0 , it is usually selected as an relatively ideal mode for non-destructive evaluation in plates.

2.3

Case study of Lamb wave propagation in an aluminum plate

In this section, ﬁnite element simulation is employed to study the Lamb wave behavior in an
aluminum plate using FEAP [36], and the results are compared with published experimental data
provided by D. W. Greve, et al [19]. The material properties of the aluminum plate are shown in
Table 2.2.
In the experiment, two PZT transducers are attached to the aluminum plate by silver epoxy. The
thickness of the plate is 1.59 mm, and the distance between the transducers is 20 cm. By applying
voltage on the PZT transducers, shear force is introduced to the plate due to the piezoelectric
16

effect. Based on the experiment, a numerical model is created, and the geometry of which is set to
be 0.8 m in x-direction, 0.5 m in y-direction and 1.59 mm in z-direction. The mesh sizes in x, y,
and z directions are 1 mm, 1 mm and 0.795 mm, respectively. To simplify the numerical model,
a point shear force is applied at x = 0.3 m, and the signals are measured at x = 0.5 m, which is
20 cm away from the excitation. The shape of excitation is shown in Figure 2.4. It is given by a
windowed sinusoidal signal expressed as:


 V sin(ωt)
 0
V (t) =


 0

sin(ωt)
10

2

t < 10π
ω

(2.4)

otherwise

Comparison is conducted between the numerical and the experimental results to verify the
accuracy of the ﬁnite element model. Figure 2.5 shows the comparison between the two results.
From the ﬁgure, we observe that, S0 mode travels faster than the A0 mode. In addition, slight
differences in arrival time are observed for both S0 and A0 modes in the comparison. It is noticed
that the magnitude of the A0 mode is much larger than that obtained from the experimental study.

17

2000
A0

ξ

1500

1000
S0
500

0
0

0.2

0.4
0.6
0.8
Frequency (MHz)

1

(a)

10000

Vph (m/s)

8000
6000

S0

4000
A0

2000
0
0

0.5
Frequency (MHz)

1

(b)
Figure 2.3: Dispersion curves for (a) wavenumber; (b) phase velocity.

18

Normalized Amplitude

1
0.5
0
−0.5
−1
0

1

2
3
Time (s)

4

5
−5

x 10

Figure 2.4: Input waveform

Normalized Amplitude

1
Exprimental
Simulation
0.5
0
−0.5
−1
0

0.2

0.4

0.6
Time (s)

0.8

1
x 10

Figure 2.5: Comparison of experimental data and numerical simulation.

19

−4

Chapter 3
Analytical Calculation of Wave Propagation
in Pipe-like Structures
In this section, the characteristic equation of Lamb wave propagating in the pipe shown in
Figure 1.5 (b) is derived. The characteristic equation, also called dispersion equation, is obtained
from ﬁnding the possible wave modes in a given wave guide [29]. The dispersion curves contain
information of the geometry, the material properties of the waveguide, and the frequency of the
input excitation. At a speciﬁc frequency, solutions of the characteristic equation might be obtained
numerically, and each solution corresponds to the wave speed of each wave mode. By tracking the
frequency, solutions can be plotted as continuous curves which are called the dispersion curves.
From the dispersion curves, the fundamental modes for a conﬁguration might be estimated, and
the corresponding wave velocities can be calculated.

3.1

Derivation of Characteristic Equation

3.1.1 Mathematical Model
Figure 1.5 (b) shows the mathematical model of the problem to be solved. An inﬁnite hollow
cylinder of a single layer made of an isotropic material is placed in vacuum. The shear traction is
applied on the circumference of the pipe at z-coordinates l and −l. The inside and outside radii
20

of the pipe are denoted as ci and co , respectively. For the surface excitation, the Morlet wavelet
which contains a wide range of frequencies, is given as the input tangential traction. The steady
state solution for a sinusoidal excitation is obtained, and is utilized to express the transient solution
by the wide-band excitation.

3.1.2 Governing Equation

The governing equation of wave propagation in an isotropic material, known as Navier’s displacement equation of motion, is written as [27]

(λ + 2µ)∇∇ · u + µ∇2 u = ρ(

∂ 2u
),
∂t2

(3.1)

where λ and µ are Lam´ constants of the material, ρ is the mass density, u is the displacement
e
vector, and ∇ is the three dimensional differential operator. Using Helmholtz decomposition, we
can express the displacement ﬁelds in terms of the scalar potential φ and the vector potential H as

u = ∇φ + ∇ × H.

(3.2)

Substituting Eq. (3.2) into Eq. (3.1), we have

∇[(λ + 2µ)∇2 φ − ρ

∂ 2φ
∂ 2H
] + ∇ × [µ∇2 H − ρ
] = 0.
∂t2
∂t2
21

(3.3)

To satisfy Eq. (3.3) in any condition, we have
∂ 2φ
,
∂t2
2
2 ∇2 H = ∂ H ,
c2
∂t2
c 2 ∇2 φ =
1

(3.4)

where H consists of Hr , Hθ and Hz , and the longitudinal velocity c1 and the shear velocity c2
are given, respectively as
c1 =

λ + 2µ
ρ

c2 =

µ
.
ρ

(3.5)

Rewriting Gazis’s [15] potential ﬁeld in exponential forms, we have
φ = f (r)cos(nθ)ei(ξz−ωt) ,
Hr = gr (r)sin(nθ)ei(ξz−ωt) ,

(3.6)

Hθ = gθ (r)cos(nθ)ei(ξz−ωt) ,
Hz = gr (r)sin(nθ)ei(ξz−ωt) ,
where ω is the radial frequency, ξ is the wave number, n indicates the number of waves in the
circumferential direction. It should be noted that the applied shear traction excites waves that
propagate only along the axial direction of the pipe. Therefore, n = 0, and Hr and Hz are zeros
as well. The Laplacian operator ∇2 for scalar potential φ can be expressed in the cylindrical
coordinate system as
∇2 φ =

∂ 2 φ 1 ∂φ ∂ 2 φ
+
+
,
r ∂r
∂r2
∂z 2

(3.7)

while for the vector potential H, the Laplacian operator can be rewritten as:

∇2 H = ∇ · (∇H),
22

(3.8)

where the gradient of H, ∇H is expressed as:

∇H =

1 ∂Hr
∂Hr
∂Hr
er ⊗ er + (
− Hθ )er ⊗ eθ +
er ⊗ ez
∂r
r ∂θ
∂z
∂H
∂Hθ
1 ∂H
+ θ eθ ⊗ er + ( θ + Hr )eθ ⊗ eθ +
e ⊗ ez
∂r
r ∂θ
∂z θ
∂Hz
1 ∂Hz
∂Hz
+
ez ⊗ er +
ez ⊗ eθ +
ez ⊗ ez .
∂r
r ∂θ
∂z

(3.9)

Denote Eq. (3.10) in matrix form as:


∂Hr
 ∂r


S =  ∂Hθ
 ∂r

∂Hz
∂r


1 ( ∂Hr − H ) ∂Hr
r ∂θ
θ
∂z 

1 ( ∂Hθ + H ) ∂Hθ 

r
r ∂θ
∂z 

∂Hz
1 ∂Hz
r ∂θ
∂z

(3.10)

Here S is a second-order tensor. Divergence of S can be obtained as:

∇·S =

∂S
∂Srr 1 ∂Srθ
+ [
+ (Srr − Sθθ )] + rθ er
∂r
r ∂θ
∂z
∂Sθr 1 ∂Sθθ
∂S
+
+ [
+ (Sθr + Srθ )] + θz eθ
∂r
r ∂θ
∂z
∂Szr 1 ∂Szθ
∂Szz
+
ez .
+ [
+ Szr ] +
∂r
r ∂θ
∂z

(3.11)

Since in our case, only longitudinal wave modes are considered, Eq. (3.12) in circumference direction can be obtained as:
2
2
2 H = ∂ Hθ + 1 ∂Hθ + ∂ Hθ − Hθ .
∇ θ
r ∂r
∂r2
∂z 2
r2

23

(3.12)

Table 3.1: Bessel functions in different regions [26]
ω
ω
ω
ω
when c < ξ < c
when ξ > c
when ξ < c
1
1
2
2
α1 = α2 ; β1 = β 2 α1 = −α2 ; β1 = β 2 α1 = −α2 ; β1 = −β 2
γ1 = 1; γ2 = 1
γ1 = −1; γ2 = 1
γ1 = −1; γ2 = −1
Zn (αr) = Jn (αr)
Zn (αr) = In (αr)
Zn (αr) = In (αr)
Wn (αr) = Yn (αr)
Wn (αr) = Kn (αr)
Wn (αr) = Kn (αr)
Zn (βr) = Jn (βr)
Zn (βr) = Jn (βr)
Zn (βr) = In (βr)
Wn (βr) = Yn (βr)
Wn (βr) = Yn (βr)
Wn (βr) = Kn (βr)

To simplify the derivation, the spatial Fourier transform along z-direction is applied to Eq. (3.4).
With the property of Fourier transform

ˆ
∂f
ˆ
= iξ f , Eq. (3.4) can be written as
∂x

ˆ
ˆ
∂ 2φ 1 ∂ φ
+
+
r ∂r
∂r2
ˆ
ˆ
∂ 2 Hθ 1 ∂ Hθ
+
+
r ∂r
∂r2

ω2
c2
2

ω2
− ξ2
2
c1
1
− ξ2 −
r2

ˆ
φ = 0,
(3.13)
ˆ
φ = 0.

The general solutions of Eq. (3.13) are
ˆ
φ = [AZ0 (αr) + BW0 (αr)]e−iωt ,

(3.14)

ˆ
Hθ = [CZ1 (βr) + DW1 (βr)]e−iωt .
where α2 = ω 2 /c2 − ξ 2 , β 2 = ω 2 /c2 − ξ 2 , the notations Z0 , Z1 , W0 and W1 are four different
1
2
types of Bessel functions. Table 3.1 shows how to choose appropriate parameters and Bessel
functions to make the solution stable. In Table 3.1, Jn and Yn are the Bessel functions of the ﬁrst
kind and the second kind, respectively, and In , Kn are the modiﬁed Bessel functions.
24

3.1.3 Displacement Field
With only longitudinal modes excited, the displacement ﬁelds can be obtained as
∂φ ∂Hθ
−
,
∂r
∂z
∂φ 1 ∂(Hθ r)
uz =
+
.
∂z
r ∂r
ur =

(3.15)

Taking the Fourier transform of Eq. (3.15) and substituting Eq. (3.14), we have

ur = −γ1 α1 Z1 (α1 r)A − α1 W1 (α1 r)B − iξZ1 (β1 r)C − iξW1 (β1 r)D,
ˆ

(3.16)

uz = iξZ0 (α1 r)A + iξW0 (α1 r)B + β1 Z0 (β1 r)C + γ2 β1 W0 (β1 r)D.
ˆ
Here, the term e−iωt is omitted for simplicity, and it will be brought back in the ﬁnal expressions
of ur and uz .

3.1.4 Stress Fields in Cylindrical Coordinate
The relationships between strains and displacements are expressed as
∂ur
,
∂r
1 ∂ur ∂uz
ǫrz = (
+
),
2 ∂z
∂r
ǫrr =

(3.17)

and the stress-strain relations are given by Hooke’s Law as
σrr = λ∇2 φ + 2µǫrr ,
σrz = 2µǫrz .
25

(3.18)

Stress ﬁelds σrr and σrz are obtained by substituting Eq. (3.17) to Eq. (3.18) as
λ
∂uz
∂ur
) + ur + λ
,
∂r
r
∂z
∂ur ∂uz
σrz = µ(
+
).
∂z
∂r
σrr = (λ + 2µ)(

(3.19)

By taking the Fourier transform of Eq. (3.19) and substituting Eq. (3.16) to Eq. (3.19), the terms
σrr and σrz are written as
ˆ
ˆ
2µα1
2µα1
Z1 (α1 r)]A + [(ξ 2 − β 2 )W0 (α1 r) +
W1 (α1 r)]B
r
r
iξ
iξ
+ 2µ[−iξβ1 Z0 (β1 r) + Z1 (β1 r)]C + 2µ[−iγ2 ξβ1 W0 (β1 r) + W1 (β1 r)]D,
r
r
σrz = −2iγ1 ξµα1 Z1 (α1 r)A − 2iξµα1 W1 (α1 r)B + µ(ξ 2 − β 2 )Z1 (β1 r)C
ˆ
σrr = [(ξ 2 − β 2 )Z0 (α1 r) + γ1
ˆ

+ µ(ξ 2 − β 2 )W1 (β1 r)D.
(3.20)

26

3.1.5 Boundary Conditions

As shown in Figure 1.5 (b), a set of shear traction is applied at the coordinates z = l and z = −l
on the outer surface of the pipeline. The boundary conditions can be expressed as

σrr (r = co ) = 0,
σrz (r = co ) = [δ(z − l) − δ(z + l)]e−iωt ,

(3.21)

σrr (r = ci ) = 0,
σrz (r = ci ) = 0.
By replacing r in Eq. (3.20) with co and ci , respectively, and by applying Fourier transform to the
boundary condition [27], we obtain
  
 A  m11
  
  
 B  m
   21
M  = 
  
 C  m31
  
  
D
m41

m12 m13
m22 m23
m32 m33
m42 m43

 
 
m14   A 
 0
 
 
 
 
 1
m24  B 
 
 
= −2isin(ξl)   ,
 
 
 
 0
m34   C 
 
 
 
 
m44
D
0

(3.22)

where M is the coefﬁcient matrix, and the components are written as
2µα
m11 = (ξ 2 − β 2 )Z0 (α1 co ) + γ1 co 1 Z1 (α1 co ),
2µα
m12 = (ξ 2 − β 2 )W0 (α1 Co ) + co 1 W1 (α1 co ),
iξ
m13 = 2µ[−iξβ1 Z0 (β1 r) + co Z1 (β1 co )],
iξ
m14 = 2µ[−iγ2 ξβ1 W0 (β1 co ) + co W1 (β1 co )],

27

m21 = −2iγ1 ξµα1 Z1 (α1 co ),
m22 = −2iξµα1 W1 (α1 co ),
m23 = µ(ξ 2 − β 2 )Z1 (β1 co ),
m24 = µ(ξ 2 − β 2 )W1 (β1 co ).
(3.23)

The expressions of m31 to m44 in the matrix M can be obtained by replacing co with ci . Then,
the constants A, B, C, and D might be solved by applying Cramer’s rule as

A=

a
,
∆(ξ)

B=

where



0


1

a = −i2sin(ξl)det 

0


0

b
,
∆(ξ)

C=

c
,
∆(ξ)

d
,
∆(ξ)

(3.24)

∆ = det(M).

(3.25)

D=



m12 m13 m14 


m22 m23 m24 

,

m32 m33 m34 


m42 m43 m43

Similarly, the expressions for b, c and d might be obtained by substituting [0 1 0 0]T to the second
column, third column, and fourth column of M, respectively. Here, the dispersion equation is
written as ∆ = 0.

3.1.6 Residue Theorem
Once the constants A, B, C and D in Eq. (3.22) are calculated, the displacement ﬁelds urr and urz
might be obtained by applying inverse Fourier transform to Eq. (3.16) as

1
ur =
2π

+∞
−∞

ei(ξz−ωt)
[−γ1 α1 a(ξ)Z1 (α1 r) − α1 b(ξ)W1 (α1 r)
∆(ξ)

− iξc(ξ)Z1 (β1 r) − iξd(ξ)W1 (β1 r)]dξ,
uz =

1
2π

+∞
−∞

ei(ξz−ωt)
[iξa(ξ)Z0 (α1 r) + iξb(ξ)W0 (α1 r)
∆(ξ)

+ β1 c(ξ)Z0 (β1 r) + γ2 β1 d(ξ)W0 (β1 r)]dξ.
28

(3.26)

The next step is to evaluate the integration above by carrying out residue theorem. Residue
theorem [23] is one of the most important tool employed in general wave propagation problems
[18]. It is effective to calculate the line integrals of analytical functions in complex domain, and
has been utilized popularly to obtain analytical solutions for various kinds of waves.
Before applying residue theorem, we rewrite Eq. (3.26) as:

1
ur =
2π

uz =

1
2π

+∞
−∞
+∞
−∞

ei(ξz−ωt)
f (ξ)dξ,
∆(ξ)
(3.27)
ei(ξz−ωt)
∆(ξ)

g(ξ)dξ.

Since the two expressions in Eq. (3.27) are similar, we use the expression of ur to show the
application of residue theorem. Rewrite ur as:

1
2π

+∞
−∞

ei(ξz−ωt)
ei(ξz−ωt)
ei(ξz−ωt)
1
1
f (ξ)dξ =
f (ξ)dξ −
f (ξ)dξ
∆(ξ)
2π C
∆(ξ)
2π C ′ ∆(ξ)

(3.28)

where C is the closed contour in the complex plane, and C ′ is the semicircle of C. There are two
main aspects we need to consider in applying residue theorem. The ﬁrst issue is to select the closed
contour in the complex plane, which could be the upper or lower contour as shown in Figure 3.1.
Choose the upper closed contour for example, let ξ = Reiθ , and denote the integration along
the semicircle C ′ as I. Then, I can be expressed as:
π
I = lim
R→∞

0

f (Reiθ )ei(Rzcosθ−ωt) e−Rzsinθ
iReiθ dθ,
∆(Reiθ )

29

(3.29)

The important term in Eq. (3.29) is e−Rz sin θ . In our problem, we are interested in the
displacement ﬁelds at z > a. From Eq. (3.29), it is easy to ﬁnd that in order to make the integration
along the semicircle to be zero, sin θ is required to be positive. Therefore, we select the upper side
closed contour to evaluate the integration.
The second issue we need to consider is the poles located on the real axial of the complex plane.
Based on the previous description of the integration on the semicircle C ′ , rewrite Eq. (3.28) in the
form of residues, we have:
+∞
−∞

ei(ξz−ωt)
f (ξ)dξ = 2πi
∆(ξ)

Res,

(3.30)

where the residues are expressed as:
ˆ
ei(ξz−ωt) ˆ
f (ξ),
Res =
ˆ
∆′ (ξ)

(3.31)

Due to the physical meaning of the wave number, ξ’s are set to be positive. Therefore, we excluded
ˆ
the negative poles and have the outward propagating waves in the form of ei(ξz−ωt) . Similarly,
we evaluate the integration of uz . Finally, the expressions of ur , uz are given as

ur =
ˆ
ξ
uz =
ˆ
ξ

ˆ
iei(ξz−ωt)
ˆ
ˆ
ˆ ˆ
ˆ ˆ
[−γ1 α1 a(ξ)Z1 (α1 r) − α1 b(ξ)W1 (α1 r) − iξc(ξ)Z1 (β1 r) − iξd(ξ)W1 (β1 r)],
ˆ
∆′ (ξ)
ˆ
iei(ξz−ωt)
ˆ
ˆ ˆ
ˆ
ˆ
[iξa(ξ)Z0 (α1 r) + iξb(ξ)W0 (α1 r) + β1 c(ξ)Z0 (β1 r) + γ2 β1 d(ξ)W0 (β1 r)],
ˆ
∆′ (ξ)
(3.32)

ˆ
where ξ is the solution of the dispersion equation, and Eq. (3.32) describes the response under

30

a steady state excitation. To get the response under a transient excitation, it is required to apply
Fast Fourier Transform (FFT) [6] to decompose the input signal into a set of sinusoidal inputs
with different frequencies and amplitudes, and then superpose the calculated displacement by each
input to get the ﬁnal response.

3.2

Dispersion Curves

3.2.1 Name of Dispersion Curves
There are three types of modes in a cylindrical system, which are the longitudinal (L), the ﬂexural
(F) and the torsional (T) modes. Figure 3.2 shows the three types of modes. For each mode
type, a two index system, e.g. L (M,N), is used to indicate the wave modes. The ﬁrst integer M
denotes the circumferential order of the mode, while the second integer N is the counter value
[26]. The modes spanning from zero frequency are named as the ﬁrst mode, and the others are
given the names consequently in the order of their cutoff frequency. All the longitudinal modes of
circumferential order M = 0 are axisymmetic.

3.2.2 Dispersion Curves for Pipes with Different Radius and Wall Thickness
In the previous section, we have obtained the expression of the dispersion equation ∆ = 0. By
numerical calculations, the solutions of the equation might be plotted. To obtain the solutions of the
dispersion equation, we ﬁx the frequency and search the wave numbers that satisfy det(M ) = 0.
Due to the nature of Bessel functions, the determinant of M is sensitive to the wavenumber.
Therefore, it is not trivial to ﬁnd the wavenumbers to make the determinant to vanish. In this
thesis, the wavenumber step is set to be ∆ξ = 0.3 to calculate the determinants of the coefﬁcient
31

matrix M . Then, the linear interpolation technique is used to obtain zeros when the determinants
change the sign.
Figure 3.3 shows the dispersion curves for pipes that have the outer radius of 136 mm with
the wall thicknesses of 1 mm, 2 mm, 4 mm and 6 mm, respectively. It is observed that dispersion
curves are sensitive to the geometry of the pipe. More wave modes exist under a certain frequency
if the thickness is larger. Another important observation is that the dispersion curves show the
dispersion of velocity versus frequency. In NDE techniques, excitation signals usually consist of a
wide range of frequencies. During the wave propagation, the shape of the response might change
since the wave velocities depend on the frequency, and some waves travel faster than the others
[7]. If the velocities are too close, waves cannot be identiﬁed clearly. Therefore, it is important to
excite wave modes which have distinct phase and group velocities.
Figure 3.4 shows the dispersion curves for pipes have same wall thickness t = 1 mm and
different outer radius, which are 13 mm, 25 mm, 50 mm and 136 mm respectively. From Figure 3.4,
it is observed that the ﬁrst longitudinal wave mode L(0,1) has the wave velocity close to the wave
speed

E/ρ [26] in a bar, which is 5159 m/s in this case. With the frequency increasing, the

wave velocity of L(0,1) decreases, then gradually increase and approach to a certain velocity. With
the increasing of the outer radius of the pipe, L(0,1) drops at lower frequency and the dispersion
curves for the pipe get closer to that for plates of the same thickness. From Figure 3.4, we can see
that the dispersion curves are not sensitive to the change of radius if the outer radius is larger than
50 mm.

32

3.3

Analytical Solution

By numerical calculation, dispersion curves for a pipe that has the outer radius of 136 mm and
the thickness of 1 mm are obtained as shown in Figure 3.5. As mentioned above, for a hollow
cylinder, the fundamental longitudinal mode L(0, 1) begins at zero frequency, and as the frequency
increases, the phase velocity increases and converges to a certain value. The second mode L(0, 2)
begins at a certain frequency with an inﬁnite phase velocity. For higher frequency, the velocity
signiﬁcantly reduces to Young’s velocity

E/(ρ(1 − ν 2 )) [26]. As shown in Figure 3.5, the

L(0, 2) mode velocity approaches to Young’s velocity in the frequency range from 0 to 1 MHz,
while L(0, 1) is more dispersive in the same frequency range. In addition, as an axisymmetric
mode in pipe-like structures, the L(0, 2) mode wave travels through the thickness of the pipe wall,
which makes it useful for detecting the circumferential defects [11].
A pipe structure is considered as shown in Figure 1.5(b). Tangential surface traction is applied
on the outer surface of the pipe at z = 0.1 m and z = −0.1 m, respectively, and a Morlet signal
is applied as an input as shown in Figure 3.6 (a). The amplitudes of Fourier coefﬁcients of the
input are plotted in Figure 3.6 (b). The temporal response is calculated analytically at z= 0.2 m
(0.1 m away to the excitation coordinate at z = 0.1 m). By substituting the wavenumbers for each
excitation frequency to Eq. (3.32), the displacement ﬁelds are calculated, and the longitudinal and
radial displacements are plotted in Figures 3.7 (a) and 3.7(b), respectively.

33

Im(ξ)
C’

R
θ

Re(ξ)

C

(a)

Im(ξ)

C

θ
Re(ξ)
R

C’

(b)
Figure 3.1: Contours for complex domain integration: (a) upper semicircle, and (b) lower semicircle

34

Longitudinal

Torsional

Flexural

Figure 3.2: Different wave modes in a cylindrical system.

35

8000
Vph (m/s)

ph

V

10000

8000
(m/s)

10000

L(0,2)

6000
4000

6000

L(0,3)

L(0,2)

4000
L(0,1)

L(0,1)

2000

2000
0
0

0.2 0.4 0.6 0.8
Frequency (MHz)

0
0

1

0.2 0.4 0.6 0.8
Frequency (MHz)

(a)

(b)

10000

10000
L(0,3)

6000
4000

L(0,3)

L(0,2)
L(0,1)

2000
0
0

L(0,5)

L(0,6)

8000

L(0,4)
Vph (m/s)

V

ph

(m/s)

8000

1

6000
4000

L(0,4)

L(0,2)
L(0,1)

2000
0.2 0.4 0.6 0.8
Frequency (MHz)

0
0

1

(c)

0.2 0.4 0.6 0.8
Frequency (MHz)

1

(d)

Figure 3.3: Dispersion curves for pipes of the outer radius co = 136 mm with the thicknesses (a)
t = 1 mm and (b) t = 2 mm, (c) t = 4 mm and (d) t = 6 mm, respectively.

36

8000

6000

Vph (m/s)

ph

V

10000

8000
(m/s)

10000

L(0,2)

4000
L(0,1)

2000
0
0

0.2 0.4 0.6 0.8
Frequency (MHz)

6000
4000

0
0

1

1

8000

L(0,2)

ph

6000

(m/s)

10000

8000

4000
2000
0
0

V

(m/s)

0.2 0.4 0.6 0.8
Frequency (MHz)

(b)

10000

ph

L(0,1)

2000

(a)

V

L(0,2)

4000
L(0,1)

L(0,1)

0.2 0.4 0.6 0.8
Frequency (MHz)

L(0,2)

6000

2000
0
0

1

(c)

0.2 0.4 0.6 0.8
Frequency (MHz)

1

(d)

Figure 3.4: Dispersion curves of pipes of the thickness t = 1 mm with the outer radii (a) co =
13 mm and (b) co = 25 mm, (c) co = 50 mm and (d) co = 136 mm, respectively

37

10000

Vph (m/s)

8000
6000

L(0,2)

4000
L(0,1)

2000
0
0

0.5
Frequency (MHz)

1

(a)

3000
2500

ξ

2000

L(0,1)

1500
L(0,2)

1000
500
0
0

0.2

0.4
0.6
0.8
Frequency (MHz)

1

(b)
Figure 3.5: Dispersion Curve of a pipe with ro = 136 mm and t = 1 mm: (a) vph versus f and
(b) ξ versus f .

38

1

σrz (Pa)

0.5
0
−0.5
−1
0

0.2

0.4
0.6
Time (s)

0.8

1
−4

x 10

(a)

Normalized Amplitude

1
0.8
0.6
0.4
0.2
0
0

0.2

0.4
0.6
0.8
Frequency (MHz)

1

(b)
Figure 3.6: (a) Tangential traction in time domain and (b) magnitudes of Fourier coefﬁcients.

39

−12

6

x 10

L(0,2)

4
Uz (m)

2
0
−2

L(0,1)

−4
−6
0

0.4

0.8
1.2
Time (s)

1.6

2
−4

x 10

(a)
−12

6

x 10

4

L(0,1)

Ur (m)

2
0
−2

L(0,2)

−4
−6
0

0.4

0.8
1.2
Time (s)

1.6

2
−4

x 10

(b)
Figure 3.7: Displacements on the outer surface of a pipe with co = 136 mm and t = 1 mm: (a) uz
and (b) ur on the outer surface.

40

Chapter 4
Numerical Simulation

4.1

Finite Element Formulations

The ﬁnite element formulation for the wave propagation system is expressed by the equation of
motion as [5]:
B
σij,j + fi = ρ¨i
u

(4.1)

B
where σij is the stress component, fi is the body force, ρ is the mass density and ui is the
displacement component.
The stress boundary condition and the displacement boundary condition are expressed respectively as:
Sf
σij nj = Fi ,

(4.2)

ui = u S u .
i
Applying principle of virtual work on Eq. (4.1), we have

V

B
(σij,j + fi − ρ¨i )¯i dV = 0.
u u
41

(4.3)

By applying divergence theorem, which is expressed as:

(σij ui ),j = σij,j ui + σij ui,j .
¯
¯
¯

(4.4)

Eq. (4.3) is rewritten as:

S

σij ui nj dS +
¯

V

B¯
fi ui dV −

V

ρ¨i ui dV −
u ¯

V

σij ui,j dV = 0.
¯

(4.5)

Substituting Eq. (4.2) to Eq. (4.5), the weak formulation is obtained as:

S

Sf
Fi ui dS +
¯

V

B¯
fi ui dV −

V

ρ¨i ui dV −
u ¯

V

σij ui,j dV = 0.
¯

(4.6)

By applying Hook’s Law,
σij = Cijkl ǫkl

(4.7)

and considering the symmetries of the stress tensor, we have

S

Sf
¯
Fi ui dS +

V

B¯
fi ui dV −

V

ρ¨i ui dV −
u ¯

V

Cijkl ǫkl ǫij dV = 0.
¯

(4.8)

The displacement ﬁeld u and strain vector ǫ is expressed by introducing the shape function Hu
and the strain-displacement matrix B as:

u = Hu U,
ǫ = BU.

42

(4.9)

where U is the nodal displacement matrix. Then, we have Eq. (4.8) in the matrix form as

¯
UT
S

¯
HT FS dS + UT
u

V

¯
HT f B dV − UT
u

V

¨
¯
ρHT Hu dV U − UT
u

V

BT CBdV U = 0.
(4.10)

¯
By eliminating UT , we have

S

HT FS dS +
u

V

HT f B dV −
u

V

¨
ρHT Hu dV U −
u

V

BT CBdV U = 0.

(4.11)

Rewriting Eq. (4.11) as:
¨
MU + KU = F,

(4.12)

where M is the mass matrix, K is the stiffness matrix and F is the force vector. They are expressed
respectively as:

M=
K=
F=

4.2

V

ρHT Hu dV,
u
BT CBdV,

(4.13)

V
S

HT FS dS +
u

V

HT f B dV.
u

Comparison of Numerical and Analytical Results

Numerical simulations are carried out using the ﬁnite element method. A pipe of the length L =
3 m, the inner radius ci = 0.135 m, and the thickness t = 1 mm is discretized with eight-node solid
elements. The material is linear elastic, Young’s Modulus E = 206 × 109 P a, and Poison’s Ratio
ν = 0.3. The excitations are applied at z = 0.1 m and z = -0.1 m, and the measuring point is located
43

at z = 0.2 m.
Figure 4.1 (a) shows an example of discretized ﬁnite element model and Figure 4.1 (b) shows
an element of the model. There are two sets of mesh sizes used in this study. Figure 4.1 (b)
demonstrated the smaller sizes, which in the longitudinal, circumferential, and radial directions
are 0.25 mm, 10.7 mm, and 0.5 mm, respectively. Another set of mesh sizes are 1 mm in axial
direction, 10.7 mm in circumferential direction and 0.5 mm in radial direction. The time step is
selected to be 3E-8 s to conﬁrm the stability of the solution. Figure 4.2 shows the comparison of
the analytical and numerical results using two different mesh sizes. From the ﬁgure, we observe
that the result from smaller mesh sizes matches better with the analytical solution than that from
the larger sizes. In both Figure 4.2 (a) and (b), the ﬁrst two wave packages of L(0,2) and L(0,1)
modes, respectively, are developed by the excitation at z = 0.1 m, while the following two waves
are from excitation at z = −0.1 m. It is observed that L(0,1) mode waves are more dispersive and
the magnitudes are smaller than those of L(0,2) mode waves.
Note that the mesh size in the circumferential direction dose not inﬂuence the magnitude and
the arrival time of the signal signiﬁcantly. It is because that the excitations are applied symmetrically in circumference, which makes the excited waves do not propagate in that direction. However, the arrive times of waves are quite sensitive to the mesh size in axial direction. Different mesh
sizes, 1 mm and 0.25 mm, were used to enhance the accuracy of arrival time. For 1 mm mesh size,
the arrival times of L(0,2) from the numerical simulation are close to those from the analytical
solution, while the arrival times of L(0,1) from the two methods vary by about 5% for the signal
excited at z = −0.1 m. For 0.25 mm mesh size, the wave packages of L(0,2) from analytical
and numerical methods are well matched for both signals came from near and far excitations. In
Figure 4.2 (b), the calculated arrival times of L(0, 1) mode waves by the excitation at z = −0.1 m

44

(a)

m

0.5 mm

25
0.

m

10.
7m

m

(b)

Figure 4.1: (a) An example of discretized FE model, and (b) an element of the model.

45

are about 122.8 µs and 122 µs, respectively by the analytical and numerical calculations, and the
variation in group velocity of L(0,1) mode is less than 1%.

4.3

Lamb Waves in Pipes with Defects

Due to the higher accuracy of the smaller mesh sizes, it is used for the simulations in this section.
For the investigation of the defects in pipe-like structures, the comparison of the signals from an
intact structure and a damaged structure might enable the quantitative estimation of the damage.
As shown in Figure 4.3, a defect is modeled by deleting ten elements along the circumferential
direction at z = 0.4 m. The defect is through the thickness of the pipe, and the defect width is
1 mm in the axial direction of the pipe. Excitations are applied at z = 0.1 m and z = −0.1 m, and
the signals are measured at 3 different locations, which are z = 0.25 m, z = 0.45 m and z = 0.6 m.
Figure 4.4 shows the received signals from the intact pipe and the damaged pipe. The waves
measured at a point (z = 0.25 m) between the excitation location (z = 0.1 m) and the defect
(z = 0.4 m) are plotted in Figure 4.4 (a). It is observed that the waves are reﬂected from the
defect, and the magnitudes of the reﬂected waves are similar to the magnitudes of the incident
waves in the intact pipe. The waves shown in Figure 4.4(b) are measured at z = 0.45 m (0.05 m
away from the defect). Compared with the waves measured in the intact pipe, the magnitudes are
reduced signiﬁcantly although the times-of-ﬂight are of slight difference. The waves still can reach
to the measuring point, but the reﬂection and diffraction around the defect reduce the wave energy
signiﬁcantly. Figure 4.4 (c) shows the displacements measured at z = 0.6 m (0.2 m away from the
defect). The inﬂuence of the defect is reduced, compared to the result shown in Figure 4.4 (b). The
magnitudes of the displacements still decrease, but the ratio of the signal from the damaged pipe
to the signal from the intact pipe is larger than the ratio observed at z = 0.45 m.
46

−12

6

x 10

FE Simulation
Analytical

Uz (m)

4
2
0
−2
−4
−6
0

0.4

0.8

1.2

1.6

Time (s)

2
−4

x 10

(a)
−12

6

x 10

FE Simulation
Analytical

Uz (m)

4
2
0
−2
−4
−6
0

0.4

0.8

1.2
Time (s)

1.6

2
−4

x 10

(b)
Figure 4.2: Comparison of displacement ﬁeld uz on the outside surface of a pipe with co = 136 mm
and t= 1 mm using mesh size in z direction of: (a) 1 mm, and (b) 0.25 mm.

47

1
2
3
r

z=-0.1
z=0.1
z=0.25
z=0.4
z=0.45
z=0.6

θ

1, 2, 3 are observing points

z

Figure 4.3: Schematic of a pipe with a defect at z= 0.4 m.

4.4

Parametric Study of Responses for Different Defect Sizes

In this section, we use the ﬁnite element method to simulate Lamb wave propagation in damaged
pipes with different sizes of defects, including the differences in defect width and depth. Our goal is
to investigate how the defect size inﬂuences the response signals. Figure 4.5 shows the comparison
of the signals resulting from pipes with defect widths of 2 mm and 1 mm, respectively. Both
defects span through the wall thickness of the pipes, and their sizes in circumferential direction
are 107 mm. The FE model used in this comparison is the same as the model used in the previous
section. The signal is recorded at the location z = 0.6 m. It is observed that the width of the defect
has negligible inﬂuences on the arrival times as well as on the magnitudes of the waves. Although
the width of the defect increases, the dimension is still very small compared with the distance
between the measuring point and the defects, thus the change is not large enough to induce any
changes into the received signals.
48

Figure 4.6 shows the comparison of the signals obtained from intact pipes, and the pipes with
different defect depth of 0.25 mm, 0.5 mm, 0.75 mm and 1 mm respectively. In order to investigate
the waves from defects of different depths, the thickness of the pipe is needed to be divided into 4
layers. Smaller mesh sizes require more time and computer memory to conduct the simulations.
Therefore, a pipe of smaller size in the longitudinal direction is used in this section: the thickness
of the pipe is 1 mm, while the length of the pipe is modiﬁed to 1.4 m. The mesh sizes of the
model are 0.25 mm in radial direction, 0.25 mm in axial direction and 10.7 mm in circumferential
direction. The defect is located at z = 0.2 m, and the measuring point is at z = 0.35 m.
From Figure 4.6, we observe that the depth of the defect is inversely proportional to the magnitude of the received incident signals. It is easy to understand that the deeper the defect, more
waves are interfered and the magnitudes of the signals are decreased. Meanwhile, we observe that
the depth of the defect does not have much inﬂuence on the arrival times of the signals. Another
interesting thing is that when the depth of the defect is smaller than the thickness of the pipe, additional wave modes are generated in the pipe walls. Those wave modes might be generated by
the diffractions of the incident wave modes on the defects. Figure 4.6 (d) shows the comparisons
between the 4 signals from pipes with defects of different depths on them. It is observed that the
magnitudes of refracted signals increase a lot when the defect depth gets deeper from 0.25 mm to
0.5 mm, while they remain almost the same when the defect depth goes from 0.5 mm to 0.75 mm.
The possible reason is that the particle vibration on the surface of the pipe is much smaller than
the vibration inside the pipe wall, which makes the signals larger when they are refracted in the
middle of the pipe wall.

49

−12

6

x 10

Intact
Defect at z=0.4 m

z

U (m)

4

Reflected L(0,2)

2
0
−2

Reflected L(0,1)

−4
−6
0

0.4

0.8

1.2

1.6

Time (s)

2
−4

x 10

(a)
−12

6

x 10

Intact
Defect at z=0.4 m

z

U (m)

4
2
0
−2
−4
−6
0

0.4

0.8

1.2

1.6

Time (s)

2
−4

x 10

(b)
−12

6

x 10

Uz (m)

4

Intact
Defect at z=0.4 m

2
0
−2
−4
−6
0

0.4

0.8

1.2
Time (s)

1.6

2
−4

x 10

(c)
Figure 4.4: Comparison of displacements measured in pipes (one is intact, and the other is damaged): Displacements uz are measured (a) at z= 0.25 m, (b) at z= 0.45 m, and (c) at z= 0.6 m.
50

−12

6

x 10

Uz (m)

4

1 mm width
2 mm width

2
0
−2
−4
−6
0

0.4

0.8

1.2
Time (s)

1.6

2
−4

x 10

Figure 4.5: Comparison of displacements measured at z = 0.6 m on the outer surface of pipes with
defect width of 1 mm vs 2 mm.

51

−12

−12

x 10

6
Intact
4
0.25 mm
2
0
−2
−4
−6
0 0.2 0.4 0.6 0.8 1
Time (s) x 10−4

Uz (m)

z

U (m)

x 10

6
Intact
4
0.5 mm
2
0
−2
−4
−6
0 0.2 0.4 0.6 0.8 1
Time (s) x 10−4

(a)

(b)

−12

−12

x 10

6
Intact
4
0.75 mm
2
0
−2
−4
−6
0 0.2 0.4 0.6 0.8 1
−4
Time (s)
x 10

Uz (m)

z

U (m)

x 10

6
Intact
4
1 mm
2
0
−2
−4
−6
0 0.2 0.4 0.6 0.8 1
−4
Time (s)
x 10

(c)

(d)

−12

z

U (m)

x 10

6
4
2
0
−2
−4
−6
0

0.25 mm
0.5 mm
0.75 mm
1 mm

0.2

0.4
0.6
Time (s)

0.8

1
−4

x 10

(e)
Figure 4.6: Comparison of displacements measured at z = 0.35 m on the outer surface of pipes
with: (a) non-defect vs defect depth of 0.25 mm; (b) non-defect vs defect depth of 0.5 mm; (c)
non-defect vs defect depth of 0.75 mm; (c) non-defect vs defect depth of 1 mm; (e) defect depth
of 0.25 mm, 0.5 mm, 0.75 mm and 1 mm.
52

Chapter 5
Summary and Conclusions
This thesis focuses on investigating the characteristics of Lamb wave propagation in pipe-like
structures. Both theoretical analysis and numerical simulation are conducted to study the Lamb
wave behavior under wide-band excitations which are symmetrically applied on the outer surface
of a pipe in circumferential direction. The analytical and numerical results are compared, and the
accuracy of the ﬁnite element model is veriﬁed for further investigation of Lamb waves in damaged
pipelines.
In Chapter 1, a brief introduction of Structural Health Monitoring and Non-Destructive Evaluation is conducted, and the motivation and objectives of this thesis are elaborated as well. The
detailed contents of Chapter 1 include:
• Deﬁne the concept of damage in structures;
• Explain the necessities of SHM and DNE methodologies;
• Introduce three commonly used NDE methods;
• Present the advantages of Lamb wave as a relatively ideal tool for ultrasonic testing;
• Introduce the developments of the theories and the applications of Lamb waves in NDE area;
• Explain the motivation and objectives of this research.
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Lamb wave propagation in plates is studied in Chapter 2. The main work conducted in Chapter
2 includes:
• Introduce two types of Lamb wave modes in plates;
• Explain two criteria of choosing proper wave modes for structural integrity inspection;
• Outline the derivation of characteristic equations of Lamb waves in plates;
• Plot the dispersion curves of a steel plate of 1 mm thick using PACshare DispersionPlus
Curves;
• Verify the accuracy of the numerical simulation of wave propagation in an aluminum plate
by comparing with published experimental data.
Chapter 3 and Chapter 4 investigated wave propagation in pipe-like structures both analytically
and numerically. Main contents in these two chapters are shown as following:
• Show the detailed derivation of dispersion equation of Lamb waves in pipe-like structures;
• Plot the dispersion curves of Lamb waves in pipelines, and compared them between pipes
with different wall thicknesses and outer radii;
• Calculate the displacement ﬁelds on the outer surface of an intact pipeline;
• Simulate Lamb wave propagation in an intact pipeline and compared the numerical results
with the analytical results;
• Conduct parametric study to investigate the inﬂuences of defect sizes on Lamb wave behaviors in pipe-like structures.
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The derivation of dispersion equation of Lamb waves in pipe-like structures is followed the
initial work conducted by Gazis in 1959. After the dispersion equation is derived, the dispersion
curves are plotted for pipes with different geometries by solving the dispersion equations. The
comparisons show the sensitivity of the dispersion curves to the geometry of the pipelines. It is
observed that, as the outer radius of a pipe increases, the wave velocity of L(0,1) mode decreases
at lower frequency and the dispersion curves of a pipe get closer to those of a plate with the same
thickness and material properties. On the other hand, as the wall thickness of a pipe increases,
more wave modes exist in the pipe. To obtain the displacement ﬁelds generated by Lamb waves
in an intact pipe, a steel pipe with 1 mm wall thickness is selected for analytical calculation and
numerical simulation. Excitations are applied as shear tractions at z = -0.1 m and z = 0.1 m on
the outer surface of the pipeline and the received signals, which are the displacement ﬁelds, are
measured at location z = 0.2 m, where z denotes the coordinate in axial direction of the pipe.
The excitation is a wide-band shear traction with center frequency fc = 500 kHz. The comparison
between the results from the two methods shows good agreement, especially for the wave mode
L(0,2), and the accuracy of the numerical model is veriﬁed.
After the accuracy is conﬁrmed, ﬁnite element simulations are carried out to study Lamb waves
in damaged pipelines, and the results are compared with the waves calculated in the intact pipe.
From the comparison, it is observed that waves are reﬂected from the edges of the defects. The
magnitudes of the waves are more sensitive to the defect than the time delays, and the effect is less
eminent when the distance from the defect to the measuring point increases. A parametric study
is conducted to compare the signals from damaged pipes with different defect widths and depths.
The signals obtained from the damaged pipes with defect widths of 1 mm and 2 mm are studied
in this thesis. From the signals, it is observed that the longitudinal modes in the pipeline are not

55

sensitive to the widths of the defects, and the signals resulting from the two defects have trivial
difference both in magnitude and arrival time. On the other hand, signals from pipes with different
defect depths show signiﬁcant differences. The magnitudes of the incident waves are inversely
proportional to the depths of defects. In addition, from the comparisons one can observe that more
wave modes are introduced to the damaged pipelines due to the diffractions by the defects. The
magnitudes of the waves generated inside the pipe walls are larger than those generated near the
surface of the pipes.

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57

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