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TNO report

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OG-RPT-DTS-2009-01162

Wall Thickness Tomography in Complex Geometries with Ultrasonic Guided Waves

Date

15 April 2009

Author(s)

ing. E.A. Luiten

Supervisors

prof. dr. ir. A. Gisolf dr. ir. A.W.F. Volker ir. J.G.P. Bloom

Project number

033.16422/01.02

All rights reserved. No part of this report may be reproduced and/or published in any form by print, photoprint, microfilm or any other means without the previous written permission from TNO. All information which is classified according to Dutch regulations shall be treated by the recipient in the same way as classified information of corresponding value in his own country. No part of this information will be disclosed to any third party. In case this report was drafted on instructions, the rights and obligations of contracting parties are subject to either the Standard Conditions for Research Instructions given to TNO, or the relevant agreement concluded bet ween the contracting parties. Submitting the report for inspection to parties who have a direct interest is permitted.

© 2009 TNO

T +31 15 269 20 00 F +31 15 269 21 11 [email protected]

2

TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

Faculty of Applied Sciences

Wall Thickness Tomography in Complex Geometries with Ultrasonic Guided Waves Master Thesis Project

Erik Luiten

Name: Studentnumber:

1208357

Programme:

Applied Physics Acoustical Imaging & Sound Control

Supervising Tutor:

prof. dr. ir. A. Gisolf

Research Group:

2nd reviewer:

dr. ir. D.J. Verschuur

3rd reviewer:

dr. F. Bociort

Start date:

01/02/2008

End date:

15/04/2009

Subject to confidentiality agreements?

No

TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

3

Summary To ensure health and integrity of pipelines in the oil and gas industry maintenance and inspection is needed. Current inspection methods require people to go in the field and locally measure the integrity of the pipeline. This is very labor intensive, because the people need to gain access to the pipeline by removing insulation, placing scaffolding or they have to dig up the pipeline. The largest costs however are from a stop or reduction of the production. To reduce manual labor and the costs of stopping the production process a permanent monitoring system is an attractive alternative. This permanent monitoring system should remotely inspect the pipeline integrity without stopping or reducing the production. TNO is currently developing the Ultrasonic Corrosion Monitor which is based on wall thickness tomography of the travel times of ultrasonic guided waves. The ultrasonic corrosion monitor is able to detect a wall thickness loss 1 mm over an area of 15x15 cm in plates and straight pipes with the sources and receivers separated up to 4 m. In this thesis the technology for of the ultrasonic corrosion monitor is extended to pipe bends. For wall thickness tomography a forward model is needed to calculate the travel times of guided waves in a pipe bend. The forward model is based on ray tracing and is validated with finite difference simulations. The difference in travel times between the forward model and finite difference simulations is within 0.25 μs. To obtain the wall thickness in a pipe bend a tomographic inversion algorithm is used to fit the forward modeled travel times to the measured travel times. The tomographic inversion algorithm is tested with a synthetic dataset. The synthetic dataset is obtained by ray tracing every frequency component of a source wavelet through the pipe bend. The wall thickness profile can be obtained from this synthetic dataset. However the quality of the obtained wall thickness profile is currently a trade off between a good approximation of the defect depth and a good approximation of the defect shape. From the finite difference simulations it appeared that the ultrasonic guided waves focus on the outer curve of the pipe bend. This focusing effect gives a phase shift to the wave field of -π/2. With ray tracing this effect is not taken into account and an algorithm should be developed which automatically corrects the data for this phase shift. When this is developed the forward model and tomographic inversion algorithm can be tested on a dataset obtained by finite difference simulations or by measurements on a real pipe.

TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

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Contents Summary ........................................................................................................................ 3 1 1.1 1.2 1.3 1.4

Introduction.................................................................................................................... 6 Integrity control of pipelines in the Oil and Gas industry ................................................ 6 The Ultrasonic Corrosion Monitor ................................................................................... 6 Research goals ................................................................................................................. 9 Thesis outline ................................................................................................................... 9

2 2.1 2.2 2.3 2.4 2.5

Theory of guided wave propagation for integrity monitoring ................................. 10 Derivation of the wave equations .................................................................................. 10 Guided waves in plates .................................................................................................. 12 Guided waves in hollow cylinders ................................................................................. 16 Optimal wave mode selection ........................................................................................ 18 Summary ........................................................................................................................ 22

3 3.1 3.2 3.3 3.4 3.5

Forward modeling........................................................................................................ 23 Ray bending due to velocity changes ............................................................................. 23 Ray tracing on a pipe ..................................................................................................... 26 Ray tracing on a pipe bend............................................................................................. 27 Travel times in a pipe bend ............................................................................................ 36 Summary ........................................................................................................................ 39

4 4.1 4.2 4.3 4.4 4.5

Validation of Ray tracing as a forward model .......................................................... 40 The finite difference method .......................................................................................... 40 Interpretation of simulated wave field ........................................................................... 41 Ray tracing of simulated wave field .............................................................................. 45 Comparison of travel times with the simulated wave field ............................................ 46 Summary ........................................................................................................................ 51

5 5.1 5.2 5.3 5.4 5.5

Performance of the Tomographic Inversion algorithm ............................................ 52 Objective function.......................................................................................................... 53 Parameterization of the forward model .......................................................................... 55 Weighting of travel times............................................................................................... 57 Dispersion correction and reference frequency.............................................................. 60 Summary ........................................................................................................................ 61

6 6.1 6.2 6.3

Inversion results ........................................................................................................... 62 Modeled data.................................................................................................................. 63 Inversion Results............................................................................................................ 67 Summary ........................................................................................................................ 74

7

Conclusions & Discussion............................................................................................ 76

8

References..................................................................................................................... 78

9

Signature ....................................................................................................................... 80

TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

Appendices A Derivation of finite difference equations B Example of the tomographic inversion process

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TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

1

Introduction

1.1

Integrity control of pipelines in the Oil and Gas industry

6

In the oil and gas industry a lot of piping and pipelines are used for the transport of materials. Shell Pernis has about 160.000 km of piping in use, according to the Dutch website of Shell. To ensure reliable and safe operation the pipelines and piping systems have to be checked on a periodic basis. The interval between checks depends on the expected rate of degradation of the system. Every time the integrity is tested, the whole or at least a part of the production process has to be stopped. Also the piping can be in hard to reach places, e.g., buried in the soil, insulated or because of high temperatures. As a consequence checking the integrity can be very cumbersome and expensive. Hence it is economically impossible to inspect everything. One of the biggest integrity issues in piping systems is corrosion. Corrosion can grow relatively quick, up to several millimeters per year. Some forms of corrosion can be unpredictable in when and where it will strike. This is mainly Corrosion Under Insulation (CUI). Corrosion under insulation commonly affects large surface areas of the pipeline and can grow unnoticed, because it happens under insulation. TNO started a large project to develop a system to permanently monitor the integrity of a pipeline or critical piping sections while the production continues. One part of this project is wireless networks to enable remote and permanent monitoring. Another part is the technology for the detection of corrosion, for instance electrochemical, magnetic and ultrasonic detection methods. 1.2

The Ultrasonic Corrosion Monitor This master thesis is within the ultrasonic corrosion monitoring part of the project, also called the Ultrasonic Corrosion Monitor (UCM). For permanent ultrasonic corrosion monitoring TNO uses ultrasonic guided wave tomography. Ultrasonic guided waves are waves trapped in the pipe wall which follow the structure of the pipe. The phase velocity of the certain guided wave modes depends on the thickness of the wave guide it follows. Therefore the travel time of a guided wave is a measure of the wall thickness. The time it takes for the guided wave to travel from a source to a receiver over a known distance is a measure for the wall thickness between the source and receiver, this setup is shown in the top of Figure 1. However the travel time is a measure for the integral wall thickness over the travel path of the guided wave and the local wall thickness between the source and receiver cannot be determined by a single measurement. To obtain information on the local wall thickness, guided wave travel times over multiple paths are needed. With non-linear inversion of the obtained travel times a wall thickness profile of the object can be obtained. The UCM uses two rings of transducers placed at a distance L from each other on a pipe, as shown in Figure 1. One ring acts as N receivers and the other ring acts as N individual sources. 

http://www.shell.nl/home/content/nld/aboutshell/shell_businesses/pernis/about/pernis_general.html

TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

7

Figure 1, Measurement principles for the ultrasonic corrosion monitor using ultrasonic guided wave transmission measurements. Top is one measurement over a certain distance and the local wall thickness cannot be estimated. Bottom multiple sources and receivers are used, the local wall thickness can be estimated by tomographic inversion.

Tomographic inversion of the acquired data is done according to the non-linear inversion scheme given in Figure 2. Based on the initial wall thickness model the travel times of a guided wave mode are calculated and a dispersion correction is applied on the data. Then the travel times are extracted from the data and an objective function is defined from the difference in travel times between the model and data. The objective function is iteratively minimized by updating the wall thickness profile of the model. When the travel time differences are minimized the wall thickness profile describing the measurements is obtained.

Figure 2, Tomographic inversion scheme.

TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

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In the tomographic inversion algorithm the data is corrected for dispersion, otherwise determining the travel times from the data will be difficult. The dispersion of the signal comes from the fact that the phase velocity of the guided waves depends on frequency and wall thickness. The dispersion correction is done by sweeping all frequency components of the signal to a certain reference frequency as though the whole signal propagated with the properties of the reference frequency. In Figure 3 is an example of a non dispersive signal, dispersive signal and a signal corrected for dispersion.

Figure 3, Dispersion correction on a dispersive signal.

TNO is already able to use ultrasound guided wave tomography on plates and pipes. The estimated wall thickness profile from the tomographic inversion is represented in a figure of the pipe or plate. An example of a wall thickness map of a pipe with two defects is shown in Figure 4, where the pipe is unfold. On the vertical axis is the unfold circumference and on the horizontal axis is the axial direction of the pipe. The color indicates the wall thickness in mm.

Figure 4, Estimated wall thickness from experiments. The color indicates the wall thickness in mm.

The ultrasonic corrosion monitor is still under development and currently TNO is perfecting their know how on ultrasonic guided wave tomography on straight pipes.

TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

9

Simulations on a pipe have proven this technology and experiments are done on a pipe in a laboratory environment. The pipe is tested on water loading and on different defect sizes in surface area and depth, but also on the surface roughness of the defect. Current issues in tomography on pipes are underestimation of defect depth, and machine profiling. The machine profile is a small and smooth thickness variation of ±0.1 mm caused by the fabrication process and stretches over the whole surface area of the pipe. The machine profile makes it more difficult to distinguish information of „small‟ defects from the machine profile. The detection threshold set by TNO is a 10% wall thickness loss over an area of 15x15 cm. An area of 15x15 cm seems like a large corrosion spot, but corrosion under insulation can easily reach these dimensions. For a plate with a wall thickness of 8 mm at a reference frequency of 150 kHz the expected time delay due to the defect becomes:

 1 1  1   1 d  Ldefect      0.15    0.13 s c  5277 5253   def cnom  1.3

Research goals A next step in the development of the UCM is to apply the same technology used for plates and pipes on pipe bends. This thesis is concerned with the investigation and adaptation of the current available technology on ultrasonic guided wave tomography at TNO for plates and straight pipes to pipe bends. The challenge with pipe bends is that the propagation of ultrasonic guided waves in bends is very complex in comparison to propagation in plates and pipes. With a pipe bend it is possible for waves to take a „short cut‟ through the inner curve of the pipe bend. Therefore it is possible for waves travelling under a large angle from source to receiver to arrive in front of waves travelling under a small angle.

1.4

Thesis outline To start with ultrasonic guided wave tomography in pipe bends the theory of ultrasonic guided waves for integrity monitoring is treated in Chapter two. Here a description of the phase velocity as a function of frequency and wall thickness is obtained and the optimal wave mode for wall thickness tomography is selected. In chapter three the forward model to calculate the travel times of ultrasonic guided waves in pipe bends is discussed. The forward model is validated in chapter four by a comparison with data from finite difference simulations. The performance of the inversion algorithm is tested on a straight pipe in chapter five. A straight pipe is used, because the data and results on a pipe are easier to interpret than for a pipe bend and the calculation times are much shorter for a straight pipe. Tomographic inversion on synthetic data from a pipe bend is discussed in chapter six. The last chapter is concerned with the conclusions and discussion on the method proposed in this thesis for wall thickness tomography using ultrasonic guided waves in bends.

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TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

2

Theory of guided wave propagation for integrity monitoring An important part of the Ultrasonic Corrosion Monitor is the forward modeling of the ultrasonic guided waves. In this chapter the physics of ultrasonic guided wave propagation is discussed. First the basic wave equations describing elastic waves in isotropic media are derived. The wave equations are used in the potential technique to obtain equations for the motion of guided waves in plates. A second method to obtain the same equations, the partial wave technique, is qualitatively discussed and gives a better insight in the physics involved in guided waves. Both methods can be used to derive a relation between the phase velocity of guided waves and the frequency and wall thickness. A comparison is made between guided wave motion in plates and cylinders. Finally the optimal wave mode for the use in ultrasonic guided wave tomography is discussed.

2.1

Derivation of the wave equations In this section the wave equations describing elastic waves in isotropic media will be derived, because they form the basis for the description of guided wave propagation in pipe walls. The derivation of elastic waves in isotropic media starts with the equations of motion (generalized Newton‟s law) [Berkhout, A.J., 1987], 3

 j 1

 ij

 2 ui   2   fi , j t

(1)

where fi is one of the three components of the source function in terms of force per volume. ij is the stress tensor, ρ the mass density, and ui the particle displacement component in i,j = x,y or z. The linearized stress-strain relation (generalized Hooke‟s law) is also required to derive elastic waves in isotropic media, and reads 

 u i u j     sij ,  i   j

 ij   ij   u   

(2)

where  and  are the first and second Lamé parameters, which are material constants, sij is a source term, and δij the Kronecker delta. Writing equations (1) and (2) in the frequency (ω) domain gives

 ij   2U i   Fi , j 1 j 3



(3)

for the equations of motion, and 

 U i U j      Sij ,  j  i  

 ij   ij  U   

for the stress-strain relation.

(4)

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TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

Assumed is a source free, homogeneous and isotropic medium, Sij = 0 and Fi = 0. After taking the gradient of equation (4), and extracting the stress tensor from equations (3) and (4), the differential equation for the particle displacement is:

  2    U    i     U   2 U  0 . 







(5)



Using a scalar potential Φ and a vector potential Ψ, the particle displacement can be written as  1          . 2   



U

(6)

The divergence of the displacement can be written as 

1

 U 

 2

2 .

The curl of the displacement can be written as 

 U 

1





  . 2

Generally the factor 1/(ρω2) is omitted. Now equation (5) can be written as 2 2         2  2                   0 .   2      

(7)

This equation can be solved by the two uncoupled solutions

 2 

2 cL2

  0,

(8)

for the compressional wave, and 

     



 2

cT2

  0,

(9)

for the shear-wave, where:

cL 

  2 

, the compressional wave velocity

cT 

 

, the shear wave velocity.

(10)

Using that the divergence of the vector potential is zero: 

    0, and            2         

(11)

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TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

The wave equations can than be written as:

  2

  2

2 cL2

2 cT2

 .

(12)



In the following paragraph the mathematics and principles of guided waves will be discussed. 2.2

Guided waves in plates In this paragraph it is discussed what ultrasonic guided waves are and what their propagation properties are. First the free plate problem is described and then the potential technique is used to arrive at the dispersion equations describing phase velocity for ultrasonic guided waves. With the dispersion equations the phase velocity as a function of wall thickness and frequency can be determined. Sir Horace Lamb was the first to describe guided waves in a free plate of infinite size [Lamb, H., 1917], as shown in Figure 76. The plate is infinite in the x –and y-direction. The surfaces ( z   d

2

) for the plate are considered to be traction free surfaces.

Figure 5, free plate problem.

Guided waves occur due to an excitation at some point in the plate. The excitation produces P –and S-waves, which encounter the upper and lower boundaries of the plate. At the surfaces mode conversions and reflections take place and to produce guided waves the wave number of the P –and S-waves must be the same in the x direction. This means that the P –and S-waves must reflect under certain but different angles and after some travel in the plate the P –and S-waves form “wave packets”. The wave packets are commonly known as guided waves. Another important condition is that the wave length of the P –and S-waves is much larger than the thickness of the plate. 2.2.1

The potential technique In this section the free plate problem will be solved analogous to [Auld, B.A., 1973] [Achenbach, J.D., 1973] and [Rose, J.L., 2004]. The underlying assumption is the assumption of plane strain, therefore the displacement can be written in terms of potentials as in equation (6).

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TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

The boundary conditions on the interface of the free plate at z   d

2

 h are

 yz   yz  0  zz     2   zz   xx     2 

U x U z   0. z x

(13)

 U x U z   0 x   z

 xz   xz   

The ~ denotes that Fourier domain is used and not the time domain. The waves are assumed to be travelling in the x-direction. Also the wave fields are assumed to be uniform in the y-direction. Therefore the strain component yy equals zero. The SH waves (horizontal polarized shear waves, in the y-direction) are not considered. The solutions are obtained by using the uncoupled wave equations (12). To solve the wave equations in (12) the solutions are assumed to be of the form

   '  z  ei kx t     '  z  ei kx t  where k 

 c phase

,

(14)

and cphase is the phase velocity of the guided waves.

The only unknown is the function depending on the z-direction. In the z-direction the waves will act like a standing wave. Equations (14) are valid solutions of the wave equations, the waves travel in the x-direction with fixed distributions in the transverse direction. After substitution of the solutions (14) in the wave equations (12) the transverse distribution can be written as:

 '  z   A1 sin  pz   A2 cos  pz   '  z   B1 sin  qz   B2 cos  qz 

,

(15)

where:

p2 

2 c L2

 k 2 and q 2 

2 cT2

k2,

(16)

where k is the horizontal wave number (wave number in the x direction). Using equation (6) the particle displacement can be calculated independently for the x – and z direction.

     ik   x z z .    Uz     ik  z x z Ux 

(17)

The solutions are grouped according to the particle displacement in the x-direction (Ux). When Ux is symmetric around the mid plane (z=0) the group consists of symmetric

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TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

modes, when Ux is anti symmetric around the mid-plane the group consists of antisymmetric modes. For simplicity the exponential term, e i kx t  as seen in the solutions (14), is omitted in the rest of the calculations. The particle displacement for the symmetric modes can be written as follows in matrix notation:

U x   ik cos  pz  q cos  qz    A2  U     p sin pz ik sin qz   B  .      1   z 

(18)

In the same way the particle displacement of the anti-symmetric modes can be written as

U x  ik sin  pz  q sin  qz    A1  U    p cos pz ik cos qz   B  .      2   z 

(19)

This separation of guided waves in symmetric and anti symmetric modes can only be used for isotropic plates. For general anisotropic media the partial wave technique can be used as seen in [Solie, L.P. & Auld, B.A., 1973]. To calculate the coefficients A1, A2, B1, B2 and the dispersion relations, the displacement for the symmetric mode and anti-symmetric mode must be substituted in the boundary conditions (13). In the following the symmetric mode is used as an example. The stress relations for boundary conditions for the symmetric mode read:

2ikp sin  ph    k 2  q 2  sin  qh    A   xz     2     2 2 2   2ikq cos  qh    B1   zz      k  p   2 p  cos  ph  

(20)

To ensure solutions other then the trivial solution the determinant for the coefficient matrix must vanish:

 2ikp sin  ph    k 2  q 2  sin  qh        0 det      k 2  p 2   2 p 2  cos  ph  2ikq cos  qh       

 4qp 2 k 2 sin  ph  cos  qh      k 2  p 2   2 p 2   k 2  q 2  cos  ph  sin  qh  After some rewriting the obtained solution for the determinant reads:

tanqh  4qpk 2 .  tan ph   k 2  p 2  2p 2 k 2  q 2









(21)

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TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

Using equations (10) and (16), equation (21) can be simplified to:

tan  qh  4qpk 2 ,  2 tan  ph   q2  k 2 

(22)

which is called the dispersion equation for the symmetric modes. The dispersion equation for the anti-symmetric mode can be calculated in the same way and reads

q2  k 2   tan  qh  .  tan  ph  4k 2 pq 2

(23)

For the ultrasonic corrosion monitor it is essential to know the phase velocity of the guided waves as a function of frequency and plate thickness. Unfortunately this can not be achieved analytically, and numerical methods must be applied to find the relation between phase velocity and the product of frequency and plate thickness. In the next section the partial wave technique will be discussed to arrive at the same dispersion equations. The partial wave technique gives a more correct explanation of the physics involved and it can be used for anisotropic media. 2.2.2

Partial wave technique The partial wave technique is another method to arrive at the dispersion equations. In this thesis the partial wave technique is only described in general terms. The mathematics behind the partial wave technique are not treated, because it is quite extensive and will lead to the same dispersion relations as with the potential technique. For the mathematical derivation and explanation the reader is referred to [Solie, L.P. & Auld, B.A., 1973]. The partial wave method has two advantages in favor of the potential technique. The first advantage is that it provides more insight into the physical nature of the waves. The dispersion equations are derived from the fact that reflections and mode conversions at the plate surfaces do occur and not from the uncoupled wave equations. The second advantage of the partial wave technique is that it also works for general anisotropic media (non-piezoelectric) and no plain strain is assumed. Hence a dispersion equation for the horizontal polarized guided waves can be obtained. However the horizontal polarized guided waves are not considered in this thesis. A schematic representation of partial waves reflecting back and forth between the boundaries of the plate is given in Figure 6 for isotropic media. Every component of the particle displacement in the wave field is written as a complex exponential. The components are six different (partial) waves, up and down in three dimensions. For anisotropic media the waves do not separate in purely horizontal and vertical polarized sets. In Figure 6 (a) the incidence and reflection of horizontal polarized shear waves are schematically shown (two partial waves). In Figure 6 (b) the reflections and wave conversions of shear –and compressional waves are depicted (four partial waves).

TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

Figure 6, Partial waves used in the isotropic plate problem [Solie, L.P. & Auld, B.A., 1973]

2.3

Guided waves in hollow cylinders In hollow cylinders guided waves can travel in the circumferential direction and the axial direction of the cylinder. In this thesis only the waves traveling in axial direction (z-direction in Figure 7) will be considered for a cylinder. In this section the guided waves in cylinders are only considered qualitatively. A mathematical derivation and description of the dispersion equations for hollow cylinders can be found in [Gazis, D.C., 1959] and [Rose, J.L., 2004].

Figure 7, Infinite hollow circular cylinder, inner radius a and outer radius b [Rose, J.L., 2004].

In this thesis three different wave modes propagating in the axial direction will be considered, these are longitudinal, flexural and torsional modes, see Figure 8.

Figure 8, wave modes in cylinders [NDTnews.org, 2007]

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TNO report | OG-RPT-DTS-2009-01162 | Master Thesis Erik Luiten

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The longitudinal and torsional modes are symmetric around the axis of symmetry (axisymmetric), the flexural modes are non-axisymmetric, notation for these modes: Longitudinal modes: L(0,m) Torsional modes: T(0,m) Flexural modes: F(n,m)

(axisymmetric modes) (axisymmetric modes) (non-axisymmetric modes)

Where n is the circumferential order and m is the mode order. In [Li, J., Rose, J.L., 2006] a plate approximation is researched for the longitudinal modes L(0,1) instead of the A0 mode in a plate, and the L(0,2) instead of the S0 mode. From [Li, J., Rose, J.L., 2006] it is seen that when the diameter is much larger then the wall thickness of the pipe the cylinder modes L(0,1) and L(0,2) can be approximated by the plate modes A0 and S0. As an example guided wave motion in a pipe with outside diameter (OD) of 355.6 mm and wall thickness of 8 mm is compared to the guided wave motion in a plate with a wall thickness of 8 mm. With Disperse 2.0 the dispersion curves are calculated for both guided wave motion in a cylinder and the guided wave motion in a plate. The relative difference in phase velocity is shown in Figure 9 on the vertical axis in a logarithmic scale and on the horizontal axis is the frequency. Only the frequency is plotted on the horizontal axis because both the wall thickness of the plate and the pipe are both 8 mm.

Figure 9, Relative difference in phase velocity between the fundamental modes in a plate A0 and S0 and the L(0,1) and L(0,2) in a hollow circular cylinder.

From Figure 9 it is seen that the difference in phase velocity above 100 kHz is about 0.01% for the S0 mode and